Webs and $q$-Howe dualities in types $\mathbf{B}\mathbf{C}\mathbf{D}$

TL;DR

This paper introduces web categories for orthogonal and symplectic Lie algebras, enabling quantum versions of classical Howe dualities in types B, C, D, and generalizes the Brauer category.

Contribution

It defines new web categories for Lie algebras and coideal subalgebras, extending the Brauer category and establishing quantum Howe dualities for types B, C, D.

Findings

Defined web categories for orthogonal and symplectic Lie algebras.

Proved quantum versions of classical Howe dualities.

Generalized the Brauer category to new algebraic contexts.

Abstract

We define web categories describing intertwiners for the orthogonal and symplectic Lie algebras, and, in the quantized setup, for certain orthogonal and symplectic coideal subalgebras. They generalize the Brauer category, and allow us to prove quantum versions of some classical type $\mathbf{B}\mathbf{C}\mathbf{D}$ Howe dualities.

Tables

Table 1. Table 1 . Our conventions for types 𝐂 m subscript 𝐂 𝑚 \mathbf{C}_{m} and 𝐃 m subscript 𝐃 𝑚 \mathbf{D}_{m} . Here we also specify the type 𝐀 m − 1 subscript 𝐀 𝑚 1 \mathbf{A}_{m-1} Cartan datum by considering the subgraphs of D 𝐷 D and the submatrix of A 𝐴 A as indicated (for both types).

o 0.9c|X[c]|X[c]	$𝔰{𝔭}_{2m}$	$𝔰{𝔬}_{2m}$
$D$
$A$
$\overrightarrow{d}$	$(1,\dots ,1,2)$	$(1,\dots ,1)$
$\Pi$
$\Phi$	$\{\pm 2{\epsilon }_i,\pm {\epsilon }_i\pm {\epsilon }_j\mid i\ne j\in I\}$	$\{\pm {\epsilon }_i\pm {\epsilon }_j\mid i\ne j\in I\}$
$X$	${\mathbb{Z}}^m$	${\mathbb{Z}}^m\oplus {\left(\frac{1}{2}+\mathbb{Z}\right)}^m$
$X^{+}$	$\begin{array}{c}\{({\lambda }_1,\dots ,{\lambda }_m)\in X\mid \\ {\lambda }_1\ge \cdots \ge {\lambda }_m\ge 0\}\end{array}$	$\begin{array}{c}\{({\lambda }_1,\dots ,{\lambda }_m)\in X\mid \\ {\lambda }_1\ge \cdots \ge {\lambda }_{m-1}\ge \left|{\lambda }_m\right|\}\end{array}$

Equations

$${\bigwedge }_q^{\bullet }({\mathrm{V}}_q^{}\otimes {\mathbb{C}}_q^k)\cong {⨁}_{a_i\in {\mathbb{Z}}_{\ge 0}}{\bigwedge }_q^{a_1}{\mathrm{V}}_q^{}\otimes \cdots \otimes {\bigwedge }_q^{a_k}{\mathrm{V}}_q^{}.$$

$${\mathbf{U}}_q({\mathfrak{gl}}_n)\text{rotatebox}-90.0\circlearrowright {⨁}_{a_i\in {\mathbb{Z}}_{\ge 0}}{\bigwedge }_q^{a_1}{\mathrm{V}}_q^{}\otimes \cdots \otimes {\bigwedge }_q^{a_k}{\mathrm{V}}_q^{}\text{rotatebox}90.0\circlearrowleft {\mathbf{U}}_q({\mathfrak{gl}}_k).$$

$${\mathbf{U}}_q({\mathfrak{gl}}_n)\text{rotatebox}-90.0\circlearrowright \underset{k\text{ times}}{\underbrace{{\bigwedge }_q^{\bullet }{\mathrm{V}}_q^{}\otimes \cdots \otimes {\bigwedge }_q^{\bullet }{\mathrm{V}}_q^{}}}\text{rotatebox}90.0\circlearrowleft {\mathbf{U}}_q({\mathfrak{gl}}_k).$$

$${\mathbf{U}}_q({\mathfrak{gl}}_n)$$

$$5$$

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$$\begin{array}{rl}& [s{]}_i=\frac{q_i^s-q_i^{-s}}{q_i^{}-q_i^{-1}}\in \mathbb{C}(q),[\mathsf{z};s{]}_i=\frac{{\mathsf{z}}^{}{q}_i^s-{\mathsf{z}}^{-1}{q}_i^{-s}}{q_i^{}-q_i^{-1}}\in \mathbb{C}(q)[{\mathsf{z}}^{\pm 1}],\\ [t{]}_i!& =[t{]}_i[t-1{]}_i\dots [1{]}_i\in \mathbb{C}(q),{\left[\genfrac{}{}{0ex}{}{s}{t}\right]}_i=\frac{[s{]}_i[s-1{]}_i\dots [s-t+1{]}_i}{[t{]}_i[t-1{]}_i\dots [1{]}_i}\in \mathbb{C}(q).\end{array}$$

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\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-19.10445pt40.66977pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}2.25pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt32.00935pt\text{pgfsys@lineto}0.0pt35.56595pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-1.0729pt40.66977pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}b\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.025.39235pt7.59981pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}g\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-10.38454pt25.38278pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}f\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}=\stackrel{(f\otimes \mathrm{id})}{\underset{(\mathrm{id}\otimes g)}{\text{∘}}}$$

$$\text{leavevmode}to34.6pt\text{vbox}to51.78pt\text{pgfpicture}\text{makeatletter}\text{lower}-11.66599ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}2.25pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@lineto}0.0pt3.5566pt\text{pgfsys@curveto}0.0pt9.15111pt3.15726pt12.04875pt7.11319pt16.00467pt\text{pgfsys@lineto}7.11319pt21.33957pt\text{pgfsys@lineto}7.11319pt28.45276pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-1.32147pt-7.68716pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-7.10095pt2.26492pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.011.5712pt20.0479pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a+b\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.02.77435pt33.55658pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a+b\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}2.25pt\text{pgfsys@invoke}\text{pgfsys@moveto}14.22638pt0.0pt\text{pgfsys@lineto}14.22638pt3.5566pt\text{pgfsys@curveto}14.22638pt9.15111pt11.0691pt12.04875pt7.11319pt16.00467pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.013.15347pt-7.68716pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}b\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.018.68439pt2.26492pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}b\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}:a\otimes b\to 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a\otimes b,$$

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text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}2.25pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@lineto}0.0pt28.45276pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-1.32147pt-7.68716pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-3.4048pt33.55658pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a-1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}2.25pt\text{pgfsys@invoke}\text{pgfsys@moveto}17.78297pt0.0pt\text{pgfsys@lineto}17.78297pt28.45276pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.016.71007pt-7.68716pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}b\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.013.51561pt33.55658pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}b+1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt3.5566pt\text{pgfsys@lineto}17.78297pt7.11319pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt8.89148pt\text{pgfsys@lineto}17.78297pt12.44807pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt21.33957pt\text{pgfsys@lineto}8.89148pt19.56126pt\text{pgfsys@lineto}17.78297pt17.78297pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}\stackrel{\text{eqref}eq:square}{=}\frac{1}{[2]}\text{leavevmode}to32.12pt\text{vbox}to51.78pt\text{pgfpicture}\text{makeatletter}\text{lower}-11.66599ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}2.25pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@lineto}0.0pt28.45276pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-1.32147pt-7.68716pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-3.4048pt33.55658pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a-1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}2.25pt\text{pgfsys@invoke}\text{pgfsys@moveto}17.78297pt0.0pt\text{pgfsys@lineto}17.78297pt28.45276pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.016.71007pt-7.68716pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}b\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.013.51561pt33.55658pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}b+1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt5.33488pt\text{pgfsys@lineto}17.78297pt8.89148pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt16.00467pt\text{pgfsys@lineto}17.78297pt12.44807pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt19.56126pt\text{pgfsys@lineto}17.78297pt23.11786pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}+\frac{[a-b-2]}{[2]}\text{leavevmode}to32.12pt\text{vbox}to51.78pt\text{pgfpicture}\text{makeatletter}\text{lower}-11.66599ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}2.25pt\text{pgfsys@invoke}\text{pgfsys@moveto}0.0pt0.0pt\text{pgfsys@lineto}0.0pt28.45276pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-1.32147pt-7.68716pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-3.4048pt33.55658pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}a-1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}2.25pt\text{pgfsys@invoke}\text{pgfsys@moveto}17.78297pt0.0pt\text{pgfsys@lineto}17.78297pt28.45276pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.016.71007pt-7.68716pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}b\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.013.51561pt33.55658pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}b+1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\tex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$$\text{leavevmode}to23.39pt\text{vbox}to37.35pt\text{pgfpicture}\text{makeatletter}\text{lower}-5.2705ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{pgfsys@moveto}-7.11319pt21.33957pt\text{pgfsys@curveto}-7.11319pt17.4009pt-3.93867pt14.22638pt0.0pt14.22638pt\text{pgfsys@curveto}3.93867pt14.22638pt7.11319pt17.4009pt7.11319pt21.33957pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-8.36319pt25.51839pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.05.86319pt25.51839pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-0.69444pt-1.29167pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}:\varnothing \to 1\otimes 1\text{and}\text{leavevmode}to23.39pt\text{vbox}to37.35pt\text{pgfpicture}\text{makeatletter}\text{lower}-32.08055ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{pgfsys@moveto}-7.11319pt-21.33957pt\text{pgfsys@curveto}-7.11319pt-17.4009pt-3.93867pt-14.22638pt0.0pt-14.22638pt\text{pgfsys@curveto}3.93867pt-14.22638pt7.11319pt-17.4009pt7.11319pt-21.33957pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-8.36319pt-28.10173pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.05.86319pt-28.10173pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}1\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@transformcm}1.00.00.01.0-0.69444pt-1.29167pt\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}:1\otimes 1\to \varnothing ,$$

$$\text{leavevmode}to14.63pt\text{vbox}to14.63pt\text{pgfpicture}\text{makeatletter}\text{lower}10.46979ptto0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{definecolor}pgfstrokecolorrgb\mathrm{0,0,0}\text{pgfsys@color@rgb@stroke}000\text{pgfsys@invoke}\text{pgfsys@color@rgb@fill}000\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{nullfont}to0.0pt\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@beginscope}\text{pgfsys@invoke}\text{pgfsys@setlinewidth}0.4pt\text{pgfsys@invoke}\text{pgfsys@moveto}-7.11319pt17.78297pt\text{pgfsys@curveto}-7.11319pt13.85443pt-3.92854pt10.66978pt0.0pt10.66978pt\text{pgfsys@curveto}3.92854pt10.66978pt7.11319pt13.85443pt7.11319pt17.78297pt\text{pgfsys@curveto}7.11319pt21.71152pt3.92854pt24.89616pt0.0pt24.89616pt\text{pgfsys@curveto}-3.92854pt24.89616pt-7.11319pt21.71152pt-7.11319pt17.78297pt\text{pgfsys@stroke}\text{pgfsys@invoke}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{pgfsys@discardpath}\text{pgfsys@invoke}\text{lxSVG@closescope}\text{pgfsys@endscope}\text{hss}\text{lxSVG@closescope}\text{endpgfpicture}=\left[\mathsf{z};0\right].$$

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Full text

1. 1 Introduction
2. 2 A reminder on the $\mathbf{A}$-web category
3. 3 *The *-web category
4. 4 *The *-web category
5. 5 Representation theoretical background
6. 6 Connecting webs and representation categories
7. 7 Main results

1. Introduction

Throughout the whole paper we fix $k,n\in{\mathbb{Z}}_{\geq 0}$, and we assume that $n$ is even whenever we write $\mathfrak{sp}_{n}$.

1A. The framework

Consider the following question: Given some Lie algebra $\mathfrak{g}$, can one give a generator-relation presentation for the category of its finite-dimensional representations, or for some well-behaved subcategory?

Maybe the best-known instance of this is the case of the monoidal category generated by the vector representation $\mathrm{V}$ of $\mathfrak{sl}_{2}$, or by the corresponding representation $\mathrm{V}_{q}^{\phantom{1}}$ of its quantized enveloping algebra $\mathbf{U}_{q}(\mathfrak{sl}_{2})$. Its generator-relation presentation is known as the Temperley–Lieb category and goes back to work of Rumer–Teller–Weyl [RTW32] and Temperley–Lieb [TL71] (the latter in the quantum setting).

In pioneering work, Kuperberg [Kup96] extended this to all rank $2$ simple Lie algebras and their quantum enveloping algebras. However, it was not clear for quite some time how to extend Kuperberg’s constructions further (although some partial results were obtained). Then, in seminal work [CKM14], Cautis–Kamnitzer–Morrison gave a generator-relation presentation of the monoidal category generated by (quantum) exterior powers of the vector representation $\mathrm{V}_{q}^{\phantom{1}}$ of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$.

Their crucial observation was that a classical tool from representation and invariant theory, known as skew Howe duality [How89, How95], can be quantized and used as a device to describe intertwiners of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$. This skew $q$-Howe duality is based on the $\mathbf{U}_{q}(\mathfrak{gl}_{n})$-module decomposition

[TABLE]

Here ${\mathbb{C}}_{q}={\mathbb{C}}(q)$ is the function field in one variable $q$ over the complex numbers, and ${\textstyle\bigwedge_{q}^{\bullet}}$ denotes the quantum exterior algebra in the sense of [BZ08]. Having (1-1), one obtains commuting actions

[TABLE]

These two actions generate each other’s centralizer, and the bimodule decomposition can be explicitly given. Moreover, by studying the kernel of the $\mathbf{U}_{q}(\mathfrak{gl}_{k})$-action, one can then completely describe the intertwiners of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$. In fact, as explained in [CKM14], they allow a nice diagrammatic interpretation via so-called $\mathbf{A}$-webs, which are basically defined by using the $\mathbf{U}_{q}(\mathfrak{gl}_{k})$-action.

The results from [CKM14] were then extended to various other instances. But, to the best of our knowledge, all generalizations so far stay in type $\mathbf{A}$.

The idea which started this paper was to extend Cautis–Kamnitzer–Morrison’s approach to types $\mathbf{B}\mathbf{C}\mathbf{D}$. However, the main obstacle immediately arises: while the quantization of skew Howe duality is fairly straightforward in type $\mathbf{A}$, it is not even clear in other types how one can define commuting actions as in (1-2). The underlying problem hereby is that ${\textstyle\bigwedge_{q}^{\bullet}}(\mathrm{V}_{q}^{\phantom{1}}\otimes{\mathbb{C}}_{q}^{k})$ is not flat if $\mathrm{V}_{q}^{\phantom{1}}$ is the vector representation in types $\mathbf{B}\mathbf{C}\mathbf{D}$ (while this holds in type $\mathbf{A}$, cf. [BZ08] and [Zwi09, Corollary 4.26]). This means that ${\textstyle\bigwedge_{q}^{\bullet}}(\mathrm{V}_{q}^{\phantom{1}}\otimes{\mathbb{C}}_{q}^{k})$ does not have the same dimension as its classical counterpart ${\textstyle\bigwedge^{\bullet}}(\mathrm{V}\otimes{\mathbb{C}}^{k})$. Hence, there is no hope for an isomorphism as in (1-1) outside type $\mathbf{A}$, and we cannot follow the approach of [CKM14].

