Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid

TL;DR

This paper explores the structure of graded right coideal subalgebras in Nichols algebras of semisimple Yetter-Drinfeld modules, establishing a correspondence with the Weyl groupoid's morphisms and proving a conjecture in the finite case.

Contribution

It constructs an order-preserving bijection between morphisms of the Weyl groupoid and right coideal subalgebras, linking algebraic and combinatorial structures in Nichols algebras.

Findings

Established a bijective correspondence for finite Weyl groupoids.

Proved the Nichols algebra is decomposable under certain conditions.

Confirmed a conjecture of Kharchenko for the Borel part of quantized enveloping algebras.

Abstract

We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving and order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the above map is bijective. In the special case of the Borel part of quantized enveloping algebras our result implies a conjecture of Kharchenko. Key words: Hopf algebra, quantum group, root system, Weyl group

Equations

$$\mathcal{B}(M)=\mathcal{B}(M_1\oplus \cdots \oplus M_{\theta })$$

$$a_{ij}^M=-\max \{m|(\mathrm{ad}{M}_i{)}^m(M_j)\ne 0\}<\infty .$$

$$R_i(M{)}_j=(\mathrm{ad}{M}_i{)}^{-a_{ij}^M}(M_j)\text{ for all }j\ne i,$$

$${\Omega }_i^M:K_i^M\#\mathcal{B}(M_i^{\ast })\to \mathcal{B}(R_i(M))$$

$$\mathcal{C}=\mathcal{C}(I,\mathcal{X},(r_i{)}_{i\in I},(A^X{)}_{X\in \mathcal{X}}),$$

$$s_i^X({\alpha }_j)={\alpha }_j-a_{ij}^X{\alpha }_i\text{ for all }j\in I.$$

$$\mathrm{Hom}(\mathcal{G},X)={\cup }_{Y\in \mathrm{Ob}(\mathcal{G})}\mathrm{Hom}(Y,X)\text{(disjoint union).}$$

$$\mathbf{\Delta }{}^{X\mathrm{re}}=\{w({\alpha }_i)|i\in I,w\in \mathrm{Hom}(\mathcal{W}(\mathcal{C}),X)\}$$

$$A^{X_1}=\left(\begin{array}{cc}2& -1\\ -3& 2\end{array}\right),A^{X_2}=\left(\begin{array}{cc}2& -1\\ -4& 2\end{array}\right),A^{X_3}=\left(\begin{array}{cc}2& -1\\ -4& 2\end{array}\right).$$

$$\mathbf{\Delta }{}^{X_1\mathrm{re}}=$$

$$\pm 1^3{2}^4,\pm 1^3{2}^5,\pm 1^4{2}^5,\pm 1^4{2}^7,\pm 1^5{2}^7,\pm 1^5{2}^8\},$$

$$\mathbf{\Delta }{}^{X_2\mathrm{re}}=$$

$$\pm 12^4,\pm 12^5,\pm 1^2{2}^5,\pm 1^2{2}^7,\pm 1^3{2}^7,\pm 1^3{2}^8\},$$

$$\mathbf{\Delta }{}^{X_3\mathrm{re}}=$$

$$\pm 12^3,\pm 1^2{2}^3,\pm 12^4,\pm 1^3{2}^4,\pm 1^2{2}^5,\pm 1^3{2}^5\},$$

$$\mathcal{R}=\mathcal{R}(\mathcal{C},(\mathbf{\Delta }{}^X{)}_{X\in \mathcal{X}})$$

$$\ell (w)=\min \{m\in {\mathbb{N}}_0|\text{there exist i1,…,im∈I with w=idXsi1⋯sim}\}.$$

$${\beta }_k={\mathrm{id}}_X{s}_{i_1}\cdots s_{i_{k-1}}({\alpha }_{i_k}),k\in \{1,2,\dots ,m\},$$

$$\underset{\text{2mi,jX factors}}{\underbrace{s_i^{r_j(r_i{r}_j{)}^{m_{i,j}^X-1}(X)}{s}_j^{(r_i{r}_j{)}^{m_{i,j}^X-1}(X)}\dots s_j^{r_i{r}_j(X)}{s}_i^{r_j(X)}{s}_j^X}}={\mathrm{id}}_X$$

$$m_{i,j}^X=\#(\mathbf{\Delta }{}^X\cap ({\mathbb{N}}_0{\alpha }_i+{\mathbb{N}}_0{\alpha }_j)).$$

$${\beta }_k=s_{i_1}^{r_{i_1}(X)}{s}_{i_2}^{r_{i_2}{r}_{i_1}(X)}\cdots s_{i_{k-1}}^{r_{i_{k-1}}\cdots r_{i_2}{r}_{i_1}(X)}({\alpha }_{i_k})\in \mathbf{\Delta }{}^X,k\in \{1,2,\dots ,m\},$$

$${\Lambda }^X(i_1,\dots ,i_m)=({\beta }_k{)}_{k\in \{1,2,\dots ,m\}},$$

$$\displaystyle\Lambda_{+}^{X}(i_{1},\dots,i_{m})=\big{\{}\lambda\in\boldsymbol{\Delta}{}^{X}_{+}\,|\,\#\{k\in\mathbb{Z}\,|\,1\leq k\leq m,\lambda=\pm\beta_{k}\}\text{ is odd}\big{\}}.$$

$$\displaystyle\Lambda_{+}^{X}(i_{1},\dots,i_{m})=\begin{cases}s_{i_{1}}^{Y}(\Lambda_{+}^{Y}(i_{2},\dots,i_{m}))\cup\{\alpha_{i_{1}}\}&\text{if $\alpha_{i_{1}}\notin\Lambda_{+}^{Y}(i_{2},\dots,i_{m})$,}\\ s_{i_{1}}^{Y}\big{(}\Lambda_{+}^{Y}(i_{2},\dots,i_{m})\setminus\{\alpha_{i_{1}}\}\big{)}&\text{if $\alpha_{i_{1}}\in\Lambda_{+}^{Y}(i_{2},\dots,i_{m})$.}\end{cases}$$

$$\displaystyle\mathrm{Hom}(Y,Z)=\big{\{}$$

$$\displaystyle n\in\mathbb{N}_{0},\,j_{1},\dots,j_{n}\in I,r_{j_{1}}\cdots r_{j_{n}}(Y)=Z\big{\}}$$

$$\displaystyle\big{(}\Lambda^{Z}_{+}(j_{1},\dots,j_{n}),\mathrm{id}_{Z}s_{j_{1}}\cdots s_{j_{n}}\big{)}\circ\big{(}\Lambda^{Y}_{+}(k_{1},\dots,k_{p}),\mathrm{id}_{Y}s_{k_{1}}\cdots s_{k_{p}}\big{)}\qquad$$

$$\displaystyle=\big{(}\Lambda^{Z}_{+}(j_{1},\dots,j_{n},k_{1},\dots,k_{p}),\mathrm{id}_{Z}s_{j_{1}}\cdots s_{j_{n}}s_{k_{1}}\cdots s_{k_{p}}\big{)}$$

$${\Lambda }_{+}^Z(j_1,\dots ,j_n)={\Lambda }_{+}^Z(j_1^{\prime },\dots ,j_{n^{\prime }}^{\prime }),{\Lambda }_{+}^Y(k_1,\dots ,k_p)={\Lambda }_{+}^Y(k_1^{\prime },\dots ,k_{p^{\prime }}^{\prime }),$$

$${\Lambda }_{+}^Z(j_1,\dots ,j_n,k_1,\dots ,k_p)=$$

$${\Lambda }_{+}^Z(\underset{2m_{i,j}^Z\text{ entries}}{\underbrace{i,j,i,j,\dots ,i,j}})=$$

$$(\mathcal{V},(({\Lambda }_{+}^Z(i),s_i^{r_i(Z)}){)}_{i\in I,Z\in \mathcal{X}})$$

$${\Lambda }_{+}^Z(i,i,j_1,\dots ,j_n)={\Lambda }_{+}^Z(j_1,\dots ,j_n),{\mathrm{id}}_Z{s}_i{s}_i{s}_{j_1}\cdots s_{j_n}={\mathrm{id}}_Z{s}_{j_1}\cdots s_{j_n}$$

$$s_{i_1}^{r_{i_1}(X)}\cdots s_{i_{k-1}}^{r_{i_{k-1}}\cdots r_{i_1}(X)}({\alpha }_{i_k})\in \mathbf{\Delta }{}^X,$$

Full text

The work of I.H. is supported by DFG within a Heisenberg fellowship

Introduction

It is well-known that quantum groups do not have “enough” Hopf subalgebras. Instead the larger class of right (or left) coideal subalgebras should be studied. A right coideal subalgebra $E\subset A$ of a Hopf algebra $A$ with comultiplication $\varDelta$ is a subalgebra of $A$ with $\varDelta(E)\subset E\otimes A$.

1. Right coideal subalgebras of quantized enveloping algebras $\boldsymbol{U^{\geq 0}}$. Let $\mathfrak{g}$ be a semisimple complex Lie algebra, $\Pi$ a basis of its root system with respect to a fixed Cartan subalgebra, and $U=U_{q}(\mathfrak{g})$ the quantized enveloping algebra of $\mathfrak{g}$ in the sense of [Jan96, Ch. 4]. We assume that $q$ is not a root of unity. Let $U^{+}$ and $U^{0}$ be the subalgebras of $U$ generated by the sets $\{E_{\alpha}\,|\,\alpha\in\Pi\}$ and $\{K_{\alpha},K_{\alpha}^{-1}\,|\,\alpha\in\Pi\}$, respectively, and let $U^{\geq 0}=U^{+}U^{0}$. For any element $w$ of the Weyl group $W$ of $\mathfrak{g}$ let $U^{+}[w]\subset U^{+}$ be the subspace defined in [Jan96, 8.24] in terms of root vectors constructed via Lusztig’s automorphisms. We prove in Thm. 7.3, see also Cor. 6.13, the following:

The map $w\mapsto U^{+}[w]U^{0}$ defines an order preserving bijection between $W$ and the set of all right coideal subalgebras of $U^{\geq 0}$ containing $U^{0}$, where right coideal subalgebras are ordered by inclusion and $W$ is ordered by the Duflo order. If $E_{1}\subset E_{2}$ are right coideal subalgebras of $U^{\geq 0}$ containing $U^{0}$, then $E_{2}$ is free over $E_{1}$ as a right module.