To overcome this problem, we consider alternative quantizations of $\mathfrak{so}_{n}$ and $\mathfrak{sp}_{n}$, namely as so-called coideal subalgebras $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})\subset\mathbf{U}_{q}(\mathfrak{gl}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})\subset\mathbf{U}_{q}(\mathfrak{gl}_{n})$, see [Let99] or [KP11]. For their vector representations, the decomposition (1-1) does hold, since they are subalgebras of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$. Hence, we get commuting actions of $\mathbf{U}_{q}(\mathfrak{gl}_{k})$ and of the $\mathbf{A}$-webs. However, since these coideals are proper subalgebras of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$, such commuting actions do not generate each other’s centralizer, cf. (1-10). Consequently, the $\mathbf{A}$-web category does not give rise to full functors to the representation categories of the coideal subalgebras $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$.

In order to get full functors, we define extended web categories, which we call - and -web categories, and prove that they act on the representation categories of the coideal subalgebras. We will then show that these extended web categories are closely connected to $\mathbf{U}_{q}(\mathfrak{so}_{2k})$ and $\mathbf{U}_{q}(\mathfrak{sp}_{2k})$ (these are the usual quantized enveloping algebras!), recovering some versions of $q$-Howe duality in types $\mathbf{B}\mathbf{C}\mathbf{D}$.

Note that our approach goes somehow the opposite way with respect to [CKM14]: instead of using $q$-Howe duality to obtain a web calculus, we use our web categories to prove quantized Howe dualities. The idea of reversing Cautis–Kamnitzer–Morrison’s path comes from the paper [QS15], where it was first deployed to quantize a different kind of Howe duality in type $\mathbf{A}$ (in which the vector representation appears together with its dual). This idea was of considerable importance for this work, and indeed many diagrammatic proofs in our paper are inspired by [QS15].

1B. Main results and proof strategy

As before, we denote by $\mathrm{V}_{q}^{\phantom{1}}$ the vector representation of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$, as well as of its coideal subalgebras $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$. We denote by ${\textstyle\bigwedge_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$ the exterior algebra and by ${\textstyle\mathrm{Sym}_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$ the symmetric algebra of $\mathrm{V}_{q}^{\phantom{1}}$.

Quantizing Howe dualities in types $\mathbf{B}\mathbf{C}\mathbf{D}$

As recalled above, the quantum version of skew Howe duality [LZZ11, Theorem 6.16] states that there are commuting actions generating each other’s centralizer:

[TABLE]

The corresponding bimodule decomposition is multiplicity-free and can be explicitly given. An analog statement holds if we replace ${\textstyle\bigwedge_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$ with ${\textstyle\mathrm{Sym}_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$ (although one has to be slightly more careful since the representation becomes infinite-dimensional).

As observed by Howe [How89, How95], in the classical setting there are four versions of (1-3) in types $\mathbf{B}\mathbf{C}\mathbf{D}$. Our main result is a quantization of Howe’s $\mathbf{B}\mathbf{C}\mathbf{D}$-dualities. In this quantization, notably, on the right-hand side the enveloping algebras $\mathbf{U}(\mathfrak{sp}_{2k})$ and $\mathbf{U}(\mathfrak{so}_{2k})$ become their quantum enveloping algebras, but on the left-hand side they get replaced by the coideal subalgebras $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$.

Theorem A**.**

There are commuting actions:

[TABLE]

In (1-4) and (1-5) for $n$ odd, and in (1-6) and (1-7), the two actions generate each other’s centralizer. Hence, the corresponding bimodule decompositions are multiplicity-free. Moreover, all the above de-quantize to the associated classical dualities of Howe.

In (1-4) and (1-5) for $n$ even one has to add an additional intertwiner on the right-hand side in order to get a full action (see Remark 1.2).

Our $q$-Howe dualities are related to (1-3) as follows:

[TABLE]

and similarly in the other three cases (1-5), (1-6) and (1-7).

Explaining the strategy

Our main tools are certain diagrams made out of trivalent graphs with edge labels from ${\mathbb{Z}_{>0}}$, which we call $\mathbf{A}$-, - and -webs.

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}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}. These categories are not monoidal, but they come with a left action of the monoidal category $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$, cf. Remark 1.1.

We will define these web categories in Section 2, 3 and 4. All the reader needs to know about them at the moment is summarized in Figure 1.

Let $\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n})$, $\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{so}_{n})$ and $\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{sp}_{n})$ denote the categories of finite-dimensional representations of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$, $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$, respectively.

Following **[CKM14]**, skew $q$-Howe duality gives rise to a $\mathbf{U}_{q}(\mathfrak{gl}_{n})$-equivariant action of $\mathbf{U}_{q}(\mathfrak{gl}_{k})$ on the $k$-fold tensor product of ${\textstyle\bigwedge_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$’s as in (1-3).

This induces a functor \Phi_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}\colon\dot{\mathbf{U}}_{q}(\mathfrak{gl}_{k})\to\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n}). By the definition of $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$, this can also be used to define a functor \Gamma_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}\colon\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}\to\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n}). In fact, there is a third functor $\Upsilon_{\!\mathfrak{gl}}\colon\dot{\mathbf{U}}_{q}(\mathfrak{gl}_{k})\to\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$ such that \Phi_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}=\Gamma_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}\circ\Upsilon_{\!\mathfrak{gl}}. It follows by skew $q$-Howe duality that all functors \Phi_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}, \Gamma_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} and $\Upsilon_{\!\mathfrak{gl}}$ are full. The same works in the symmetric case (cf. **[RT16]** and **[TVW15]**) where ${\textstyle\bigwedge_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$ is replaced by ${\textstyle\mathrm{Sym}_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$: again one constructs full functors \Phi_{\mathbf{A}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} and \Gamma_{\mathbf{A}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} such that \Phi_{\mathbf{A}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}}=\Gamma_{\mathbf{A}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}}\circ\Upsilon_{\!\mathfrak{gl}}.

Our goal is to have an analogous picture in types $\mathbf{B}\mathbf{C}\mathbf{D}$: we want to have functors \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}, \Gamma_{\mathbf{C}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}, \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}}, \Gamma_{\mathbf{C}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}}, $\Upsilon_{\!\mathfrak{so}}$ and $\Upsilon_{\!\mathfrak{sp}}$ and commuting diagrams as in Figure 2.

To summarize (after appropriate parameter substitution in the symmetric case):

Theorem B**.**

There are ladder and presentation functors as in Figure 2. These define the various Howe functors therein and hence, the actions in Theorem A. All of these functors are full in types $\mathbf{B}\mathbf{C}$.

As before, fullness in type $\mathbf{D}$ can be achieved by a slight modification, cf. Remark 1.2. The connection of the various webs and Howe dualities is summarized in Figure 3.

Moreover, we will explain in Section 7 how Theorem A and B (in particular, the commuting diagrams from Figure 3) generalize the (quantum) Brauer category.

1C. Some further remarks

Remark 1.1.

The coideals $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$ are not Hopf subalgebras of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$, because they are not closed under comultiplication. Hence, $\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}^{\prime}(\mathfrak{sp}_{n})$ do not inherit a monoidal structure. But since $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$ are left coideal subalgebras of $\mathbf{U}_{q}^{\prime}(\mathfrak{gl}_{n})$, there is a left action of $\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n})$ on them. In the web language this translates to the left-right partitioning as in Figure 1.

We stress that all these phenomena disappear if one de-quantizes.

Remark 1.2.

Let $\mathrm{O}_{n}$ be the orthogonal group, and $\mathrm{V}$ its vector representation. Brauer [Bra37] defined the Brauer algebra, which surjects onto $\operatorname{End}_{\mathrm{O}_{n}}(\mathrm{V}^{\otimes k})$, for all $k$. But, as Brauer observed (see also [LZ06, §5.1.3]), if one wants to replace $\mathrm{O}_{n}$ by the special orthogonal group $\mathrm{SO}_{n}$, then this is not true anymore since:

1. $\blacktriangleright$

If $n$ is odd, then $\operatorname{End}_{\mathrm{O}_{n}}(\mathrm{V}^{\otimes k})=\operatorname{End}_{\mathrm{SO}_{n}}(\mathrm{V}^{\otimes k})$ for all $k$. 2. $\blacktriangleright$

If $n$ is even, then $\operatorname{End}_{\mathrm{O}_{n}}(\mathrm{V}^{\otimes k})=\operatorname{End}_{\mathrm{SO}_{n}}(\mathrm{V}^{\otimes k})$ if and only if $n\geq 2k+1$.

(Morally, one “Brauer diagram generator” is missing for $\mathrm{SO}_{n}$ if $n$ is even, see also [Gro99] and [LZ16].) As a consequence, surjectivity fails in general for $\mathrm{SO}_{n}$ in type $\mathbf{D}$.

We will see in Section 7 that the Brauer algebra is closely related to our web categories. Hence, to have surjectivity or fullness in general, we would have to add this extra Brauer diagram generator to our web categories. However, since this is not the main point of our construction, we prefer to avoid technicalities. Hence, we obtain slightly weaker statements in type $\mathbf{D}$ than in types $\mathbf{B}\mathbf{C}$.

Remark 1.3.

The algebras on the right-hand side of our $q$-Howe dualities basically define the web categories, which on the other hand correspond to the representation categories of the algebras on the left-hand side.

Indeed, our webs have a representation theoretical incarnation via the functors $\Gamma$ from Figure 2. For example, the start and end dots as in Figure 1 correspond (in the de-quantized setting) to the fact that ${\textstyle\bigwedge^{2}}\mathrm{V}$ (in type $\mathbf{C}$) respectively ${\textstyle\mathrm{Sym}^{2}}\mathrm{V}$ (in types $\mathbf{B}\mathbf{D}$) are not indecomposable, but contain a copy of the trivial module.

1D. Conventions

We work over the ring ${\mathbb{C}}(q)[{\mathsf{z}}^{\pm 1}]$ of Laurent polynomial over the complex function field. We call $q$ and ${\mathsf{z}}$ generic parameters. We also consider specializations of ${\mathbb{C}}(q)[{\mathsf{z}}^{\pm 1}]$ obtained by setting ${\mathsf{z}}$ equal to some non-zero value in the field ${\mathbb{C}}(q)$. (The cases of overriding importance for us are the specializations of the form ${\mathsf{z}}=\pm q^{\pm n}$ and there is no harm to think of ${\mathsf{z}}=\pm q^{\pm n}$ throughout.)

In this setup, let $d_{i}\in{\mathbb{Z}_{\geq 0}}$ and set $q_{i}=q^{d_{i}}$. The (${\mathsf{z}}$-)quantum number, the quantum factorial, and the quantum binomial are given by (here $s\in{\mathbb{Z}}$ and $t\in{\mathbb{Z}_{\geq 0}}$)

[TABLE]

By convention, $[0]_{i}!=1={\textstyle\genfrac{[}{]}{0.0pt}{}{s}{0}}_{i}$. Note that $[0]_{i}=0={\textstyle\genfrac{[}{]}{0.0pt}{}{0\leq s<t}{t}}_{i}$ and $[-s]_{i}=-[s]_{i}$. In case $d_{i}=1$ we write $[s]=[s]_{1}$ etc. for simplicity of notation.

Let ${\mathbb{K}}$ be a ring. All our categories are assumed to be additive and ${\mathbb{K}}$-linear (except in Definition 2.2 and 3.1), and all our functors are assumed to be ${\mathbb{K}}$-linear (and hence, additive). Which specific choice of ${\mathbb{K}}$ we mean will be clear from the context.

1E. Acknowledgements

We like to thank Pedro Vaz for freely sharing his ideas and observations, some of which started this project. We also thank Jonathan Comes, Michael Ehrig, Hoel Queffelec, Catharina Stroppel and Arik Wilbert for some useful discussions. Special thanks to the referee for helpful comments and suggestions.

The Hausdorff Center for Mathematics (HCM) in Bonn and the GK 1821 in Freiburg sponsored some research visits of the authors during this project. Both this support and the hospitality during our visits are gratefully acknowledged.

D.T. likes to thank the wastebasket in his office for supporting a summer of calculations involving crazy quantum scalars – most of which ended in utter chaos. Luckily, the symbol ′ came around at one point.

2. A reminder on the $\mathbf{A}$-web category

In this section we recall the construction of $\mathbf{A}$-webs in the spirit of **[CKM14]**. (Note that, in contrast to **[CKM14]**, we use unoriented diagrams. This is due to the fact that the representations which we consider later in Section 5 are self-dual.)

2A. The $\mathbf{A}$-web category

We start by fixing conventions:

Convention 2.1**.**

For us the composition $\circ$ in diagram categories will be given by vertical stacking, while the monoidal product $\otimes$ will be given by horizontal juxtaposition, and identities are given by parallel vertical strands. We read our diagrams from bottom to top and left to right, e.g.:

[TABLE]

Here $f,g$ are some morphisms in the categories in question. Moreover, as in the illustration above, we tend to omit the symbol $\otimes$ between objects.

Definition 2.2**.**

We say a (strict) monoidal category $\boldsymbol{{\mathcal{M}}}=(\boldsymbol{{\mathcal{M}}},\otimes,\mathbbm{1})$ is generated by two finite sets of objects $\mathtt{O}_{\boldsymbol{{\mathcal{M}}}}$ and morphisms $\mathtt{M}_{\boldsymbol{{\mathcal{M}}}}$ if any object, respectively morphism, is a $\otimes$, respectively a $\circ$-$\otimes$, composite of objects, respectively morphisms, from the two fixed sets (we allow the empty composition). If we further fix a set of relations $\mathtt{R}_{\boldsymbol{{\mathcal{M}}}}$ among the morphisms of $\boldsymbol{{\mathcal{M}}}$, then $\boldsymbol{{\mathcal{M}}}$ is meant to be the quotient of the monoidal category freely generated by the fixed generators modulo these relations. See e.g. [Kas95, Section XII.1] for details.

Let ${\mathbb{K}}$ be some ring. For a monoidal, ${\mathbb{K}}$-linear category these notions are to be understood verbatim by enriching the morphism spaces formally in free ${\mathbb{K}}$-modules.

The additive closure of $\boldsymbol{{\mathcal{M}}}$ means that we allow formal direct sums of objects from $\boldsymbol{{\mathcal{M}}}$, and formal matrices of morphisms from $\boldsymbol{{\mathcal{M}}}$. See e.g. [BN05, Definition 3.2] for details. (Beware that Bar-Natan uses a different nomenclature than we do.)

The monoidal category of $\mathbf{A}$-webs

Definition 2.3**.**

The $\mathbf{A}$-web category $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$ is the additive closure of the (strict) monoidal, ${\mathbb{C}}(q)$-linear category generated by objects $a$ for $a\in{\mathbb{Z}}_{>0}$ (note that the monoidal unity is given by the empty sequence $\varnothing$), and morphisms

[TABLE]

(which we call merge and split), modulo the relations:

[TABLE]

1. $\vartriangleright$

Associativity and coassociativity

[TABLE] 2. $\vartriangleright$

The (thin) square switch

[TABLE]

Every diagram representing a morphism in $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$ will be called an $\mathbf{A}$-web. Note that the interchange law (2-1) allows us to use topological height moves among $\mathbf{A}$-webs, as well as other topological manipulations which keep an upward-directedness of $\mathbf{A}$-webs (i.e. no critical points, when one sees $\mathbf{A}$-webs as embedded graphs), and we do so in the following. In fact, we simplified our illustrations by sometimes drawing them in a topological fashion, a shorthand which we will use throughout. However, we stress that all our web calculi are rigidly built from generating sets.

Convention 2.4**.**

We call the label of an edge the thickness of the edge in question. Although we do not allow edges labeled [math] or negative labeled edges, it is convenient in illustrations to allow edges which are potentially zero – these are to be erased to obtain the corresponding $\mathbf{A}$-web – or negative – which set the $\mathbf{A}$-web to be zero. Edges labeled $1$, called thin, will play an important role and we illustrate them thinner than arbitrary labeled edges, cf. (A2). Moreover, edges of thickness $2$ also play a special role and are displayed slightly thicker than thin edges. We sometimes omit the edge labels: if they are omitted, then they can be recovered from the illustrated ones, or are $1$ or $2$ whenever they correspond to thinner edges.

Later it will be convenient to consider $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$ as a ${\mathbb{C}}(q)[{\mathsf{z}}^{\pm 1}]$-linear category, denoted by $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$, which can be easily achieved via scalar extension.

Remark 2.5.

Note that the thick square switches, i.e.

[TABLE]

where $e\in{\mathbb{Z}_{\geq 0}}$, as well as the divided power collapsing, i.e.

[TABLE]

can be deduced from the above relations since we work over ${\mathbb{C}}(q)$. An example is:

[TABLE]

The first step here is called explosion. This is a general feature for (many) web categories: the web calculus is basically determined by what happens in the case of thin labels, as the thick ones can be reduced to the thin ones via explosion. We will see this phenomenon turning up later on as well.

Note also that the so-called digon removals, i.e.

[TABLE]

are special cases of the square switches.

Remark 2.6.

By one of the main results of [CKM14], we have a list of additional relations which we call the $\mathbf{A}$-web Serre relations. We just give a blueprint example (cf. [CKM14, Lemma 2.2.1]):

[TABLE]

Since we work over ${\mathbb{C}}(q)$, thick versions of these hold as well. We leave it to the reader to write them down, keeping in mind that they are “web versions” of the higher order Serre relations (5A) of type $\mathbf{A}$. (We refer to these specifying $s,t$ as therein.)

The braiding

Recall that $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$ is a braided category. There is some freedom in the choice of scaling of the braiding. For us the most convenient choice for thin overcrossings (left crossing in (2-5)) and thin undercrossing (right crossing in (2-5)) is:

[TABLE]

Recall that a braiding on $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$ is, via explosion, uniquely determined by specifying (2-5) (see e.g. **[QS15, Lemma 5.12]). That is, we also get thick over- and undercrossings and one can inductively compute how these are expressed in terms of the $\mathbf{A}$-web generators from (****Agen**).

Remark 2.7.

The category $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$ has a $q$-anti-linear (that is, flipping $q\leftrightarrow q^{-1}$) involution $\Psi$ given by switching the crossings and an anti-involution $\omega$ given by taking the vertical mirror of a diagram. In particular, it suffices to give relations involving one type of crossing, and we will do so below.

We remark that the naturality of the braiding is equivalent to the following pitchfork relations, which hold for all values of $a$, $b$ and $c$:

[TABLE]

We additionally need the following relations:

Lemma 2.8**.**

For all $a,b,c$ the trivalent twists hold in $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$:

[TABLE]

Proof.

These relations are easily verified inductively by using explosion. ∎

3. The -web category

Next, we define a web category which, as we will see, will describe exterior $\mathbf{B}\mathbf{D}$-webs as well as symmetric $\mathbf{C}$-webs. We call its morphisms -webs.

3A. Categories with a monoidal action

We will define webs of types $\mathbf{B}\mathbf{C}\mathbf{D}$ as morphisms of categories with a left monoidal action of the monoidal category $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$, as formalized by the following definition, following **[HO01, Section 2]** or **[EGNO15, Sections 7.1 and 7.3]**.

Definition 3.1**.**

Let $\boldsymbol{{\mathcal{M}}}=(\boldsymbol{{\mathcal{M}}},\otimes,\mathbbm{1})$ be a (strict) monoidal category, and $\boldsymbol{{\mathcal{C}}}$ be a category. A (left) action of $\boldsymbol{{\mathcal{M}}}$ on $\boldsymbol{{\mathcal{C}}}$ is a bifunctor $\otimes\colon\boldsymbol{{\mathcal{M}}}\times\boldsymbol{{\mathcal{C}}}\rightarrow\boldsymbol{{\mathcal{C}}}$ with natural isomorphisms $(X\otimes Y)\otimes C\cong X\otimes(Y\otimes C)$ and $\mathbbm{1}\otimes C\cong\mathbbm{1}\cong C\otimes\mathbbm{1}$ for $X,Y\in\boldsymbol{{\mathcal{M}}}$, $C\in\boldsymbol{{\mathcal{C}}}$ satisfying the usual coherence conditions (see e.g. [HO01, Section 2], [EGNO15, Definition 7.11] or [Wei13, Definition IV.4.7]). We will then say that $\boldsymbol{{\mathcal{C}}}$ is an $\boldsymbol{{\mathcal{M}}}$-category.

In case $\boldsymbol{{\mathcal{M}}}$ and $\boldsymbol{{\mathcal{C}}}$ are both ${\mathbb{K}}$-linear over a ring ${\mathbb{K}}$, we additionally assume that $\otimes$ is ${\mathbb{K}}$-bilinear on morphisms.

The additive closure of an $\boldsymbol{{\mathcal{M}}}$-category is to be understood verbatim as in Definition 2.2, where we additionally extend the action of $\boldsymbol{{\mathcal{M}}}$ to direct sums component-wise.

Without assuming that ${\mathcal{M}}$ has generators/relations: We say $\boldsymbol{{\mathcal{C}}}$ is generated by two finite sets $\mathtt{O}_{\boldsymbol{{\mathcal{C}}}}$ of objects and $\mathtt{M}_{\boldsymbol{{\mathcal{C}}}}$ of morphisms if every object is of the form $X\otimes C$, where $X\in\boldsymbol{{\mathcal{M}}}$ and $C$ is a $\otimes$ composite of objects from $\mathtt{O}_{\boldsymbol{{\mathcal{C}}}}$, and similarly for morphisms. If we further fix a set of relations $\mathtt{R}_{\boldsymbol{{\mathcal{C}}}}$ among the morphisms of $\boldsymbol{{\mathcal{C}}}$, then $\boldsymbol{{\mathcal{C}}}$ is meant to be the quotient of the $\boldsymbol{{\mathcal{M}}}$-category freely generated by the fixed generators modulo the left $\boldsymbol{{\mathcal{M}}}$-ideal spanned by these relations. (This definition can be spelled out in details analogously to e.g. [Kas95, Section XII.1].)

3B. The diagrammatic -web category

-webs

In this section we work over ${\mathbb{C}}(q)[{\mathsf{z}}^{\pm 1}]$, if not stated otherwise. For the definition of the quantum numbers see (1-11).

Definition 3.2**.**

The -web category \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} is the additive closure of the ${\mathbb{C}}(q)[{\mathsf{z}}^{\pm 1}]$-linear $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$-category generated by the object $\varnothing$ and by the cup and cap morphisms

[TABLE]

modulo the following relations:

[TABLE]

1. $\vartriangleright$

The circle removal

[TABLE] 2. $\vartriangleright$

The bubble removal

[TABLE] 3. $\vartriangleright$

The lasso move

[TABLE] 4. $\vartriangleright$

The lollipop relations

[TABLE] 5. $\vartriangleright$

The merge-split sliding relations

[TABLE]

Remark 3.3.

Thanks to relation (4), it is irrelevant whether we use overcrossings or undercrossings in (3). Moreover, one directly sees that the symmetries $\Psi$ and $\omega$ from Remark 2.7 extend to \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} (where we assume that $\Psi$ also flips ${\mathsf{z}}\leftrightarrow{\mathsf{z}}^{-1}$). Abusing notation, we denote these symmetries by the same symbols.

Remark 3.4.

Beware that a cup or a cap in a diagram representing a morphism in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} is only allowed if there are no strands on its right, cf. Figure 1. Here are some additional examples:

[TABLE]

In particular, there are no zig-zag-type relations:

[TABLE]

and also other types of isotopy-like relations do not hold. We will meet the representation theoretical interpretation of this left-right partitioning in Section 5, see also Remark 5.9.

Topological versions of the -web relations

Next, we give some alternative, topologically more meaningful, relations to our defining relations from above.

[TABLE]

Lemma 3.5**.**

The bubble removal (2) is equivalent to

[TABLE]

Lemma 3.6**.**

The lasso move (3) is equivalent to

[TABLE]

Lemma 3.7**.**

The lollipop relations (4) are equivalent to

[TABLE]

Lemma 3.8**.**

The merge-split sliding relations (5) are equivalent to

[TABLE]

*We give the proofs of Lemma 3.5, 3.6, 3.7 and 3.8 after we have commented on the topological nature of the *-web calculus.

Note that, by using the involution $\Psi$ and the anti-involution $\omega$, we obtain many more equivalent relations.

Why -webs do not form a monoidal category

*The first thing to note is that the *-web calculus is only partially topological: Some topological manipulations are allowed, e.g. (b), but its similar looking counterparts do not necessarily hold. For example, we have

[TABLE]

Moreover, one is not allowed to use certain isotopies cf. Remark 3.4. In particular, there is no interchange law (2-1); and (a) and (c) are different relations (“turning your head is forbidden”).

Furthermore, one may be tempted to define arbitrary cups and caps as in the following picture:

[TABLE]

However, this is dangerous since the diagram

[TABLE]

would be ambiguous, as it could be any of the following two pictures:

[TABLE]

Unfortunately, these are not equal. (We note that, in the setting of categories with a monoidal action, the first diagram is the correct meaning, and we already used this before, namely in (d), cf. Remark 3.4.)

To summarize, one has the whole power of topological manipulations for the $\mathbf{A}$-web part, but for cups and caps one has to be extremely careful. For example, (b) and (d) are the only Reidemeister type $2$ moves involving cups and caps which hold.

*All of these problems disappear if one de-quantizes, and the resulting *-web category at $q=1$ is a genuine monoidal category. Hence, \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} gives an example of a deformation of a monoidal category which is not monoidal anymore. This is related, as we shall see in Section 5, to the well-understood fact that the quantization of the inclusion $\mathfrak{so}_{n}\subseteq\mathfrak{gl}_{n}$ cannot be realized as an inclusion of Hopf algebras, but only as the inclusion of a coideal subalgebra.

Actually, in the de-quantized case the resulting web category is not just monoidal, but also gets a topological flavor by defining thick cup and cap morphisms via explosion, cf. Remark 2.5, and cups and caps between $\mathbf{A}$ web strands as in (3-1). The corresponding web categories will satisfy all reasonable kinds of isotopies. This is very much in the spirit of the original “web categories” introduced by Kuperberg **[Kup96]**.

3C. Some useful lemmas

Until the end of this section we will work in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}.

Proof of Lemma 3.5, 3.6, 3.7

and 3.8.

This basically follows by expanding the crossings using (2-5). However, we give the necessary calculations for Lemma 3.5, 3.7 and 3.8 in full detail since they serve as a blueprint for all our calculations in the present and the next section. (Verifying the equivalence between (3) and (b), which inspired the name lasso move, is lengthy but easy, and follows along the same lines.) The main idea is to use the fact that the $\mathbf{A}$-web calculus is topological, and then carefully arrange the diagrams to apply the defining relations from Definition 3.2.

**(2) implies (a): **

Here is the calculation:

[TABLE]

Note hereby that we used a topological manipulation on an $\mathbf{A}$-web part.

**(4) implies (c): **

Let us verify the right equation:

[TABLE]

using the same trick as before.

**(5) implies (d): **

This can be shown as above: expanding the expressions in (d) gives four terms, two of which are equal, two of which are zero. The main topological manipulation one needs is of the form

[TABLE]

where we recall that two cups next to each are actually a shorthand for the left diagram in (3-2). To this we can then apply (4).

The other implications follow similarly. ∎

Our next aim it to derive some diagrammatic relations which, as we will see later, correspond to relations in the quantum group $\mathbf{U}_{q}(\mathfrak{so}_{2k})$. In the proofs of the following lemmas, we will repeatedly use the defining relations of \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}, as well as the topological and braided structure of $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$ (in particular, (2-6) and (2-7)). At each step, we will indicate the most important relations that we use.

Lemma 3.9**.**

For all $a,b$ we have

[TABLE]

Proof.

Using the naturality of the braiding and the defining relations as well as the relations for $\mathbf{A}$-webs, we compute:

[TABLE]

The last step is just a tedious calculation with quantum numbers. ∎

Lemma 3.10**.**

For all $a,b$ we have

[TABLE]

Proof.

We have

[TABLE]

Lemma 3.11**.**

We have

[TABLE]

Proof.

We get by the definition of the braiding:

[TABLE]

Lemma 3.12**.**

For all $a,b$ we have

[TABLE]

Proof.

By associativity (A1), we have

[TABLE]

and hence we may assume $a=0$. Now, we have

[TABLE]

Lemma 3.13**.**

For all $a,b,c$ we have

[TABLE]

Proof.

Observing that (3-7) is equivalent to

[TABLE]

the proof follows from the $\mathbf{A}$-web Serre relations (by applying the corresponding relation for $s=1$, $t=2$ to the marked part), cf. Remark 2.6. ∎

Lemma 3.14**.**

For all $a,b,c$ we have

[TABLE]

Proof.

First note that

[TABLE]

Thus, the statement follows from the thick square switch relations (2-2). ∎

4. The -web category

*In this section, which is structured exactly as the previous one, we define another web category which will play a complimentary role to the *-web category, as it describes exterior $\mathbf{C}$-webs and symmetric $\mathbf{B}\mathbf{D}$-webs. We call its morphisms -webs.

4A. The diagrammatic -web category

-webs

Again, we work over ${\mathbb{C}}(q)[{\mathsf{z}}^{\pm 1}]$, and we define:

Definition 4.1**.**

The -web category \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} is the additive closure of the ${\mathbb{C}}(q)[{\mathsf{z}}^{\pm 1}]$-linear $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$-category generated by the object $\varnothing$ and by the start/end dot morphisms

[TABLE]

modulo the following relations:

1. [TABLE]
2. $\vartriangleright$

The barbell removal

[TABLE] 3. $\vartriangleright$

The thin K removal

[TABLE] 4. $\vartriangleright$

The thick K opening

[TABLE] 5. $\vartriangleright$

The merge-split sliding relations

[TABLE]

where the cup and cap morphisms are defined as

[TABLE]

Remark 4.2.

As before for -webs, the dot morphisms are only allowed if there are no strands to their right, cf. Remark 3.4 (see also below). For example,

[TABLE]

In particular, we get the same restrictions on topological manipulations as for -webs, and again there will be a representation theoretical explanation of this in Section 5, see also Remark 5.13. Moreover, the category \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} has the same (anti)-involutions as \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} (cf. Remark 2.7), which we, abusing notation, denote also by $\Psi$ and $\omega$.

Topological versions of the -web relations

For completeness, we give some topologically meaningful versions of the relations above.

[TABLE]

Lemma 4.3**.**

The barbell removal (1) is equivalent to

[TABLE]

Lemma 4.4**.**

The thin K removal (2) is equivalent to

[TABLE]

Lemma 4.5**.**

The thick K opening (3) is equivalent to

[TABLE]

Lemma 4.6**.**

The following relations hold:

[TABLE]

Lemma 4.7**.**

The merge-split sliding relations (5) are equivalent to

[TABLE]

Proof of Lemmas 4.3, 4.4, 4.5, 4.6

and 4.7.

Again, the equations can be checked by expanding the crossings using (2-5) (although it requires some time and patience to verify that (c) is equivalent to (3)). Let us check one as an example, showing that (a) and (2) imply (b):

[TABLE]

∎

Again, by using $\Psi$ and $\omega$, we obtain many more equivalent relations.