Recall that if $w_{1},w_{2}$ are elements in $W$, then $w_{1}\leq_{D}w_{2}$ in the (right) Duflo order if and only if any reduced expression of $w_{1}$ can be extended to a reduced expression of $w_{2}$ beginning with $w_{1}$.

In particular, the number of right coideal subalgebras of $U^{\geq 0}$ containing $U^{0}$ is equal to the order of the Weyl group $W$. This last statement was conjectured by Kharchenko in [Kha09] for simple Lie algebras $\mathfrak{g}$. The conjecture was proven for $\mathfrak{g}$ of type $A_{n}$ [KS08], $B_{n}$ [Kha09] and $G_{2}$ [Pog09] by combinatorial calculations using Lyndon words. In these papers right coideal subalgebras are classified in terms of certain subsets of positive roots.

The subspaces $U^{+}[w]\subset U^{+}$ are familiar objects in quantum groups. Among others, they are used by Lusztig [Lus93] to establish a PBW basis for $U^{+}$, by De Concini, Kac and Procesi [CKP95] to introduce quantum Schubert calculus, and are identified by Yakimov [Yak09] as quotients of quantized Bruhat cell translates [Jos95, Gor00]. It was essentially well-known that $U^{+}[w]U^{0}$ is a right coideal subalgebra of $U^{\geq 0}$: proofs and indications in this direction can be found in [LS90], [CKP95, 2.2], [CP92, 9.3], [AJS94]. The arguments often use case by case considerations and reduction to the rank two case, and sometimes they work only in the $h$-adic setting. The algebras $U^{+}[w]$ are known to depend only on $w$ and not on the chosen reduced expression of $w$, see e. g. [Lus93, 40.2.1] and [Jan96, 8.21]. With our systematic approach to graded right coideal subalgebras we offer a new way to study $U^{+}$ without the usual case by case considerations, and intrinsically characterize the algebras $U^{+}[w]$ and their ordering with respect to inclusion. With the necessary modifications, our results also apply to the small quantum groups of semisimple Lie algebras where $q$ is a root of unity, and to multiparameter versions of $U$, see Cor. 6.17 and Rem. 7.4.

2. Right coideal subalgebras of Nichols algebras. The paper is written in the very general context of Nichols algebras $\mathcal{B}(M)$ of semisimple Yetter-Drinfeld modules $M\in{}^{H}_{H}\mathcal{YD}$, where $H$ is an arbitrary Hopf algebra with bijective antipode. Nichols algebras, also called quantum symmetric algebras, see [Ros98], appear as fundamental objects in the classification theory of Hopf algebras [AS98, AS02, AS05], in particular of Hopf algebras which are generated by group-like and skew-primitive elements. For example, in the setting of quantized enveloping algebras, $M=\oplus_{\alpha\in\Pi}\Bbbk E_{\alpha}$ is a Yetter-Drinfeld module over $U^{0}$, and $\mathcal{B}(M)=U^{+}$. Finite-dimensional Nichols algebras of diagonal type are classified in [Hec09]. Recently, much progress in the understanding of finiteness properties of Nichols algebras of nonabelian group type has been achieved, see e. g. [AFGV09b, AFGV09a], [HS08] and references therein.

In rather general situations (if $M$ admits all reflections, see Sect. 6) one can associate a Weyl groupoid $\mathcal{W}(M)$ to $M$, see [AHS08], [HS08]. In case of the Borel part of a quantized Kac-Moody algebra $\mathfrak{g}$, $\mathcal{W}(M)$ is essentially the Weyl group of $\mathfrak{g}$. Under the assumption that $\mathcal{W}(M)$ is finite, we prove in Cor. 6.13 a PBW-theorem for the Nichols algebra $\mathcal{B}(M)$ and its right coideal subalgebras, where the subalgebra generated by a root vector in the quantum group case is replaced by the Nichols algebra of a finite-dimensional irreducible Yetter-Drinfeld module. As a consequence we can show that the real roots associated with $\mathcal{W}(M)$ satisfy the axioms of a root system in the sense of [HY08b], see also [HS08]. In Thm. 6.14 we provide generalizations of results of Levendorskii and Soibelman [LS90, LS91] on coproducts and commutators of root vectors. Our proofs are new even for $U^{+}$, since they are free of case by case calculations, and do not use the braid relations for Lusztig’s automorphisms.