Why -webs do not form a monoidal category

*Again, as for **-webs, the **-web category is not monoidal. As will become clear later, this is related to the fact that the inclusion $\mathfrak{sp}_{n}\hookrightarrow\mathfrak{gl}_{n}$ can only be quantized naturally as an inclusion of a coideal subalgebra. However, de-quantization gives again a genuine monoidal, topologically flavored category of *-webs.

4B. Some more useful lemmas

Until the end of the section we work in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,{\mathsf{z}}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}, and we derive some diagrammatic relations which, as we will see later, correspond to relations in the quantum group $\mathbf{U}_{q}(\mathfrak{sp}_{2k})$.

The philosophy is again to “manipulate the $\mathbf{A}$-web part and to keep dots where they are”.

Lemma 4.8**.**

For all $a$ we have

[TABLE]

Proof.

We compute:

[TABLE]

A tedious but straightforward computation gives the claimed coefficients. ∎

Lemma 4.9**.**

For all $a,b$ we have

[TABLE]

Proof.

Clear by associativity (A1). ∎

Lemma 4.10**.**

For all $a,b$ we have

[TABLE]

Proof.

First, we note that (4-5) is equivalent to

[TABLE]

Next, we can apply the $\mathbf{A}$-web Serre relations (cf. Remark 2.6) to the marked part (for $s=2$, $t=3$) and we are done. ∎

Lemma 4.11**.**

We have

[TABLE]

Proof.

The first equation is equivalent to the merge-split sliding relation (4) through the chain of equalities

[TABLE]

Using (2-4), the second equality is an immediate consequence of the first one. ∎

Lemma 4.12**.**

We have

[TABLE]

Proof.

We compute, using (4-6), that

[TABLE]

Noting that $\left[3\right]-1=\left[2\right]_{2}$, we are done. ∎

Lemma 4.13**.**

For all $a,b$ we have

[TABLE]

Proof.

The proof is a repeated application of the $\mathbf{A}$-web Serre relations (cf. Remark 2.6). We always indicate where we apply these and for what values of $s,t$.

We start by applying these for $s=1$, $t=4$ as follows.

[TABLE]

Similarly, but for $s=1$, $t=3$, we can rewrite the second term as

[TABLE]

Combining these gives

[TABLE]

Next and as before, this time with $s=1$, $t=1$, we get for the second term

[TABLE]

Again, by combining this with the above we get

[TABLE]

We can rewrite this as

[TABLE]

On the other side, using now the $s=1$, $t=1$ case, we have

[TABLE]

Putting everything together, we get the claimed equality. ∎

5. Representation theoretical background

In this section we fix our conventions for the quantum enveloping algebras and recall the definition of the coideal subalgebras $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$. We will also consider their vector representations, the associated exterior and symmetric powers, and construct some intertwiners.

5A. Quantum enveloping algebras

Let $\mathfrak{g}$ be a reductive Lie algebra with simple roots $(\alpha_{i})_{i\in I}$, simple coroots $(h_{i})_{i\in I}$ and weight lattice $X$. Denote by $a_{ij}=\langle h_{i},\alpha_{j}\rangle$ the entries of the Cartan matrix, and by $d_{i}\in{\mathbb{Z}_{\geq 0}}$ the minimal values such that the matrix $(d_{i}a_{ij})_{i,j\in{\mathrm{I}}}$ is symmetric and positive definite, see also below.

Throughout, all indices are always from the evident sets, e.g. if we write ${\mathrm{E}}_{i}$, then we always assume that $i\in{\mathrm{I}}$.

Definition 5.1**.**

The quantum enveloping algebra $\mathbf{U}_{q}(\mathfrak{g})$ of $\mathfrak{g}$ is the associative, unital ${\mathbb{C}}(q)$-algebra generated by $\mathbf{q}^{h}$ for $h\in X^{*}$, and by ${\mathrm{E}}_{i}$, ${\mathrm{F}}_{i}$ for $i\in{\mathrm{I}}$, subject to:

[TABLE]

[TABLE]

[TABLE]

The latter two relations are the so-called Serre relations. Here, ${\mathrm{K}}_{i}=\mathbf{q}^{d_{i}h_{i}}$ and the quantum binomials are as in (1-11).

Root and weight conventions

Before we can give our key examples of Definition 5.1, we fix some conventions which will be important for explicit computations.

Fix $m\in{\mathbb{Z}}_{\geq 1}$, the rank (which usually will be denoted $k$ or $n$, depending on which side of Howe duality we are, cf. Section 6). Let $\mathfrak{g}=\mathfrak{sp}_{2m}$ or $\mathfrak{g}=\mathfrak{so}_{2m}$, and we denote by $\Phi$ and $\Pi$ the sets of roots and simple roots, which we choose accordingly to Table 1. Here $\{\varepsilon_{i}\mid i\in{\mathrm{I}}\}$ for ${\mathrm{I}}=\{1,\dots,m\}$ denotes a chosen basis of the dual of the Cartan $\mathfrak{h}$, which is orthonormal with respect to the Killing form $(\cdot,\cdot)$. Correspondingly, we have the weight lattice $X$ and dominant integral weights $X^{+}$. We let also as usual $\{h_{i}\in\mathfrak{h}\mid i\in{\mathrm{I}}\}$ be the basis of $\mathfrak{h}$ determined by $\langle h_{i},\lambda\rangle=2\frac{(\alpha_{i},\lambda)}{(\alpha_{i},\alpha_{i})}$ for $\lambda\in X$. Moreover, recall that the Cartan matrix $A=(a_{ij})_{i,j\in{\mathrm{I}}}$ is defined via $a_{ij}=\langle h_{i},\alpha_{j}\rangle$. The sequence $(d_{1},\dots,d_{m})$ is chosen with $d_{i}\in{\mathbb{Z}_{\geq 0}}$ for $i=1,\dots,m$ minimal such that the matrix $(d_{i}a_{ij})_{i,j\in{\mathrm{I}}}$ is symmetric and positive definite. (The Cartan datum can also be read off from the corresponding Dynkin diagram $D$.)

We do not need to fix a Cartan datum for type $\mathbf{B}$, since in this paper we only encounter the type $\mathbf{B}$ Lie algebra $\mathfrak{so}_{n}$ (for $n$ odd) in the coideal $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$, and never in the quantum enveloping algebra $\mathbf{U}_{q}(\mathfrak{so}_{n})$.

Example 5.2.

Besides $\mathfrak{gl}_{n}$, we will consider the cases $\mathfrak{g}=\mathfrak{sp}_{2k}$ and $\mathfrak{g}=\mathfrak{so}_{2k}$ with conventions fixed above. The corresponding Serre relations for the ${\mathrm{E}}_{i}$’s are

[TABLE]

in case $\mathfrak{g}=\mathfrak{sp}_{2k}$, and for $\mathfrak{g}=\mathfrak{so}_{2k}$ they are

[TABLE]

Additionally, there are versions involving ${\mathrm{F}}_{k}$’s, and the type $\mathbf{A}$ Serre relations:

[TABLE]

where $i$ is not $k$.

As usual, we define the divided powers

[TABLE]

One can then show that the higher order Serre relations

[TABLE]

hold for $\varepsilon=\pm 1$, for all $s,t\in{\mathbb{Z}}$ with $s\geq 1$ and $t>-a_{ij}$ (see e.g. **[Lus10, Chapter 7]** and in particular Proposition 7.1.5 therein).

Moreover, recall that $\mathbf{U}_{q}(\mathfrak{g})$ has the structure of a Hopf algebra. We use the following conventions for the comultiplication $\Delta\colon\mathbf{U}_{q}(\mathfrak{g})\rightarrow\mathbf{U}_{q}(\mathfrak{g})\otimes\mathbf{U}_{q}(\mathfrak{g})$, the counit $\varepsilon\colon\mathbf{U}_{q}(\mathfrak{g})\rightarrow{\mathbb{C}}(q)$ and the antipode $S\colon\mathbf{U}_{q}(\mathfrak{g})\rightarrow\mathbf{U}_{q}(\mathfrak{g})$:

[TABLE]

The idempotented versions

Next, following **[Lus10, Chapter 23]**, we define:

Definition 5.3**.**

The idempotented quantum enveloping algebra $\dot{\mathbf{U}}_{q}(\mathfrak{g})$ is the additive closure of the ${\mathbb{C}}(q)$-linear category with:

1. $\vartriangleright$

objects $\mathsf{1}_{\lambda}$ for $\lambda\in X$, and 2. $\vartriangleright$

morphisms $\operatorname{Hom}_{\dot{\mathbf{U}}_{q}(\mathfrak{g})}(\mathsf{1}_{\lambda},\mathsf{1}_{\mu})=\mathbf{U}_{q}(\mathfrak{g})/I_{\lambda,\mu}$, where

[TABLE]

The reader unfamiliar with the idempotented version of $\mathbf{U}_{q}(\mathfrak{g})$ in its categorical disguise is referred to **[CKM14, §4.1]**, whose type $\mathbf{A}$ treatment immediately generalizes to a general $\mathfrak{g}$. Sometimes it is also convenient to regard $\dot{\mathbf{U}}_{q}(\mathfrak{g})$ as an algebra, and we use both viewpoints interchangeably.

We denoted the morphism of $\dot{\mathbf{U}}_{q}(\mathfrak{g})$ by ${\mathsf{X}}\mathsf{1}_{\lambda}=\mathsf{1}_{\mu}{\mathsf{X}}\mathsf{1}_{\lambda}\in\operatorname{Hom}_{\dot{\mathbf{U}}_{q}(\mathfrak{g})}(\mathsf{1}_{\lambda},\mathsf{1}_{\mu})$ for ${\mathsf{X}}$ being some product of ${\mathsf{E}}_{i}$’s and ${\mathsf{F}}_{i}$’s, and appropriate $\lambda$ and $\mu$. In particular,

[TABLE]

(Note that we write ${\mathrm{E}}_{i}$ etc. for elements of $\mathbf{U}_{q}(\mathfrak{g})$, and ${\mathsf{E}}_{i}\mathsf{1}_{\lambda}$ etc. for $\dot{\mathbf{U}}_{q}(\mathfrak{g})$.)

The quantum enveloping algebra $\mathbf{U}_{q}(\mathfrak{gl}_{n})$

We denote by $\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n})$ the braided monoidal category of finite-dimensional representations of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$. Let us recall some basic facts about some representations of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$.

We denote by ${\mathbb{C}}_{q}^{\phantom{1}}={\mathbb{C}}(q)$ the trivial and by $\mathrm{V}_{q}^{\phantom{1}}$ the (quantum analog of the) vector representation of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$. On the standard basis $v_{1},\dots,v_{n}$ of $\mathrm{V}_{q}^{\phantom{1}}$, the action of the generators is explicitly given by

[TABLE]

As usual, we define the ($q$-)exterior algebra of $\mathrm{V}_{q}^{\phantom{1}}$ as

[TABLE]

where $\mathrm{T}\mathrm{V}_{q}^{\phantom{1}}$ denotes the tensor algebra of $\mathrm{V}_{q}^{\phantom{1}}$ and $\mathrm{S}_{q}^{2}\mathrm{V}_{q}^{\phantom{1}}\subset\mathrm{V}_{q}^{\phantom{1}}\otimes\mathrm{V}_{q}^{\phantom{1}}$ is the ${\mathbb{C}}(q)$-linear subspace spanned by

[TABLE]

*Since $\mathrm{T}V$ is naturally graded and the ideal $\langle\mathrm{S}_{q}^{2}\mathrm{V}_{q}^{\phantom{1}}\rangle$ is homogeneous, ${\textstyle\bigwedge_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$ is also graded and decomposes as a $\mathbf{U}_{q}(\mathfrak{gl}_{n})$-module as $\bigoplus_{a\in{\mathbb{Z}_{\geq 0}}}{\textstyle\bigwedge_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$, with ${\textstyle\bigwedge_{q}^{0}}\mathrm{V}_{q}^{\phantom{1}}\cong{\mathbb{C}}_{q}^{\phantom{1}}$ and ${\textstyle\bigwedge_{q}^{1}}\mathrm{V}_{q}^{\phantom{1}}\cong\mathrm{V}_{q}^{\phantom{1}}$. We call ${\textstyle\bigwedge_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$ the *$a$th exterior power (of $\mathrm{V}_{q}^{\phantom{1}}$), and we write $v_{i_{1}}\wedge\cdots\wedge v_{i_{a}}$ for the image of $v_{i_{1}}\otimes\dotsb\otimes v_{i_{a}}$ in the quotient ${\textstyle\bigwedge_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$.

Similarly, we define the ($q$-)symmetric algebra as

[TABLE]

where $\mathrm{E}_{q}^{2}\mathrm{V}_{q}^{\phantom{1}}\subset\mathrm{V}_{q}^{\phantom{1}}\otimes\mathrm{V}_{q}^{\phantom{1}}$ is spanned by

[TABLE]

*As before, we have a $\mathbf{U}_{q}(\mathfrak{gl}_{n})$-module decomposition ${\textstyle\mathrm{Sym}_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}=\bigoplus_{a\in{\mathbb{Z}_{\geq 0}}}{\textstyle\mathrm{Sym}_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$, with ${\textstyle\mathrm{Sym}_{q}^{0}}\mathrm{V}_{q}^{\phantom{1}}\cong{\mathbb{C}}_{q}^{\phantom{1}}$ and ${\textstyle\mathrm{Sym}_{q}^{1}}\mathrm{V}_{q}^{\phantom{1}}\cong\mathrm{V}_{q}^{\phantom{1}}$. We call ${\textstyle\mathrm{Sym}_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$ the *$a$th symmetric power (of $\mathrm{V}_{q}^{\phantom{1}}$). We write $v_{i_{1}}\cdots v_{i_{a}}$ for the corresponding element of ${\textstyle\mathrm{Sym}_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$.

Clearly, ${\textstyle\bigwedge_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$ and ${\textstyle\mathrm{Sym}_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$ are ${\mathbb{C}}(q)$-linearly spanned by elements of the form

[TABLE]

Henceforth, we will always assume that the indices are increasing (strictly increasing in the exterior and weakly increasing in the symmetric case).

The multiplication of the tensor algebra $\mathrm{T}\mathrm{V}_{q}^{\phantom{1}}$ is clearly $\mathbf{U}_{q}(\mathfrak{gl}_{n})$-equivariant, and therefore induces $\mathbf{U}_{q}(\mathfrak{gl}_{n})$-equivariant multiplications on ${\textstyle\bigwedge_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$ and ${\textstyle\mathrm{Sym}_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$. Moreover, both ${\textstyle\bigwedge_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$ and ${\textstyle\mathrm{Sym}_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$ are coalgebras, with $\mathbf{U}_{q}(\mathfrak{gl}_{n})$-equivariant comultiplications. (This follows from Howe duality in type $\mathbf{A}$, see **[CKM14, Lemma 3.1.2]** for ${\textstyle\bigwedge_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$ and **[RT16, Lemma 2.21]** for ${\textstyle\mathrm{Sym}_{q}^{\bullet}}\mathrm{V}_{q}^{\phantom{1}}$.) Thus, we can define $\mathbf{U}_{q}(\mathfrak{gl}_{n})$-equivariant maps

[TABLE]

to be the corresponding (co)multiplications.

Remark 5.4.

In order to facilitate the distinction between the exterior and the symmetric power, we use the color code from [TVW15], i.e. “exterior=red” and “symmetric=green” (with “black=${\textstyle\bigwedge_{q}^{1}}\mathrm{V}_{q}^{\phantom{1}}=\mathrm{V}_{q}^{\phantom{1}}={\textstyle\mathrm{Sym}_{q}^{1}}\mathrm{V}_{q}^{\phantom{1}}$”). However, our web categories are “red and green at the same time” (cf. Figure 2), so we do not color their webs.

Example 5.5.

The base cases of the $\mathbf{U}_{q}(\mathfrak{gl}_{n})$-intertwiners from above are the ones with $a=b=1$. In these cases we omit the sub- and superscripts and we have

[TABLE]

5B. The coideal subalgebra $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$

Next, we recall the definition of the coideal subalgebra $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$, following **[KP11, Section 3]**.

Definition 5.6**.**

Let $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ be the ${\mathbb{C}}(q)$-subalgebra of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$ generated by

[TABLE]

Remark 5.7.

Despite the similar notation, $\mathbf{U}_{q}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ are different algebras. In fact, the standard embedding $\mathbf{U}(\mathfrak{so}_{n})\hookrightarrow\mathbf{U}(\mathfrak{gl}_{n})$ does not lift to the quantum level as an embedding of $\mathbf{U}_{q}(\mathfrak{so}_{n})$ into $\mathbf{U}_{q}(\mathfrak{gl}_{n})$. In contrast, $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ is, by definition, a subalgebra of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$. Both of them are, however, quantizations of the ${\mathbb{C}}$-algebra $\mathbf{U}(\mathfrak{so}_{n})$, cf. [Let99, Section 4, especially Theorem 4.8].

The algebra $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ is not a Hopf subalgebra of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$ (in particular, it is not closed under the comultiplication). Indeed, using (5-32), we get

[TABLE]

However, (5-36) shows that $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ is a so-called left coideal subalgebra.

The representation category of $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$

We denote the category of finite-dimensional representations of $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ by $\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{so}_{n})$. Via restriction, we see that the objects and morphisms from $\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n})$ are also in $\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{so}_{n})$. 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}\definecolor[named]{pgffillcolor}{rgb}{0.046875,0.5703125,0.046875}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,_{a,b}^{a{+}b}, \,\leavevmode\hbox to6.89pt{\vbox to9.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{8.53635pt}\pgfsys@curveto{0.0pt}{6.52805pt}{1.42537pt}{5.68825pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{5.69089pt}{8.53635pt}\pgfsys@curveto{5.69089pt}{6.52805pt}{4.26552pt}{5.68825pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,_{a{+}b}^{a,b} and \,\leavevmode\hbox to6.89pt{\vbox to9.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\pgfsys@color@rgb@stroke{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.046875,0.5703125,0.046875}{}\pgfsys@moveto{0.0pt}{8.53635pt}\pgfsys@curveto{0.0pt}{6.52805pt}{1.42537pt}{5.68825pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\pgfsys@color@rgb@stroke{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.046875,0.5703125,0.046875}{}\pgfsys@moveto{5.69089pt}{8.53635pt}\pgfsys@curveto{5.69089pt}{6.52805pt}{4.26552pt}{5.68825pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\pgfsys@color@rgb@stroke{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.046875,0.5703125,0.046875}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,_{a{+}b}^{a,b} are $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$-equivariant as well.

Moreover, as recalled above, $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ is not closed under comultiplication. Hence, $\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{so}_{n})$ does not inherit the structure of a monoidal category from $\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n})$. However, since $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ is a coideal subalgebra, $\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{so}_{n})$ is a $\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n})$-category in the sense of Definition 3.1.

Some intertwiners

We define ${\mathbb{C}}(q)$-linear maps

[TABLE]

[TABLE]

Lemma 5.8**.**

The ${\mathbb{C}}(q)$-linear maps   ,   ,    and    intertwine the $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$-actions.

Proof.

First we note that

[TABLE]

We already know that    and    intertwine the action of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$. Thus, via restriction, they intertwine the action of $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ as well. So it remains to show that    and    intertwine the action of $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$.

**The    case: **

One just has to show that {\mathrm{B}}_{j}\big{(}\sum_{i=1}^{n}v_{i}v_{i}\big{)}=0 for all $j=1,\dots,n$, which follows via direct and straightforward computation.

**The    case: **

The computation boils down to checking that

[TABLE]

and the claim follows.∎

Remark 5.9.

Beware that   $\otimes$   is not $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$-equivariant. To see this we note that

[TABLE]

which can be easily verified by observing that

[TABLE]

which is not equal to \,\leavevmode\hbox to18.09pt{\vbox to13.33pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-4.53247pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {}{}{}\pgfsys@moveto{17.07095pt}{-2.13387pt}\pgfsys@lineto{17.07095pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{6.4016pt}\pgfsys@curveto{0.0pt}{4.89554pt}{0.62779pt}{3.20079pt}{2.13387pt}{3.20079pt}\pgfsys@curveto{3.63992pt}{3.20079pt}{4.26773pt}{4.89554pt}{4.26773pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.78044pt}{-0.36613pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\otimes}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,({\mathrm{B}}_{i}(v_{j})). Hereby we used that ${\mathrm{B}}_{i}({\textstyle\sum_{i=1}^{n}}\;v_{i}\otimes v_{i})=0$ and (5-36). However, using almost the same calculation, one can see that   $\otimes$   is indeed $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$-equivariant. This is the representation theoretical incarnation of the left-right partitioning of the -web calculus in Section 3, cf. Figure 1.

5C. The coideal subalgebra $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$

Similarly to the orthogonal case, we define now the coideal subalgebra $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$, following **[KP11, Section 5]**.

Definition 5.10**.**

Let $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$ be the ${\mathbb{C}}(q)$-subalgebra of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$ generated by

[TABLE]

where $\mathrm{ad}({\mathrm{X}})\centerdot{\mathrm{Y}}$ denotes the right adjoint action for ${\mathrm{X}},{\mathrm{Y}}\in\mathbf{U}_{q}(\mathfrak{gl}_{n})$, cf. [Jan96, §4.18], i.e. in Sweedler notation $\mathrm{ad}({\mathrm{X}})\centerdot{\mathrm{Y}}=S({\mathrm{X}}_{(2)}){\mathrm{Y}}{\mathrm{X}}_{(1)}$.

Explicitly, the adjoint action in (5-39) is

[TABLE]

This expression is the one which we use below, e.g. in Lemma 5.12.

Remark 5.11.

As before, $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$ should not be confused with $\mathbf{U}_{q}(\mathfrak{sp}_{n})$, although they both de-quantize to $\mathbf{U}(\mathfrak{sp}_{n})$ (cf. [Let99, Section 4]).

One again checks that $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$ is a left coideal subalgebra of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$. However, we do not need the explicit formula for the comultiplication in this paper.

The representation category of $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$

We denote by $\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{sp}_{n})$ the category of finite-dimensional representations of $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$. Again, the category $\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{sp}_{n})$ is a $\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n})$-category since $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$ is a coideal subalgebra of $\mathbf{U}_{q}(\mathfrak{gl}_{n})$, and, via restriction, the objects and morphisms from $\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n})$ are also in $\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{sp}_{n})$.

Some more intertwiners

We define ${\mathbb{C}}(q)$-linear maps

[TABLE]

[TABLE]

Lemma 5.12**.**

The ${\mathbb{C}}(q)$-linear maps   ,   ,    and    intertwine the $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$-actions.

Proof.

As in the proof of Lemma 5.8 we have

[TABLE]

Hence, as before, we only need to check that    and    are $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$-equivariant.

**The    case: **

We need to show for $i$ odd that ${\mathrm{K}}_{i}^{\pm 1}$ acts on \,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower 0.57748pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{1.99179pt}\pgfsys@lineto{0.0pt}{7.6827pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{1.99179pt}\pgfsys@curveto{1.4143pt}{2.77289pt}{0.7811pt}{3.4061pt}{0.0pt}{3.4061pt}\pgfsys@curveto{-0.7811pt}{3.4061pt}{-1.4143pt}{2.77289pt}{-1.4143pt}{1.99179pt}\pgfsys@curveto{-1.4143pt}{1.2107pt}{-0.7811pt}{0.57748pt}{0.0pt}{0.57748pt}\pgfsys@curveto{0.7811pt}{0.57748pt}{1.4143pt}{1.2107pt}{1.4143pt}{1.99179pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.99179pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{1.99179pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,(1) as the identity and ${\mathrm{E}}_{i}$, ${\mathrm{F}}_{i}$ as zero, and for $i$ even that {\mathrm{B}}_{i}(\,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower 0.57748pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{1.99179pt}\pgfsys@lineto{0.0pt}{7.6827pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{1.99179pt}\pgfsys@curveto{1.4143pt}{2.77289pt}{0.7811pt}{3.4061pt}{0.0pt}{3.4061pt}\pgfsys@curveto{-0.7811pt}{3.4061pt}{-1.4143pt}{2.77289pt}{-1.4143pt}{1.99179pt}\pgfsys@curveto{-1.4143pt}{1.2107pt}{-0.7811pt}{0.57748pt}{0.0pt}{0.57748pt}\pgfsys@curveto{0.7811pt}{0.57748pt}{1.4143pt}{1.2107pt}{1.4143pt}{1.99179pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.99179pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{1.99179pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,(1))=0. The former is clear, while the latter computation essentially boils down to

[TABLE]

since ${\mathrm{E}}_{i-1}(v_{i-1}\wedge v_{i})=v_{i-1}\wedge v_{i-1}=0$ and ${\mathrm{E}}_{i+1}(v_{i+1}\wedge v_{i+2})=v_{i+1}\wedge v_{i+1}=0$.

**The    case: **

We have to show that

[TABLE]

This is clear for ${\mathrm{X}}={\mathrm{K}}_{l}^{\pm 1}$ with $l$ odd, so let us assume that ${\mathrm{X}}$ is either an ${\mathrm{E}}$, an ${\mathrm{F}}$ or a ${\mathrm{B}}$. Of course, we can also assume that $i<j$. Still, we have a few cases to check, where we only need to verify \,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,({\mathrm{X}}(v_{i}\wedge v_{j}))=0, since the other side is always zero:

1. $\blacktriangleright$**: **

If $j>i+2$, then it is easily shown that \,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,({\mathrm{X}}(v_{i}\wedge v_{j}))=0. Indeed, the only thing to observe hereby is

[TABLE]

which shows that \,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,({\mathrm{B}}_{i+1}(v_{i}\wedge v_{i+3}))=0 for $i$ odd. 2. $\blacktriangleright$**: **

If $j=i+1$ and $i$ is odd, then ${\mathrm{E}}_{i}(v_{i}\wedge v_{i+1})={\mathrm{F}}_{i}(v_{i}\wedge v_{i+1})=0$. Moreover,

[TABLE] 3. $\blacktriangleright$**: **

If $j=i+1$ and $i$ is even, then clearly \,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,({\mathrm{X}}(v_{i}\wedge v_{i+1}))=0 for ${\mathrm{X}}$ being either of ${\mathrm{E}}_{i-1},{\mathrm{E}}_{i+1},{\mathrm{F}}_{i-1},{\mathrm{F}}_{i+1}$. Moreover, one also directly sees that ${\mathrm{B}}_{i}(v_{i}\wedge v_{i+1})=0$. 4. $\blacktriangleright$**: **

If $j=i+2$ and $i$ is odd, then clearly ${\mathrm{E}}_{l}(v_{i}\wedge v_{i+2})=0$ for all $l$ odd. We also see directly that \,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,({\mathrm{F}}_{i+2}(v_{i}\wedge v_{i+2}))=0 and ${\mathrm{B}}_{i+1}(v_{i}\wedge v_{i+2})=0$. Moreover, noting that $i+1$ is even, we get

[TABLE] 5. $\blacktriangleright$**: **

Finally, if $j=i+2$ and $i$ is even, then ${\mathrm{F}}_{l}(v_{i}\wedge v_{i+2})=0$ for all $l$ odd. We also directly see that \,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,({\mathrm{E}}_{i-1}(v_{i}\wedge v_{i+2}))=0. Further, because $i$ is even, we have

[TABLE]

Moreover, noting that $i-1$ and $i+1$ are odd, we get

[TABLE]

and \,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,({\mathrm{B}}_{i+2}(v_{i}\wedge v_{i+2}))=0 follows again because \,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,(v_{i}\wedge v_{i+1})=0.∎

Remark 5.13.

Similarly as in Remark 5.9 one can show that   $\otimes$   is not $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$-equivariant, but   $\otimes$   is. Again, this is related to the left-right partitioning of the -web calculus in Section 4, cf. Figure 1.

5D. An integral form

For the purpose of later specialization, we need a version of Lusztig’s integral form for $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$. To this end, we let ${\mathscr{A}}={\mathbb{C}}[q,q^{-1},\tfrac{1}{[n]}]$. We denote by $\prescript{{\mathscr{A}}}{}{\mathbf{U}_{q}}(\mathfrak{g})$ the ${\mathscr{A}}$-form of $\mathbf{U}_{q}(\mathfrak{g})$, which is the ${\mathscr{A}}$-subalgebra generated by the ${\mathrm{E}}_{i}$’s, ${\mathrm{F}}_{i}$’s and $\mathbf{q}^{h}$’s. Note that we clearly have $\prescript{{\mathscr{A}}}{}{\mathbf{U}_{q}}(\mathfrak{g})\otimes_{{\mathscr{A}}}{\mathbb{C}}(q)=\mathbf{U}_{q}(\mathfrak{g})$.

Definition 5.14**.**

We let $\prescript{{\mathscr{A}}}{}{\mathbf{U}_{q}^{\prime}}(\mathfrak{so}_{n})\subset\prescript{{\mathscr{A}}}{}{\mathbf{U}_{q}}(\mathfrak{gl}_{n})$ be the ${\mathscr{A}}$-form of $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$, which is defined to be the ${\mathscr{A}}$-subalgebra generated by the ${\mathrm{B}}_{i}$’s from (5-35). Similarly, we define the ${\mathscr{A}}$-form of $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$ using the ${\mathrm{B}}_{i}$’s from (5-39).

Again, we clearly have that

[TABLE]

6. Connecting webs and representation categories

We are now going to define the functors from Figure 2.

6A. Actions on representations in types $\mathbf{B}\mathbf{C}\mathbf{D}$

We will now define actions of our diagrammatic web categories on representations of $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$.

The presentation functors for $\mathbf{U}_{q}(\mathfrak{gl}_{n})$

First, we recall that in type $\mathbf{A}$ we can define functors \Gamma_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}\colon\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}\rightarrow\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n}) and \Gamma_{\mathbf{A}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}}\colon\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}\rightarrow\boldsymbol{\mathcal{R}\mathrm{ep}}_{q}(\mathfrak{gl}_{n}) (sending the object $a$ to ${\textstyle\bigwedge_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$ and ${\textstyle\mathrm{Sym}_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$, respectively) using the $\mathbf{U}_{q}(\mathfrak{gl}_{n})$-intertwiners \smash{\,\leavevmode\hbox to6.89pt{\vbox to9.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{1.42537pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{5.69089pt}{0.0pt}\pgfsys@curveto{5.69089pt}{2.00829pt}{4.26552pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,_{a,b}^{a{+}b}}, \smash{\,\leavevmode\hbox to6.89pt{\vbox to9.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{8.53635pt}\pgfsys@curveto{0.0pt}{6.52805pt}{1.42537pt}{5.68825pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{5.69089pt}{8.53635pt}\pgfsys@curveto{5.69089pt}{6.52805pt}{4.26552pt}{5.68825pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,_{a{+}b}^{a,b}} and \smash{\,\leavevmode\hbox to6.89pt{\vbox to9.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\pgfsys@color@rgb@stroke{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.046875,0.5703125,0.046875}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{1.42537pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\pgfsys@color@rgb@stroke{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.046875,0.5703125,0.046875}{}\pgfsys@moveto{5.69089pt}{0.0pt}\pgfsys@curveto{5.69089pt}{2.00829pt}{4.26552pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\pgfsys@color@rgb@stroke{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.046875,0.5703125,0.046875}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,_{a,b}^{a{+}b}}, \smash{\,\leavevmode\hbox to6.89pt{\vbox to9.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\pgfsys@color@rgb@stroke{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.046875,0.5703125,0.046875}{}\pgfsys@moveto{0.0pt}{8.53635pt}\pgfsys@curveto{0.0pt}{6.52805pt}{1.42537pt}{5.68825pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\pgfsys@color@rgb@stroke{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.046875,0.5703125,0.046875}{}\pgfsys@moveto{5.69089pt}{8.53635pt}\pgfsys@curveto{5.69089pt}{6.52805pt}{4.26552pt}{5.68825pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\pgfsys@color@rgb@stroke{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.046875}{0.5703125}{0.046875}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.046875,0.5703125,0.046875}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,_{a{+}b}^{a,b}} from Section 5. By Example 5.5, we get

[TABLE]

We will use (6-1) frequently below.