We note that a PBW-theorem for right coideal subalgebras of character Hopf algebras (where the braiding is diagonal) is obtained by Kharchenko in terms of Lyndon words, see [Kha08]. For Nichols algebras of diagonal type a PBW theorem in the spirit of Lusztig was proven by the first author and Yamane [HY08a].

The main results in this paper rely on the crucial coproduct formula in Thm. 4.2. For quantum groups this formula amounts to an explicit computation of the braided coproduct of $U^{+}$ in the image of $T_{\alpha}(U^{+})$ as a subalgebra of $U$. Our formula has the advantage to involve only algebra maps, and hence it is well-suited to study coideal subalgebras.

To provide more details, let $\theta\in\mathbb{N}$, let $M_{1},\dots,M_{\theta}$ be finite-dimensional irreducible objects in ${}^{H}_{H}\mathcal{YD}$, and $M=(M_{1},\dots,M_{\theta})$. The goal is to understand the Nichols algebra

[TABLE]

as a Hopf algebra in the braided category ${}^{H}_{H}\mathcal{YD}$. Let $\mathbb{Z}^{\theta}$ be the free abelian group of rank $\theta$ with standard basis $\alpha_{1},\dots,\alpha_{\theta}$. The Nichols algebra $\mathcal{B}(M)$ is $\mathbb{Z}^{\theta}$-graded where $\deg(M_{i})=\alpha_{i}$ for all $1\leq i\leq\theta$.

First we define reflection operators $R_{i}$, $1\leq i\leq\theta$. Assume that for all $j\neq i$,

[TABLE]

Define $a_{ii}^{M}=2$. Then $(a^{M}_{ij})_{i,j\in\{1,\dots,\theta\}}$ is a generalized Cartan matrix. Let $s_{i}^{M}\in\mathrm{Aut}(\mathbb{Z}^{\theta})$ be the corresponding reflection. Define $R_{i}(M)_{i}=M_{i}^{*}$, and

[TABLE]

and let $R_{i}(M)=(R_{i}(M)_{1},\dots,R_{i}(M)_{\theta})$. Finally let $K_{i}^{M}=\mathcal{B}(M)^{\mathrm{co}\,\mathcal{B}(M_{i})},$ where the coinvariant elements are defined with respect to the projection of $\mathcal{B}(M)$ onto $\mathcal{B}(M_{i})$. By [AHS08, Thm. 3.12] there is an algebra isomorphism

[TABLE]

which is the identity on all $R_{j}(M)_{j}\subset K_{i}^{M}\#\mathcal{B}(M_{i}^{*})$. By the coproduct formula in Thm. 4.2, $\Omega_{i}^{M}$ becomes an isomorphism of $\mathbb{Z}^{\theta}$-graded braided Hopf algebras.

In Prop. 7.1 we show that in the quantum group case the inverse of $\Omega_{i}^{M}$ can be identified with Lusztig’s automorphism $T_{\alpha_{i}}$ restricted to $U^{+}$.

Assume that all iterations of the construction $M\mapsto R_{j}(M)$ are well-defined. In [AHS08], [HS08, Thm. 6.10] the Weyl groupoid $\mathcal{W}(M)$ of $M$ is defined. The objects of $\mathcal{W}(M)$ are sequences of isomorphism classes $[N]=([N_{1}],\dots,[N_{\theta}])$ where the sequence of Yetter-Drinfeld modules $(N_{1},\dots,N_{\theta})$ is obtained from $M$ by iterating the operations $R_{j}$. The morphisms are generated by elementary reflections $s_{i}^{N}:R_{i}(N)\to N$. Then our main result on right coideal subalgebras in the general case, see Thm. 6.15 and Cor. 6.17, says the following.

Assume that the Weyl groupoid of $M$ is finite. Then there exists an order preserving bijection $\varkappa^{M}$ between the set of morphisms of $\mathcal{W}(M)$ with target $[M]$ and the set of $\mathbb{N}_{0}^{\theta}$-graded right coideal subalgebras of $\mathcal{B}(M)\#H$ containing $H$, where right coideal subalgebras are ordered with respect to inclusion and the morphisms are ordered by the Duflo order.

The map $\varkappa^{M}$ also exists for non-finite $\mathcal{W}(M)$, if we assume that $\mathcal{B}(M)$ is decomposable, see Def. 6.8. Then $\varkappa^{M}$ is always injective, order preserving and order reflecting by Thm. 6.12.

Acknowledgement. The first author would like to thank S. Kolb for interesting discussions on coideal subalgebras of $U_{q}(\mathfrak{g})$.

1. Weyl groupoids and the Duflo order

Recall the definition of the Weyl groupoid of a root system from [CH09, Sect. 2], see also [HS08, Sect. 5].

Let $I$ be a non-empty finite set and $(\alpha_{i})_{i\in I}$ the standard basis of $\mathbb{Z}^{I}$. Let $\mathcal{X}$ be a non-empty set, and for all $i\in I$ and $X\in\mathcal{X}$ let $r_{i}:\mathcal{X}\to\mathcal{X}$ be a map and $A^{X}=(a^{X}_{jk})_{j,k\in I}$ a generalized Cartan matrix. The quadruple

[TABLE]

is called a Cartan scheme if

1. (C1)

$r_{i}^{2}=\mathrm{id}$ for all $i\in I$, 2. (C2)

$a^{X}_{ij}=a^{r_{i}(X)}_{ij}$ for all $X\in\mathcal{X}$ and $i,j\in I$.

Let $\mathcal{C}=\mathcal{C}(I,\mathcal{X},(r_{i})_{i\in I},(A^{X})_{X\in\mathcal{X}})$ be a Cartan scheme. For all $i\in I$, $X\in\mathcal{X}$ define $s_{i}^{X}\in\mathrm{Aut}(\mathbb{Z}^{I})$ by

[TABLE]

Recall that a groupoid is a category where all morphisms are isomorphisms. The Weyl groupoid of $\mathcal{C}$ is the groupoid $\mathcal{W}(\mathcal{C})$ with $\mathrm{Ob}(\mathcal{W}(\mathcal{C}))=\mathcal{X}$, where the morphisms are generated by all $s_{i}^{X}$ (considered as morphism in $\mathrm{Hom}(X,r_{i}(X))$ with $i\in I$, $X\in\mathcal{X}$. Then $s_{i}^{r_{i}(X)}s_{i}^{X}=\mathrm{id}_{X}$ in $\mathrm{Hom}(X,X)$. We will write $s_{i}$ instead of $s_{i}^{X}$ if $X$ is uniquely determined by the context.

For any groupoid $\mathcal{G}$ and any $X\in\mathrm{Ob}(\mathcal{G})$ let

[TABLE]

Let $\mathcal{C}$ be a Cartan scheme and let $X\in\mathcal{X}$. Following [Kac90, §5.1] we say that

[TABLE]

is the set of real roots (of $X$), where $w\in\mathrm{Hom}(\mathcal{W}(\mathcal{C}),X)$ is interpreted as an element in $\mathrm{Aut}(\mathbb{Z}^{I})$. A real root $\alpha\in\boldsymbol{\Delta}{}^{X\,\mathrm{re}}$ is called positive, if $\alpha\in\mathbb{N}_{0}^{I}$. The set of positive real roots is denoted by $\boldsymbol{\Delta}{}^{X\,\mathrm{re}}_{+}$.

Remark 1.1*.*

Weyl groupoids associated to Nichols algebras satisfy additional properties which do not follow from the axioms of Cartan schemes, see Thm. 6.15. For example, cf. [CH09, Pf. of Thm. 6.1], let $\mathcal{X}=\{X_{1},X_{2},X_{3}\}$, $I=\{1,2\}$, $r_{1}(X_{i})=X_{\sigma(i)}$, $r_{2}(X_{i})=X_{\tau(i)}$, where $\sigma=(1\,2)$, $\tau=(2\,3)$. Let

[TABLE]

Then $\mathcal{C}(I,\mathcal{X},(r_{i})_{i\in I},(A^{X})_{X\in\mathcal{X}})$ is a Cartan scheme with finitely many real roots

[TABLE]

where $k\alpha_{1}+l\alpha_{2}$ is abbreviated by $1^{k}2^{l}$ for all $k,l\in\mathbb{Z}$. Observe that $\boldsymbol{\Delta}{}^{X_{3}\,\mathrm{re}}$ contains the real root $\alpha_{1}-\alpha_{2}$, and hence $\mathcal{C}(I,\mathcal{X},(r_{i})_{i\in I},(A^{X})_{X\in\mathcal{X}})$ does not admit a root system in the sense of the following definition.

We say that

[TABLE]

is a root system of type $\mathcal{C}$ if $\mathcal{C}=\mathcal{C}(I,\mathcal{X},(r_{i})_{i\in I},(A^{X})_{X\in\mathcal{X}})$ is a Cartan scheme and $\boldsymbol{\Delta}{}^{X}\subset\mathbb{Z}^{I}$, where $X\in\mathcal{X}$, are subsets such that

1. (R1)

$\boldsymbol{\Delta}{}^{X}=(\boldsymbol{\Delta}{}^{X}\cap\mathbb{N}_{0}^{I})\cup-(\boldsymbol{\Delta}{}^{X}\cap\mathbb{N}_{0}^{I})$ for all $X\in\mathcal{X}$, 2. (R2)

$\boldsymbol{\Delta}{}^{X}\cap\mathbb{Z}\alpha_{i}=\{\alpha_{i},-\alpha_{i}\}$ for all $i\in I$, $X\in\mathcal{X}$, 3. (R3)

$s_{i}^{X}(\boldsymbol{\Delta}{}^{X})=\boldsymbol{\Delta}{}^{r_{i}(X)}$ for all $i\in I$, $X\in\mathcal{X}$, 4. (R4)

$(r_{i}r_{j})^{m_{i,j}^{X}}(X)=X$ for all $i,j\in I$ and $X\in\mathcal{X}$ such that $i\not=j$ and $m_{i,j}^{X}:=\#(\boldsymbol{\Delta}{}^{X}\cap(\mathbb{N}_{0}\alpha_{i}+\mathbb{N}_{0}\alpha_{j}))$ is finite.