Remark 6.1.

Note that \Gamma_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} is the functor from [CKM14, §3.2], while \Gamma_{\mathbf{A}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} is its cousin as in [RT16, Definition 2.18] or [TVW15, Definition 3.17].

One can check that both \Gamma_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} and \Gamma_{\mathbf{A}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} are functors of braided monoidal categories (see e.g. **[TVW15, Theorem 3.20]**) – a fact that we use silently below.

The presentation functors for $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$

We now specialize ${\mathsf{z}}=q^{n}\in{\mathbb{C}}(q)$ in the exterior and ${\mathsf{z}}=-q^{-n}\in{\mathbb{C}}(q)$ in the symmetric case. (Note that in both cases $\left[{\mathsf{z}};a\right]$ specializes to $\left[n+a\right]$ and $\left[{\mathsf{z}};a\right]_{2}$ specializes to $\left[n+a\right]_{2}$.)

We define \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}\colon\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}\rightarrow\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{so}_{n}) on objects by $a\mapsto{\textstyle\bigwedge_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$ and on the generating morphisms by the assignment

[TABLE]

*and to be \Gamma_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} on the $\mathbf{A}$-web generators (***Agen). Similarly, we define its symmetric counterpart \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}}\colon\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}\rightarrow\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{so}_{n}) on objects by $a\mapsto{\textstyle\mathrm{Sym}_{q}^{a}}\mathrm{V}_{q}^{\phantom{1}}$ and on the generating morphisms by the assignment

[TABLE]

*and to be \Gamma_{\mathbf{A}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} on the $\mathbf{A}$-web generators (***Agen). The $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$-intertwiners in (6-2) and (6-3) are defined in (5-37) and (5-38).

In order to prove that \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} and \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} are well-defined, we need to show that the defining relations of \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} are satisfied in the image. For \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}, we do this in detail in the following lemmas, where we denote by $\mathrm{id}_{a}=\mathrm{id}_{\bigwedge_{q}^{a}\!\mathrm{V}_{q}^{\phantom{1}}}$ the identity morphisms (we write $\mathrm{id}=\mathrm{id}_{1}$ for short) and all indexes are from $\{1,\dots,n\}$. Further, we abbreviate $v_{j_{1}\dotsm j_{\ell}}=v_{j_{1}}\otimes\dotsb\otimes v_{j_{l}}$.

Lemma 6.2** (Circle removal).**

We have $\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-4.46817pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-2.00829pt}{0.83716pt}{-4.26817pt}{2.84544pt}{-4.26817pt}\pgfsys@curveto{4.85373pt}{-4.26817pt}{5.69089pt}{-2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=[n]\,\mathrm{id}_{0}$.

Proof.

By definition, \,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-4.46817pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-2.00829pt}{0.83716pt}{-4.26817pt}{2.84544pt}{-4.26817pt}\pgfsys@curveto{4.85373pt}{-4.26817pt}{5.69089pt}{-2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,(1)=\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\big{(}\sum_{i=1}^{n}v_{ii}\big{)}=\sum_{i=1}^{n}q^{n+1-2i}=[n]. ∎

Lemma 6.3** (Bubble removal).**

We have (\mathrm{id}\otimes\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)(\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\otimes\mathrm{id})(\mathrm{id}\otimes\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-4.46817pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-2.00829pt}{0.83716pt}{-4.26817pt}{2.84544pt}{-4.26817pt}\pgfsys@curveto{4.85373pt}{-4.26817pt}{5.69089pt}{-2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)=[n-1]\,\mathrm{id}.

Proof.

We compute

[TABLE]

which shows the statement. ∎

Lemma 6.4** (Lasso move).**

We have

[TABLE]

Proof.

We compute

[TABLE]

Now, if $x<y$, then we get

[TABLE]

Similarly, if $x>y$, then we get

[TABLE]

So the statement is proved on $v_{x}\otimes v_{y}$ if $x\neq y$. Finally, if $x=y$, then we get

[TABLE]

and we are done. ∎

Lemma 6.5** (Lollipop relation).**

We have \,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-4.46817pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-2.00829pt}{0.83716pt}{-4.26817pt}{2.84544pt}{-4.26817pt}\pgfsys@curveto{4.85373pt}{-4.26817pt}{5.69089pt}{-2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=0 and \,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=0.

Proof.

First, if $x<y$, then \textstyle(\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)(v_{xy})=\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,(qv_{xy}-v_{yx})=0 while, if $x>y$, then \textstyle(\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)(v_{x}\otimes v_{y})=\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,(q^{-1}v_{xy}-v_{yx})=0. Next, \textstyle(\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-4.46817pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{-2.00829pt}{0.83716pt}{-4.26817pt}{2.84544pt}{-4.26817pt}\pgfsys@curveto{4.85373pt}{-4.26817pt}{5.69089pt}{-2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)(1)=\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,(\sum_{i=1}^{n}v_{ii})=0. ∎

Lemma 6.6** (Merge-split sliding relations).**

We have

[TABLE]

Proof.

First, we compute

[TABLE]

Now, it is easy to see that both (\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\otimes\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)(\mathrm{id}\otimes\leavevmode\hbox to10.11pt{\vbox to10.11pt{\pgfpicture\makeatletter\hbox{\hskip 1.5pt\lower-1.5pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{0.0pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@setlinewidth{3.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\otimes\mathrm{id})(\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\otimes\mathrm{id}\otimes\mathrm{id})(v_{wxyz}) and (\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\otimes\,\leavevmode\hbox to6.09pt{\vbox to4.67pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{0.83716pt}{4.26817pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.85373pt}{4.26817pt}{5.69089pt}{2.00829pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)(\mathrm{id}\otimes\leavevmode\hbox to10.11pt{\vbox to10.11pt{\pgfpicture\makeatletter\hbox{\hskip 1.5pt\lower-1.5pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{}{}\pgfsys@moveto{7.11319pt}{0.0pt}\pgfsys@lineto{0.0pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@setlinewidth{3.0pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{7.11319pt}{7.11319pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\otimes\mathrm{id})(\mathrm{id}\otimes\mathrm{id}\otimes\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)(v_{wxyz}) can only be non-zero if $w=y$ and $x=z$, and that they are equal in this case. This shows the first equation.

For the second equation, we compute

[TABLE]

Next, applying both \,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\otimes\mathrm{id}\otimes\mathrm{id} or \mathrm{id}\otimes\mathrm{id}\otimes\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\, to (6-4) yields

[TABLE]

which proves the lemma. ∎

The proof that (6-3) is well-defined works very similarly. It follows basically by the above, by comparison of the topological version of the relations in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} and \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,-q^{-n}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}, and by comparison of (5-37) and (5-38). We omit the details for brevity. Hence, we get:

Proposition 6.7**.**

The two functors \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} and \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} are well-defined. Moreover, we have commuting diagrams

[TABLE]

The presentation functors for $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$

Again, we specialize to ${\mathsf{z}}=q^{n}\in{\mathbb{C}}(q)$ in the exterior and to ${\mathsf{z}}=-q^{-n}\in{\mathbb{C}}(q)$ in the symmetric case.

We define \Gamma_{\mathbf{C}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}\colon\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}\rightarrow\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{sp}_{n}) on generators by the assignment

[TABLE]

and, as before, to be \Gamma_{\mathbf{A}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} on $\mathbf{A}$-web generators. Analogously, we define its symmetric counterpart \Gamma_{\mathbf{C}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}}\colon\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,-q^{-n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}\rightarrow\boldsymbol{\mathcal{R}\mathrm{ep}}^{\prime}_{q}(\mathfrak{sp}_{n}) on generators via

[TABLE]

and, as before, to be \Gamma_{\mathbf{A}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} on $\mathbf{A}$-web generators.

Again, in order to prove that (6-5) is well-defined, we need to show that the defining relations of \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} are satisfied in the image. This boils down to prove the following lemmas, which can be verified, similarly as in type $\mathbf{B}\mathbf{D}$, via involved and lengthy computations. In order to keep the number of (boring) computations in this paper in reasonable boundaries, we omit their proofs.

Lemma 6.8** (Barbell removal).**

We have \,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ\,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower 0.57748pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{1.99179pt}\pgfsys@lineto{0.0pt}{7.6827pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{1.99179pt}\pgfsys@curveto{1.4143pt}{2.77289pt}{0.7811pt}{3.4061pt}{0.0pt}{3.4061pt}\pgfsys@curveto{-0.7811pt}{3.4061pt}{-1.4143pt}{2.77289pt}{-1.4143pt}{1.99179pt}\pgfsys@curveto{-1.4143pt}{1.2107pt}{-0.7811pt}{0.57748pt}{0.0pt}{0.57748pt}\pgfsys@curveto{0.7811pt}{0.57748pt}{1.4143pt}{1.2107pt}{1.4143pt}{1.99179pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.99179pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{1.99179pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=[\tfrac{n}{2}]_{2}\mathrm{id}_{0}. $\blacksquare$

Lemma 6.9** (Thin K removal).**

We have (\mathrm{id}\otimes\,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)\circ\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ(\mathrm{id}\otimes\,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower 0.57748pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{1.99179pt}\pgfsys@lineto{0.0pt}{7.6827pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{1.99179pt}\pgfsys@curveto{1.4143pt}{2.77289pt}{0.7811pt}{3.4061pt}{0.0pt}{3.4061pt}\pgfsys@curveto{-0.7811pt}{3.4061pt}{-1.4143pt}{2.77289pt}{-1.4143pt}{1.99179pt}\pgfsys@curveto{-1.4143pt}{1.2107pt}{-0.7811pt}{0.57748pt}{0.0pt}{0.57748pt}\pgfsys@curveto{0.7811pt}{0.57748pt}{1.4143pt}{1.2107pt}{1.4143pt}{1.99179pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.99179pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{1.99179pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)=[\tfrac{n}{2}-1]_{2}\mathrm{id}. $\blacksquare$

Lemma 6.10** (Thick K opening).**

We have (\mathrm{id}_{2}\otimes\,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower-1.45363pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@lineto{0.0pt}{-0.85364pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{4.83725pt}\pgfsys@curveto{1.4143pt}{5.61835pt}{0.7811pt}{6.25156pt}{0.0pt}{6.25156pt}\pgfsys@curveto{-0.7811pt}{6.25156pt}{-1.4143pt}{5.61835pt}{-1.4143pt}{4.83725pt}\pgfsys@curveto{-1.4143pt}{4.05615pt}{-0.7811pt}{3.42294pt}{0.0pt}{3.42294pt}\pgfsys@curveto{0.7811pt}{3.42294pt}{1.4143pt}{4.05615pt}{1.4143pt}{4.83725pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.83725pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{4.83725pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)\circ\,\leavevmode\hbox to6.09pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.84544pt}{1.42271pt}{2.84544pt}{2.84544pt}{4.26817pt}\pgfsys@curveto{4.26817pt}{2.84544pt}{5.69089pt}{2.84544pt}{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{{{}{}}}{{}{}}{{{}{}}}{}{}{}{}\pgfsys@moveto{0.0pt}{11.38179pt}\pgfsys@curveto{0.0pt}{8.53635pt}{1.42271pt}{8.53635pt}{2.84544pt}{7.11362pt}\pgfsys@curveto{4.26817pt}{8.53635pt}{5.69089pt}{8.53635pt}{5.69089pt}{11.38179pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\circ(\mathrm{id}_{2}\otimes\,\leavevmode\hbox to2.83pt{\vbox to7.71pt{\pgfpicture\makeatletter\hbox{\hskip 1.4143pt\lower 0.57748pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{1.99179pt}\pgfsys@lineto{0.0pt}{7.6827pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.4143pt}{1.99179pt}\pgfsys@curveto{1.4143pt}{2.77289pt}{0.7811pt}{3.4061pt}{0.0pt}{3.4061pt}\pgfsys@curveto{-0.7811pt}{3.4061pt}{-1.4143pt}{2.77289pt}{-1.4143pt}{1.99179pt}\pgfsys@curveto{-1.4143pt}{1.2107pt}{-0.7811pt}{0.57748pt}{0.0pt}{0.57748pt}\pgfsys@curveto{0.7811pt}{0.57748pt}{1.4143pt}{1.2107pt}{1.4143pt}{1.99179pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.99179pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{1.99179pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)=\,\leavevmode\hbox to2.12pt{\vbox to11.44pt{\pgfpicture\makeatletter\hbox{\hskip 1.06055pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{10.24257pt}\pgfsys@lineto{0.0pt}{7.04176pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.06055pt}{3.20079pt}\pgfsys@curveto{1.06055pt}{3.78651pt}{0.58572pt}{4.26134pt}{0.0pt}{4.26134pt}\pgfsys@curveto{-0.58572pt}{4.26134pt}{-1.06055pt}{3.78651pt}{-1.06055pt}{3.20079pt}\pgfsys@curveto{-1.06055pt}{2.61507pt}{-0.58572pt}{2.14024pt}{0.0pt}{2.14024pt}\pgfsys@curveto{0.58572pt}{2.14024pt}{1.06055pt}{2.61507pt}{1.06055pt}{3.20079pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.20079pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{3.20079pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@moveto{1.06055pt}{7.04176pt}\pgfsys@curveto{1.06055pt}{7.62749pt}{0.58572pt}{8.10231pt}{0.0pt}{8.10231pt}\pgfsys@curveto{-0.58572pt}{8.10231pt}{-1.06055pt}{7.62749pt}{-1.06055pt}{7.04176pt}\pgfsys@curveto{-1.06055pt}{6.45604pt}{-0.58572pt}{5.98122pt}{0.0pt}{5.98122pt}\pgfsys@curveto{0.58572pt}{5.98122pt}{1.06055pt}{6.45604pt}{1.06055pt}{7.04176pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{7.04176pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{7.04176pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\pgfsys@color@rgb@stroke{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0.8671875}{0.125}{0.14453125}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,+[\tfrac{n}{2}-2]_{2}\mathrm{id}_{2}. $\blacksquare$

Lemma 6.11** (Merge-split sliding relations).**

We have

[TABLE]

Again, the proof that (6-6) is well-defined goes similarly, and we immediately obtain:

Proposition 6.12**.**

The two functors \Gamma_{\mathbf{C}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} and \Gamma_{\mathbf{C}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} are well-defined. Moreover, we have commuting diagrams

[TABLE]

6B. The ladder functor in types $\mathbf{B}\mathbf{C}\mathbf{D}$

We now define the ladder functors $\Upsilon_{\!\mathfrak{so}}$ and $\Upsilon_{\!\mathfrak{sp}}$, which relate our web categories to the quantum groups $\mathbf{U}_{q}(\mathfrak{so}_{2k})$ and $\mathbf{U}_{q}(\mathfrak{sp}_{2k})$. We stress that the definition of the ladder functors do not depend on whether we are in the exterior or the symmetric case.

The ladder functor for -webs

Let $\overline{\lambda}=\lambda+\tfrac{n}{2}$. We define the ladder functor \Upsilon_{\!\mathfrak{so}}\colon\dot{\mathbf{U}}_{q}(\mathfrak{so}_{2k})\rightarrow\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} via

[TABLE]

Here, we silently assume that $\lambda$, as an object of \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}, is the zero object if $\lambda\notin{\mathbb{Z}}^{k}_{\geq 0}$.

Lemma 6.13**.**

The ladder functor $\Upsilon_{\!\mathfrak{so}}$ is well-defined.

Proof.

We need to check that the relations of $\dot{\mathbf{U}}_{q}(\mathfrak{so}_{2k})$ are satisfied in the image.

**Assignment of the generators: **

Recall that

[TABLE]

where $\alpha_{i}$ are the simple roots. By our conventions for types $\mathbf{A}$ and $\mathbf{D}$ (cf. at the beginning of Section 5A), we see that (6-7) lands in the correct morphisms spaces.

**The $\dot{\mathbf{U}}_{q}(\mathfrak{gl}_{2k})$ relations: **

The relations involving only ${\mathsf{E}}_{i}$’s and ${\mathsf{F}}_{i}$’s with $i\neq k-1$ are clearly satisfied by the web calculus in type $\mathbf{A}$, i.e. by [CKM14, Proposition 5.2.1].

**The $\dot{\mathbf{U}}_{q}(\mathfrak{so}_{2k})$ relations: **

We just have to check case by case that the defining relations of $\dot{\mathbf{U}}_{q}(\mathfrak{so}_{2k})$ which involve ${\mathsf{E}}_{k}$’s and ${\mathsf{F}}_{k}$’s hold in the web calculus (for this purpose, recall the anti-involution $\omega$ from Remark 3.3):

1. $\blacktriangleright$**: **

The commutator relation (5-2) between ${\mathsf{E}}_{k}$ and ${\mathsf{F}}_{k}$ holds in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} by Lemmas 3.9. 2. $\blacktriangleright$**: **

The images of ${\mathsf{F}}_{k-1}$ and ${\mathsf{E}}_{k}$ commute thanks to Lemmas 3.10. Applying $\omega$ shows that the images of ${\mathsf{E}}_{k-1}$ and ${\mathsf{F}}_{k}$ commute as well. 3. $\blacktriangleright$**: **

The Serre relation (5-29) holds in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} by Lemmas 3.12. The ${\mathsf{F}}$ version of it holds by applying $\omega$. 4. $\blacktriangleright$**: **

The Serre relation (5-30) holds in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} by Lemmas 3.13. The versions involving ${\mathsf{F}}$’s hold by applying $\omega$. 5. $\blacktriangleright$**: **

The Serre relation (5-31) holds in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} by Lemmas 3.14. Again, the versions involving ${\mathsf{F}}$’s hold by applying $\omega$.

Note here that the quantum numbers work out thanks to the shift by $\tfrac{n}{2}$ in (6-7). All other relations, e.g. far-commutativity, are clearly satisfied. ∎

The ladder functor for -webs

Using the same notation as above, we define the ladder functor \Upsilon_{\!\mathfrak{sp}}\colon\dot{\mathbf{U}}_{q}(\mathfrak{sp}_{2k})\rightarrow\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} via

[TABLE]

Again, we assume that $\lambda$, as an object of \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}, is the zero object if $\lambda\notin{\mathbb{Z}}^{k}_{\geq 0}$.

Lemma 6.14**.**

The ladder functor $\Upsilon_{\!\mathfrak{sp}}$ is well-defined.

Proof.

The proof is, mutatis mutandis, as the proof of Lemma 6.13. In particular:

1. $\blacktriangleright$

The ${\mathsf{E}}_{k}$-${\mathsf{F}}_{k}$ commutator relation holds in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} by Lemma 4.8. 2. $\blacktriangleright$

The images of ${\mathsf{F}}_{k-1}$ and ${\mathsf{E}}_{k}$ commute by Lemma 4.9. That the images of ${\mathsf{E}}_{k-1}$ and ${\mathsf{F}}_{k}$ commute follows by applying $\omega$. 3. $\blacktriangleright$

The Serre relation (5-27) holds in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} by Lemma 4.10. As before, the versions involving ${\mathsf{F}}$’s follow then applying $\omega$. 4. $\blacktriangleright$

The Serre relation (5-28) holds in \smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,q^{n}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} by Lemma 4.13. As usual, the versions involving ${\mathsf{F}}$’s follow then applying $\omega$.∎

6C. The Howe functors

Note that we never used that ${\mathsf{z}}$ was specialized to $q^{n}$ in the definition of the ladder functors, and we actually get ladder functors \dot{\mathbf{U}}_{q}(\mathfrak{so}_{2k})\rightarrow\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,z}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} and \dot{\mathbf{U}}_{q}(\mathfrak{sp}_{2k})\rightarrow\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,z}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} for any $z\in{\mathbb{C}}(q)$. In particular, we also get ladder functors \dot{\mathbf{U}}_{q}(\mathfrak{so}_{2k})\rightarrow\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,-q^{-n}}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{ {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{5.33467pt}\pgfsys@curveto{0.0pt}{3.8286pt}{0.62779pt}{2.13387pt}{2.13387pt}{2.13387pt}\pgfsys@curveto{3.63992pt}{2.13387pt}{4.26773pt}{3.8286pt}{4.26773pt}{5.33467pt}\pgfsys@stroke\pgfsys@invoke{ } } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}} and \dot{\mathbf{U}}_{q}(\mathfrak{sp}_{2k})\rightarrow\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q,-q^{-n}}^{\,\leavevmode\hbox to6.13pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 1.26692pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@lineto{0.0pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@moveto{1.06693pt}{2.13387pt}\pgfsys@curveto{1.06693pt}{2.72311pt}{0.58925pt}{3.20079pt}{0.0pt}{3.20079pt}\pgfsys@curveto{-0.58925pt}{3.20079pt}{-1.06693pt}{2.72311pt}{-1.06693pt}{2.13387pt}\pgfsys@curveto{-1.06693pt}{1.54462pt}{-0.58925pt}{1.06694pt}{0.0pt}{1.06694pt}\pgfsys@curveto{0.58925pt}{1.06694pt}{1.06693pt}{1.54462pt}{1.06693pt}{2.13387pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}, which, by slight abuse of notation, we still denote by $\Upsilon_{\!\mathfrak{so}}$ and $\Upsilon_{\!\mathfrak{sp}}$.

Composing the presentation and the ladder functors, we finally obtain the Howe functors:

[TABLE]

7. Main results

We are finally ready to state and prove our main results.

7A. Quantizing Howe dualities in types $\mathbf{B}\mathbf{C}\mathbf{D}$

A brief reminder on (quantum) highest weight theory

The finite-dimensional representation theory of $\mathbf{U}_{q}(\mathfrak{g})$ at generic $q$ is fairly well-understood. In particular, all such representations are semisimple, and, if we restrict to so-called type $1$ representations (where $\mathbf{q}^{h}$ acts by powers of $q$, cf. **[Jan96, Section 5.2]**), then the simple modules are in bijection with dominant integral weights $\lambda\in X^{+}$. We denote by ${\mathrm{L}}_{q}(\mathfrak{g},\lambda)$ the corresponding simple $\mathbf{U}_{q}(\mathfrak{g})$-module.

The situation for the coideal subalgebras, on the contrary, is more difficult and less understood. For $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$, in particular, one cannot consider weights and weight representations, since there is no natural analog of a Cartan subalgebra (although see **[Let17]** for some progress in this direction). Still, we will encounter some of their representations through Howe duality.

Before we can start, we need some more notation. Let $\mathfrak{P}$ be the set of partitions (or Young diagrams). Given a partition $\lambda=(\lambda_{1},\dots,\lambda_{s})\in\mathfrak{P}$ (with $\lambda_{s}\neq 0$), we write $\ell(\lambda)=s$ for its length, and we denote by $\lambda^{\mathrm{T}}=(\lambda_{1}^{\mathrm{T}},\dots,\lambda_{t}^{\mathrm{T}})\in\mathfrak{P}$ its transpose. For the rest, we keep the notation from Section 6.

We start with the sympletic case since it is easier to state (cf. Remark 1.2).

Skew quantum Howe duality for the pair $(\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n}),\mathbf{U}_{q}(\mathfrak{sp}_{2k}))$

Theorem 7.1**.**

There are commuting actions

[TABLE]

generating each other’s centralizer. Hence, the $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$–$\mathbf{U}_{q}(\mathfrak{sp}_{2k})$-bimodule (7-1) is multiplicity-free. The $\mathbf{U}_{q}(\mathfrak{sp}_{2k})$-modules appearing in its decomposition are

[TABLE]

Proof.

We denote the space in the middle of (7-1) by $\mathrm{M}_{q}$. All $\lambda$’s appearing below will always satisfy the conditions in (7-2).

By construction, $\mathrm{M}_{q}$ is acted on by $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$ via restriction of the action by $\mathbf{U}_{q}(\mathfrak{gl}_{n})$. Using \Phi_{\mathbf{C}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} from (6-9), we see that there is a commuting action of $\mathbf{U}_{q}(\mathfrak{sp}_{2k})$. (In fact, we get an action of $\dot{\mathbf{U}}_{q}(\mathfrak{sp}_{2k})$ which then gives an action of $\mathbf{U}_{q}(\mathfrak{sp}_{2k})$ since $\mathrm{M}_{q}$ is finite-dimensional, cf. [Lus10, Section 23.1.4].)

Next, we want to use the analogous result in the non-quantized setting (see [How95] and [CW12, Corollary 5.33], but beware that the roles of $k$ and $n$ are swapped in [CW12]). It states that there is an action of $\mathbf{U}(\mathfrak{sp}_{2k})$ on $\mathrm{M}={\textstyle\bigwedge^{\bullet}}({\mathbb{C}}^{n}\otimes{\mathbb{C}}^{k})$ commuting with the natural action of $\mathbf{U}(\mathfrak{sp}_{n})$ and that these two actions generate each others centralizer. Moreover, [CW12, Corollary 5.33] gives the bimodule decomposition of $\mathrm{M}$, similar to (7-2).

Now, we can easily compare the action of $\mathbf{U}_{q}(\mathfrak{sp}_{2k})$ on $\mathrm{M}_{q}$ and the action of $\mathbf{U}(\mathfrak{sp}_{2k})$ on $\mathrm{M}$, and see that the weights and their multiplicities are the same. Hence, we can deduce that the decomposition of $\mathrm{M}_{q}$ as a $\mathbf{U}_{q}(\mathfrak{sp}_{2k})$-module is the quantum analog of the one in [CW12, Corollary 5.33]. It follows that the $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$–$\mathbf{U}_{q}(\mathfrak{sp}_{2k})$-bimodule $\mathrm{M}_{q}$ decomposes as

[TABLE]

with $\lambda$ as in (7-2) and where the ${\mathrm{L}}^{\prime}_{q}(\mathfrak{sp}_{n},\lambda)$’s denote just some $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$-modules (which are indexed by the $\lambda$’s).

We want to show that all appearing ${\mathrm{L}}^{\prime}_{q}(\mathfrak{sp}_{n},\lambda)$ are irreducible, or, equivalently, that the action gives a surjection

[TABLE]

To this end, consider the integral version $\prescript{{\mathscr{A}}}{}{\mathrm{M}_{q}}$ of the representation $\mathrm{M}_{q}$, defined as the ${\mathscr{A}}$-span of tensor products of wedges of the standard basis vectors $v_{i}$ inside $\mathrm{M}_{q}$. Note that $\prescript{{\mathscr{A}}}{}{\mathrm{M}_{q}}$ is a free ${\mathscr{A}}$-module, and this will be important for what follows.

It can be easily checked that $\prescript{{\mathscr{A}}}{}{\mathrm{M}_{q}}$ is stable under the actions of $\prescript{{\mathscr{A}}}{}{\mathbf{U}_{q}^{\prime}}(\mathfrak{sp}_{n})$ and $\prescript{{\mathscr{A}}}{}{\mathbf{U}_{q}}(\mathfrak{sp}_{2k})$. Moreover, setting $q=1$, we can identify $\prescript{{\mathscr{A}}}{}{\mathrm{M}_{q}}\otimes_{\mathscr{A}}{\mathscr{A}}/(q-1)$ with $\mathrm{M}$, and it is then clear that the action of $\prescript{{\mathscr{A}}}{}{\mathbf{U}_{q}^{\prime}}(\mathfrak{sp}_{n})\otimes_{\mathscr{A}}{\mathscr{A}}/(q-1)$ matches the natural action of $\mathbf{U}(\mathfrak{sp}_{n})$, i.e.

[TABLE]

(One could actually show that $\prescript{{\mathscr{A}}}{}{\mathbf{U}_{q}^{\prime}}(\mathfrak{sp}_{n})\otimes_{\mathscr{A}}{\mathscr{A}}/(q-1)$ and $\mathbf{U}(\mathfrak{sp}_{n})$ are isomorphic, but since we do not need it, we avoid this additional complication.) In particular, the images of these two actions agree, and their dimensions are both equal to

[TABLE]

It follows that the dimension of the image for generic $q$ cannot be strictly smaller, and in particular the dimension of the image of (7-3) has to be greater or equal than $d$. Hence, the map in (7-3) is surjective, and we are done. ∎

Symmetric quantum Howe duality for the pair $(\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n}),\dot{\mathbf{U}}_{q}(\mathfrak{so}_{2k}))$

Theorem 7.2**.**

There are commuting actions

[TABLE]

generating each other’s centralizer. Hence, the $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$–$\dot{\mathbf{U}}_{q}(\mathfrak{so}_{2k})$-bimodule (7-4) is multiplicity-free. The $\dot{\mathbf{U}}_{q}(\mathfrak{so}_{2k})$-modules appearing in its decomposition are

[TABLE]

Proof.

The proof is similar to the proof of Theorem 7.3, but using the functor \Phi_{\mathbf{C}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} and the non-quantized Howe duality from [CW12, Corollary 5.32]. (Note hereby that we cannot easily pass from $\dot{\mathbf{U}}_{q}(\mathfrak{so}_{2k})$ to $\mathbf{U}_{q}(\mathfrak{so}_{2k})$ since the ${\mathbb{C}}(q)$-vector space in (7-4) is infinite-dimensional.) ∎

Skew quantum Howe duality for the pair $(\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n}),\mathbf{U}_{q}(\mathfrak{so}_{2k}))$

Theorem 7.3**.**

There are commuting actions

[TABLE]

In case $n$ is odd they generate each other’s centralizer. In any case, the $\mathbf{U}_{q}(\mathfrak{so}_{2k})$-modules appearing in the decomposition of (7-6) are

[TABLE]

Proof.