If $\mathcal{R}(\mathcal{C},(\boldsymbol{\Delta}{}^{X})_{X\in\mathcal{X}})$ is a root system of type $\mathcal{C}$, then $\mathcal{W}(\mathcal{R}):=\mathcal{W}(\mathcal{C})$ is called the Weyl groupoid of $\mathcal{R}$. The elements of $\boldsymbol{\Delta}{}^{X}_{+}:=\boldsymbol{\Delta}{}^{X}\cap\mathbb{N}_{0}^{I}$ and $\boldsymbol{\Delta}{}^{X}_{-}:=-\boldsymbol{\Delta}{}^{X}_{+}$ are called positive and negative roots, respectively. Note that (R3) implies that $w(\boldsymbol{\Delta}{}^{Y})=\boldsymbol{\Delta}{}^{X}$ for all $X,Y\in\mathcal{X}$ and $w\in\mathrm{Hom}(Y,X)$.

Recall that a groupoid $\mathcal{G}$ is connected, if for all $X,Y\in\mathrm{Ob}(\mathcal{G})$ the set $\mathrm{Hom}(Y,X)$ is non-empty. It is finite, if $\mathrm{Hom}(\mathcal{G})$ is finite.

The following claim was proven in [CH09, Lemma 2.11].

Lemma 1.2**.**

Let $\mathcal{C}$ be a Cartan scheme and $\mathcal{R}$ a root system of type $\mathcal{C}$. Assume that $\mathcal{W}(\mathcal{R})$ is connected. Then the following are equivalent.

1. (1)

$\boldsymbol{\Delta}{}^{X}$* is finite for at least one $X\in\mathrm{Ob}(\mathcal{W}(\mathcal{R}))$. * 2. (2)

$\boldsymbol{\Delta}{}^{X\,\mathrm{re}}$* is finite for at least one $X\in\mathrm{Ob}(\mathcal{W}(\mathcal{R}))$.* 3. (3)

$\mathrm{Hom}(\mathcal{W}(\mathcal{R}),X)$* is finite for at least one $X\in\mathrm{Ob}(\mathcal{W}(\mathcal{R}))$.* 4. (4)

The groupoid $\mathcal{W}(\mathcal{R})$ is finite.

Further, (1)–(3) hold for one $X\in\mathrm{Ob}(\mathcal{W}(\mathcal{R}))$ if and only if they hold for all $X\in\mathrm{Ob}(\mathcal{W}(\mathcal{R}))$.

Remark 1.3*.*

The equivalence of (1), (2), and (4) was stated and proven in [CH09, Lemma 2.11]. Clearly, (4) implies (3). For the proof of the implication (4)$\Rightarrow$(1) in [CH09, Lemma 2.11] one needs only to assume (3), and hence all claims of Lemma 1.2 are equivalent.

Let $\mathcal{C}$ be a Cartan scheme and $\mathcal{R}$ a root system of type $\mathcal{C}$. Then $\mathcal{R}$ is called finite, see [CH09, Def. 2.20], if $\boldsymbol{\Delta}{}^{X}$ is finite for all $X\in\mathrm{Ob}(\mathcal{W}(\mathcal{R}))$. If $\mathcal{W}(\mathcal{R})$ is connected, then this is equivalent to the conditions in Lemma 1.2.

Let $\ell$ denote the length function on Weyl groupoids of root systems: for each $X\in\mathcal{X}$ and each $w\in\mathrm{Hom}(\mathcal{W}(\mathcal{R}),X)$ let

[TABLE]

Proposition 1.4**.**

[CH09, Prop. 2.12]** Let $\mathcal{C}$ be a Cartan scheme and $\mathcal{R}$ a root system of type $\mathcal{C}$. Let $X\in\mathcal{X}$, $m\in\mathbb{N}_{0}$, and $i_{1},\dots,i_{m}\in I$ such that $\ell(w)=m$ for $w=s_{i_{1}}\cdots s_{i_{m}}\in\mathrm{Hom}(\mathcal{W}(\mathcal{R}),X)$. Then the roots

[TABLE]

are positive and pairwise distinct. If $\mathcal{R}$ is finite and $w\in\mathrm{Hom}(\mathcal{W}(\mathcal{R}),X)$ is the unique longest element, then $\{\beta_{k}\,|\,1\leq k\leq\ell(w)\}=\boldsymbol{\Delta}{}^{X}_{+}$.

Prop. 1.4 implies in particular that the set of roots of a finite root system is uniquely determined by its Cartan scheme and coincides with the set of real roots.

Definition 1.5**.**

Let $\mathcal{X}$ and $I$ be non-empty sets, $(r_{i})_{i\in I}$ a family of maps $r_{i}:\mathcal{X}\to\mathcal{X}$, and $(m_{i,j}^{X})_{i,j\in I,X\in\mathcal{X}}$ a family of numbers in $\mathbb{N}\cup\{\infty\}$ such that $m_{i,i}^{X}=1$ and $(r_{i}r_{j})^{m_{i,j}^{X}}(X)=X$ for all $X\in\mathcal{X}$ and $i,j\in I$ with $m_{i,j}^{X}<\infty$. 111This slight extension of notation is compatible with (R4) and (C1). Let $\mathcal{G}$ be a groupoid with $\mathrm{Ob}(\mathcal{G})=\mathcal{X}$, and let $(s_{i}^{X})_{i\in I,X\in\mathcal{X}}$ be a family of morphisms $s_{i}^{X}\in\mathrm{Hom}(X,r_{i}(X))$. We say that $(\mathcal{G},(s_{i}^{X})_{i\in I,X\in\mathcal{X}})$ satisfies the Coxeter relations if

[TABLE]

for all $X\in\mathcal{X}$ and $i,j\in I$ with $m_{i,j}^{X}<\infty$. In particular, Eq. (1.2) means for $i=j$ that $s_{i}^{r_{i}(X)}s_{i}^{X}=\mathrm{id}_{X}$ for all $X\in\mathcal{X}$ and $i\in I$.

Let $\mathcal{W}$ be a groupoid and $(s_{i}^{X})_{i\in I,X\in\mathcal{X}}$ a family of morphisms as above. We say that $(\mathcal{W},(s_{i}^{X})_{i\in I,X\in\mathcal{X}})$ is a Coxeter groupoid, if

1. (1)

$(\mathcal{W},(s_{i}^{X})_{i\in I,X\in\mathcal{X}})$ satisfies the Coxeter relations, and 2. (2)

for each pair $(\mathcal{G},(t_{i}^{X})_{i\in I,X\in\mathcal{X}})$ satisfying the Coxeter relations (with the same $\mathcal{X}$, $I$, $(r_{i})$ and $(m_{i,j}^{X})$ as for $\mathcal{W}$) there is a unique functor $F:\mathcal{W}\to\mathcal{G}$ such that $F$ is the identity on $\mathcal{X}=\mathrm{Ob}(\mathcal{W})=\mathrm{Ob}(\mathcal{G})$ and $F(s_{i}^{X})=t_{i}^{X}$ for all $i\in I$, $X\in\mathcal{X}$.

The universal property of a Coxeter groupoid $(\mathcal{W},(s_{i}^{X})_{i\in I,X\in\mathcal{X}})$ implies that $\mathrm{Hom}(\mathcal{W})$ is generated by the morphisms $s_{i}^{X}\in\mathrm{Hom}(X,r_{i}(X))$, where $i\in I$ and $X\in\mathcal{X}$.

For the rest of this section let $\mathcal{C}=\mathcal{C}(I,\mathcal{X},(r_{i})_{i\in I},(A^{X})_{X\in\mathcal{X}})$ be a Cartan scheme and let $\mathcal{R}=\mathcal{R}(\mathcal{C},(\boldsymbol{\Delta}{}^{X})_{X\in\mathcal{X}})$ be a root system of type $\mathcal{C}$.

Theorem 1.6**.**

[HY08b, Thm. 1]** For all $i,j\in I$ and $X\in\mathcal{X}$ let

[TABLE]

Then $(\mathcal{W}(\mathcal{R}),(s_{i}^{X})_{i\in I,X\in\mathcal{X}})$ is a Coxeter groupoid with respect to $(m_{i,j}^{X})$.

Definition 1.7**.**

For all $X\in\mathcal{X}$, $m\in\mathbb{N}$ and $(i_{1},\dots,i_{m})\in I^{m}$ let $\Lambda_{+}^{X}()=\emptyset$ and

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Lemma 1.8**.**

*Let $m\in\mathbb{N}$, $i_{1},\dots,i_{m}\in I$, $X\in\mathcal{X}$, and $Y=r_{i_{1}}(X)$. Then *

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Proof.

The claim follows from Def. 1.7 and basic properties of the map $s_{i_{1}}^{Y}$. ∎

The sets $\Lambda_{+}^{X}(i_{1},\dots,i_{m})$ ultimately describe the elements of $\mathrm{Hom}(\mathcal{W}(\mathcal{R}),X)$.

Proposition 1.9**.**

Let $X\in\mathcal{X}$, $m,m^{\prime}\in\mathbb{N}_{0}$ and $i_{1},\dots,i_{m},i^{\prime}_{1},\dots,i^{\prime}_{m^{\prime}}\in I$. The following are equivalent.

1. (1)

$\Lambda_{+}^{X}(i_{1},\dots,i_{m})=\Lambda_{+}^{X}(i^{\prime}_{1},\dots,{i^{\prime}}_{m^{\prime}})$, 2. (2)

$s_{i_{1}}\cdots s_{i_{m}}=s_{i^{\prime}_{1}}\cdots s_{i^{\prime}_{m^{\prime}}}$* in $\mathrm{Hom}(\mathcal{W}(\mathcal{R}),X)$.*

Moreover, $\#\Lambda_{+}^{X}(i_{1},\dots,i_{m})=\ell(\mathrm{id}_{X}s_{i_{1}}\cdots s_{i_{m}})$.