Mutatis mutandis as in the proof of Theorem 7.1, but using the functor \Phi_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} and the non-quantized Howe duality from [CW12, Corollary 5.41]. Note that one has $\mathrm{O}(n)\cong\mathrm{SO}(n)\times{\mathbb{Z}}/2{\mathbb{Z}}$ in type $\mathbf{B}$. As explained in [CW12, above Proposition 5.35] or [LZ06, §5.1.3], the extra generator in $\mathrm{O}(n)-\mathrm{SO}(n)$ acts trivially on the de-quantized analog of (7-6). It follows that [CW12, Corollary 5.41] works in this case for $\mathrm{SO}(n)$ instead of $\mathrm{O}(n)$, and hence also for $\mathfrak{so}_{n}$, cf. also Remark 1.2. ∎

Symmetric quantum Howe duality for the pair $(\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n}),\dot{\mathbf{U}}_{q}(\mathfrak{sp}_{2k}))$

Theorem 7.4**.**

There are commuting actions

[TABLE]

In case $n$ is odd they generate each other’s centralizer. In any case, the $\dot{\mathbf{U}}_{q}(\mathfrak{sp}_{2k})$-modules appearing in the decomposition of (7-8) are

[TABLE]

Proof.

Mutatis mutandis as in the proof of Theorem 7.1, but using the functor \Phi_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} and the non-quantized Howe duality from [CW12, Corollary 5.40]. (Keeping the same remarks as in the proofs of Theorem 7.2 and 7.3 in mind.) ∎

Some concluding remarks

Remark 7.5.

We stress again that Theorem 7.3 and 7.4 can be strengthened to include the double centralizer property for type $\mathbf{D}$ as well, cf. Remark 1.2.

Remark 7.6.

In the spirit of [TVW15], one could use the Howe dualities involving the orthosymplectic Lie superalgebra $\mathfrak{osp}$, as in [How89], [CZ04] or [CW12], to give a unified treatment of the exterior and the symmetric story. Since quantization in our setup is already quite involved, we decided to not pursue this further.

Remark 7.7.

One feature of web categories is that they are “amenable to categorification”. For example, one can use foams in the sense of [Kho04], see e.g. [Bla10], [LQR15], [EST15] and [EST16] for categorifiying webs. Or category $\mathcal{O}$ as e.g. in [Sar16a] or [Sar16b]. Categorifications of Howe dualities involving coideal subalgebras (of different kinds) have already been obtained in [ES13] (which also connects to foams, cf. [ETW16]), and there are good reasons to hope that our story categorifies as well.

7B. Relation of the web categories to the (quantum) Brauer algebra

In groundbreaking work, Brauer **[Bra37]** introduced the so-called Brauer algebra, which arose naturally in his study of the centralizer of the action of the orthogonal group $\mathrm{O}(n)$ and of the symplectic group $\mathrm{Sp}(n)$ acting on the $k$-fold tensor product $\mathrm{V}^{\otimes k}$ of their vector representations. Comparing this to the de-quantized versions of Theorem 7.1, 7.2, 7.3 and 7.4 suggests that there should be a connection to our web categories. We make this more precise in the following.

Various quantizations of the Brauer algebra

The first quantization of the Brauer algebra, called BMW algebra, was introduced by Birman-Wenzl **[BW89]** and Murakami **[Mur87]**. The BMW algebra plays the role of Brauer’s algebra with respect to the actions of $\mathbf{U}_{q}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}(\mathfrak{sp}_{n})$ on their quantum tensor spaces. However, since we are looking at the centralizers of actions of $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$, and not of $\mathbf{U}_{q}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}(\mathfrak{sp}_{n})$, the BMW algebra does not fit into our picture.

In contrast, Molev **[Mol03]** defined a new quantization $\smash{\mathrm{Br}_{q,{\mathsf{z}}}^{k}}$ of the Brauer algebra, called quantum or $q$-Brauer algebra. This ${\mathbb{C}}(q)[{\mathsf{z}}^{\pm 1}]$-algebra is related by a version of $q$-Schur-Weyl duality to $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$ and $\mathbf{U}_{q}^{\prime}(\mathfrak{sp}_{n})$. Thus, $\smash{\mathrm{Br}_{q,{\mathsf{z}}}^{k}}$ is the natural candidate to be connected to our web categories.

A quantized Brauer category

First, let us quickly recall the situation in type $\mathbf{A}$:

Definition 7.8**.**

The Hecke category $\smash{\boldsymbol{\mathcal{H}}_{q}}$ is the additive closure of the (strict) monoidal, ${\mathbb{C}}(q)$-linear category generated by one object $1$ and by one morphism $T\colon 1\otimes 1\rightarrow 1\otimes 1$ modulo the relations

[TABLE]

(The second relation is known as the braid relation.)

We depict the generator $T$ by an overcrossing, cf. (2-5). Then, by sending $T$ in the evident way to the braiding of $\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}$, we get a functor

[TABLE]

which is fully faithful, see e.g. **[QS15, Proposition 5.9]** or **[TVW15, Lemma 2.25]**. Note, in particular, that crossings span $\operatorname{End}_{\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}}(1^{\otimes k})$.

Our next goal is to extend this to types $\mathbf{B}\mathbf{C}\mathbf{D}$.

Definition 7.9**.**

The quantum or $q$-Bauer category $\boldsymbol{\mathcal{B}\mathrm{r}}_{q,{\mathsf{z}}}$ is the additive closure of the ${\mathbb{C}}(q)[{\mathsf{z}}^{\pm 1}]$-linear $\smash{\boldsymbol{\mathcal{H}}_{q}}$-category generated by $\varnothing$ and by the cup and cap morphisms (depicted as in (gen)) modulo the relations (1), (a), (b), (c) and (d).

*Recall that the relations (a), (b), (c) and (d) are the topological analogs of the relations in Definition 3.2 and 4.1 (for **-webs with slightly different parameters), and that (1) is equivalent to (1) in case of *-webs. Hence, the functor $\beta_{\mathbf{A}}$ extends to two functors

[TABLE]

Connection with the quantum Brauer algebra

Let us now denote by $\smash{\mathrm{Br}_{q,{\mathsf{z}}}^{k}}$ the $q$-Brauer algebra as defined by Molev in **[Mol03, Definition 2.3]**. Precisely, the $q$-Brauer algebra is a ${\mathbb{C}}(q)[{\mathsf{z}}^{\pm 1}]$-algebra with generators $T_{i}$ for $i=1,\dots,k-1$ and additionally $e_{k-1}$. (Note that Molev uses the notation $\sigma_{i}$ instead of $T_{i}$.)

Lemma 7.10**.**

The assignment

[TABLE]

defines an algebra homomorphism $\psi_{k}\colon\smash{\mathrm{Br}_{-q^{-1},-{\mathsf{z}}^{-1}}^{k}}\rightarrow\operatorname{End}_{\boldsymbol{\mathcal{B}\mathrm{r}}_{q,{\mathsf{z}}}}(1^{\otimes k})$.

Proof.

This is immediate up to the last relation in [Mol03, Definition 2.3]. Verifying the last relation in [Mol03, Definition 2.3] is a lengthy, but straightforward computation, which can be done by using (b) and (d) repeatedly. ∎

In particular, the composite \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}\circ\beta_{\leavevmode\hbox to6.89pt{\vbox to9.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{1.42537pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{5.69089pt}{0.0pt}\pgfsys@curveto{5.69089pt}{2.00829pt}{4.26552pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{7.11362pt}\pgfsys@curveto{0.0pt}{5.10533pt}{0.83716pt}{2.84544pt}{2.84544pt}{2.84544pt}\pgfsys@curveto{4.85373pt}{2.84544pt}{5.69089pt}{5.10533pt}{5.69089pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\circ\psi_{k} defines an action of the $q$-Brauer algebra which commutes which the natural action of $\mathbf{U}_{q}^{\prime}(\mathfrak{so}_{n})$:

[TABLE]

Up to scaling conventions, this is the action defined in **[Mol03, Theorem 4.2]**. Similarly, the composite \Gamma_{\mathbf{C}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}\circ\beta_{\leavevmode\hbox to4.21pt{\vbox to7.36pt{\pgfpicture\makeatletter\hbox{\hskip 1.47272pt\lower 1.37273pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{4.26817pt}\pgfsys@lineto{2.13408pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.1pt}\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@moveto{1.42271pt}{2.84544pt}\pgfsys@curveto{1.42271pt}{3.6312pt}{0.78575pt}{4.26816pt}{0.0pt}{4.26816pt}\pgfsys@curveto{-0.78575pt}{4.26816pt}{-1.42271pt}{3.6312pt}{-1.42271pt}{2.84544pt}\pgfsys@curveto{-1.42271pt}{2.0597pt}{-0.78575pt}{1.42273pt}{0.0pt}{1.42273pt}\pgfsys@curveto{0.78575pt}{1.42273pt}{1.42271pt}{2.0597pt}{1.42271pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\circ\psi_{k} provides commuting actions

[TABLE]

(Clearly, we could have also chosen \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} and \Gamma_{\mathbf{C}}^{{\color[rgb]{0.046875,0.5703125,0.046875}\definecolor[named]{pgfstrokecolor}{rgb}{0.046875,0.5703125,0.046875}\mathrm{sym}}\!\!\!\phantom{t}} instead of \Gamma_{\mathbf{B}\mathbf{D}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}} and \Gamma_{\mathbf{C}}^{{\color[rgb]{0.8671875,0.125,0.14453125}\definecolor[named]{pgfstrokecolor}{rgb}{0.8671875,0.125,0.14453125}\mathrm{ext}}\!\!\!\phantom{y}}.)

We show now that Molev’s $q$-Brauer algebra can be identified with the endomorphism algebra of $1^{\otimes k}$ in our $q$-Brauer category:

Proposition 7.11**.**

The map $\psi_{k}$ is an algebra isomorphism, and the functors \beta_{\leavevmode\hbox to6.89pt{\vbox to9.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{1.42537pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{5.69089pt}{0.0pt}\pgfsys@curveto{5.69089pt}{2.00829pt}{4.26552pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{7.11362pt}\pgfsys@curveto{0.0pt}{5.10533pt}{0.83716pt}{2.84544pt}{2.84544pt}{2.84544pt}\pgfsys@curveto{4.85373pt}{2.84544pt}{5.69089pt}{5.10533pt}{5.69089pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} and \beta_{\leavevmode\hbox to4.21pt{\vbox to7.36pt{\pgfpicture\makeatletter\hbox{\hskip 1.47272pt\lower 1.37273pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{4.26817pt}\pgfsys@lineto{2.13408pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.1pt}\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@moveto{1.42271pt}{2.84544pt}\pgfsys@curveto{1.42271pt}{3.6312pt}{0.78575pt}{4.26816pt}{0.0pt}{4.26816pt}\pgfsys@curveto{-0.78575pt}{4.26816pt}{-1.42271pt}{3.6312pt}{-1.42271pt}{2.84544pt}\pgfsys@curveto{-1.42271pt}{2.0597pt}{-0.78575pt}{1.42273pt}{0.0pt}{1.42273pt}\pgfsys@curveto{0.78575pt}{1.42273pt}{1.42271pt}{2.0597pt}{1.42271pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} are fully faithful.

Proof.

**Surjectivity of $\psi_{k}$: **

Because crossings span the space $\operatorname{End}_{\smash{\boldsymbol{\mathcal{W}\mathrm{eb}}_{q}^{\,\leavevmode\hbox to5.47pt{\vbox to7.6pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{1.50606pt}{1.06891pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{4.26773pt}{0.0pt}\pgfsys@curveto{4.26773pt}{1.50606pt}{3.1988pt}{2.13585pt}{2.13387pt}{3.20079pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.13387pt}{3.20079pt}\pgfsys@lineto{2.13387pt}{6.4016pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}}}(1^{\otimes k})$, it is enough to show that $\operatorname{End}_{\boldsymbol{\mathcal{B}\mathrm{r}}_{q,{\mathsf{z}}}}(1^{\otimes k})$ is spanned by diagrams of the form $w_{\mathrm{top}}e^{(l)}w_{\mathrm{bot}}$, with $w_{\mathrm{bot}},w_{\mathrm{top}}\in\operatorname{End}_{\smash{\boldsymbol{\mathcal{H}}_{q}}}(1^{\otimes k})$ and diagrams

[TABLE]

This can be easily seen by induction on the number of crossings of some fixed diagram.

**Injectivity of $\psi_{k}$: **

This follows because the representations in (7-10) and (7-11) are faithful for $n\gg k$ (the precise bound is irrelevant for us). Indeed, the proof that they are faithful for $n\gg k$ follows, as in the proof of [Wen12, Theorem 3.8], by the same results in the non-quantized setting (see e.g. [AST15, Theorem 3.17], but the statement therein can already be found implicitly in the work of Brauer [Bra37]).

**Fully faithfulness of \beta_{\leavevmode\hbox to6.89pt{\vbox to9.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{1.42537pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{5.69089pt}{0.0pt}\pgfsys@curveto{5.69089pt}{2.00829pt}{4.26552pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{7.11362pt}\pgfsys@curveto{0.0pt}{5.10533pt}{0.83716pt}{2.84544pt}{2.84544pt}{2.84544pt}\pgfsys@curveto{4.85373pt}{2.84544pt}{5.69089pt}{5.10533pt}{5.69089pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} and \beta_{\leavevmode\hbox to4.21pt{\vbox to7.36pt{\pgfpicture\makeatletter\hbox{\hskip 1.47272pt\lower 1.37273pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{4.26817pt}\pgfsys@lineto{2.13408pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.1pt}\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@moveto{1.42271pt}{2.84544pt}\pgfsys@curveto{1.42271pt}{3.6312pt}{0.78575pt}{4.26816pt}{0.0pt}{4.26816pt}\pgfsys@curveto{-0.78575pt}{4.26816pt}{-1.42271pt}{3.6312pt}{-1.42271pt}{2.84544pt}\pgfsys@curveto{-1.42271pt}{2.0597pt}{-0.78575pt}{1.42273pt}{0.0pt}{1.42273pt}\pgfsys@curveto{0.78575pt}{1.42273pt}{1.42271pt}{2.0597pt}{1.42271pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}: **

Very similar arguments as for the proof of bijectivity of $\psi_{k}$ imply that the functors \beta_{\leavevmode\hbox to6.89pt{\vbox to9.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.59999pt\lower-0.59999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.0pt}{2.00829pt}{1.42537pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{5.69089pt}{0.0pt}\pgfsys@curveto{5.69089pt}{2.00829pt}{4.26552pt}{2.84808pt}{2.84544pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{2.84544pt}{4.26817pt}\pgfsys@lineto{2.84544pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{7.11362pt}\pgfsys@curveto{0.0pt}{5.10533pt}{0.83716pt}{2.84544pt}{2.84544pt}{2.84544pt}\pgfsys@curveto{4.85373pt}{2.84544pt}{5.69089pt}{5.10533pt}{5.69089pt}{7.11362pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} and \beta_{\leavevmode\hbox to4.21pt{\vbox to7.36pt{\pgfpicture\makeatletter\hbox{\hskip 1.47272pt\lower 1.37273pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{ }\pgfsys@color@gray@fill{1}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{4.26817pt}\pgfsys@lineto{2.13408pt}{4.26817pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@lineto{0.0pt}{8.53635pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.1pt}\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@moveto{1.42271pt}{2.84544pt}\pgfsys@curveto{1.42271pt}{3.6312pt}{0.78575pt}{4.26816pt}{0.0pt}{4.26816pt}\pgfsys@curveto{-0.78575pt}{4.26816pt}{-1.42271pt}{3.6312pt}{-1.42271pt}{2.84544pt}\pgfsys@curveto{-1.42271pt}{2.0597pt}{-0.78575pt}{1.42273pt}{0.0pt}{1.42273pt}\pgfsys@curveto{0.78575pt}{1.42273pt}{1.42271pt}{2.0597pt}{1.42271pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}} are fully faithful.∎

Remark 7.12.

Because of Proposition 7.11, our web categories can be seen as (vast) generalizations of the (quantum) Brauer calculus.