Proof.

Let $\mathcal{V}$ denote the category with $\mathrm{Ob}(\mathcal{V})=\mathcal{X}$ and morphisms

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for all $Y,Z\in\mathcal{X}$. Composition of morphisms is defined via concatenation:

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for all $Y,Z\in\mathcal{X}$, $n,p\in\mathbb{N}_{0}$ and $j_{1},\dots,j_{n},k_{1},\dots,k_{p}\in I$ with $Z=r_{j_{1}}\cdots r_{j_{n}}(Y)$. First we prove that $\mathcal{V}$ is indeed a category.

Let $n,n^{\prime},p,p^{\prime}\in\mathbb{N}_{0}$, $j_{1},\dots,j_{n}$, $j^{\prime}_{1},\dots,j^{\prime}_{n^{\prime}}$, $k_{1},\dots,k_{p}$, $k^{\prime}_{1},\dots,k^{\prime}_{p^{\prime}}\in I$ and $Z\in\mathcal{X}$ such that

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and $\mathrm{id}_{Z}s_{j_{1}}\cdots s_{j_{n}}=\mathrm{id}_{Z}s_{j^{\prime}_{1}}\cdots s_{j^{\prime}_{n^{\prime}}}$, where $Y=r_{j_{n}}\cdots r_{j_{1}}(Z)$. The definition of $\Lambda^{Z}_{+}$ implies that

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and hence the composition is independent of the choice of representatives. Clearly, the composition is associative, and $(\Lambda^{Z}_{+}(),\mathrm{id}_{Z})$ is the identity for any $Z\in\mathcal{X}$, and hence $\mathcal{V}$ is a category. Moreover, $\mathrm{Hom}(\mathcal{V})$ is generated by the morphisms $(\Lambda^{Z}_{+}(i),s_{i}^{r_{i}(Z)})$ with $i\in I$ and $Z\in\mathcal{X}$, which are invertible since $\Lambda^{Z}_{+}(i,i)=\emptyset$. Thus $\mathcal{V}$ is a groupoid.

Let $Z\in\mathcal{X}$ and $i,j\in I$. Assume that $m_{i,j}^{Z}<\infty$. Then

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since the entries $\beta_{k}$ of $\Lambda^{Z}(i,j,i,j,\dots,i,j)$ are the elements of $(\mathbb{Z}\alpha_{i}+\mathbb{Z}\alpha_{j})\cap\boldsymbol{\Delta}{}^{Z}$, each appearing with multiplicity one, see [HY08b, Lemma 6]. In the special case, where $i=j$, we have $\Lambda^{Z}(i,i)=(\alpha_{i},-\alpha_{i})$. Hence the pair

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satisfies the Coxeter relations in the sense of Def. 1.5. By Thm. 1.6, there is a functor from $\mathcal{W}(\mathcal{R})$ to $\mathcal{V}$ which maps $s_{i}^{r_{i}(Z)}$ to $(\Lambda^{Z}_{+}(i),s_{i}^{r_{i}(Z)})$ for each $Z\in\mathcal{X}$ and $i\in I$. This proves the implication (2)$\Rightarrow$(1) in the claim of the lemma.

We prove (1)$\Rightarrow$(2). Assume that $\Lambda_{+}^{X}(i_{1},\dots,i_{m})=\Lambda_{+}^{X}(i^{\prime}_{1},\dots,{i^{\prime}}_{m^{\prime}})$. Using that

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for all $Z\in\mathcal{X}$ and $n\in\mathbb{N}_{0}$, $i,j_{1},\dots,j_{n}\in I$, we may assume that $m^{\prime}=0$. By the first part of the lemma we may restrict to the case when $\mathrm{id}_{X}s_{i_{1}}\cdots s_{i_{m}}$ is a reduced expression, that is, $\ell(\mathrm{id}_{X}s_{i_{1}}\cdots s_{i_{m}})=m$. Then we have to show that $m=0$. The roots

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where $1\leq k\leq m$, are pairwise distinct and positive by Prop. 1.4. Hence the assumption $\Lambda_{+}^{X}(i_{1},\dots,i_{m})=\emptyset$ implies the desired claim $m=0$. The last assertion of the lemma follows from the first one and from Prop. 1.4. ∎

Let $X,Y\in\mathcal{X}$, $w\in\mathrm{Hom}(Y,X)$, $m\in\mathbb{N}_{0}$, and $i_{1},\dots,i_{m}\in I$ such that $w=s_{i_{1}}\cdots s_{i_{m}}$. Let $\Lambda^{X}_{+}(w)=\Lambda^{X}_{+}(i_{1},\dots,i_{m})$. This notation is justified by Prop. 1.9.

Corollary 1.10**.**

Let $X,Y\in\mathcal{X}$, $w\in \mathrm{H}\mathrm{o}\mathrm{m}(Y,X)$