Matrix product solution to the reflection equation associated with a coideal subalgebra of $U_q(A^{(1)}_{n-1})$

TL;DR

This paper introduces a novel matrix product solution to the reflection equation linked to a coideal subalgebra of quantum affine algebra, utilizing $q$-hypergeometric series and connecting to crystal base theory at $q=0$.

Contribution

It provides a new matrix product formula for the reflection equation solution associated with $U_q(A^{(1)}_{n-1})$ coideal subalgebras, expanding the understanding of integrable systems.

Findings

Matrix product formula involving $q$-hypergeometric series

Solution reduces to a known crystal base solution at $q=0$

Applicable to symmetric tensor representations and their duals

Abstract

We present a new solution to the reflection equation associated with a coideal subalgebra of $U_q(A^{(1)}_{n-1})$ in the symmetric tensor representations and their dual. Elements of the $K$ matrix are expressed by a matrix product formula involving terminating $q$-hypergeometric series in $q$-boson generators. At $q=0$, our result reproduces a known set-theoretical solution to the reflection equation connected to the crystal base theory.

Equations

$$K_1(x)R^{\ast }((xy{)}^{-1})K_1(y)R(x{y}^{-1})=R^{\ast \ast }(x{y}^{-1})K_1(y)R^{\ast }((xy{)}^{-1})K_1(x),$$

$$b_i=-e_i+q^2{k}_i{f}_i+\frac{q}{1-q}{k}_i\in U_q(A_{n-1}^{(1)})(i\in {\mathbb{Z}}_n).$$

$$\displaystyle K(z)^{\beta}_{\alpha}=\varrho(z)\mathrm{Tr}\bigl{(}z^{-{\bf h}}\hat{G}^{\beta_{1}}_{\alpha_{1}}\cdots\hat{G}^{\beta_{n}}_{\alpha_{n}}\bigr{)}$$

$$\displaystyle G^{j}_{i}=(-q;q)_{s}\,({\rm{\bf a}}^{\!+})^{(j-i)_{+}}{}_{2}\phi_{1}\Bigl{(}{q^{-t},-q^{-t}\atop-q^{-s}};q,q{\rm{\bf k}}\Bigr{)}({\rm{\bf a}}^{\!-})^{(i-j)_{+}},\quad s=i+j,\;\;t=\min(i,j),$$

$$[u]=\frac{q^u-q^{-u}}{q-q^{-1}},(z;q{)}_m=\prod _{k=1}^m(1-z{q}^{k-1}),{\left(\genfrac{}{}{0ex}{}{l}{m}\right)}_q=\frac{(q;q{)}_l}{(q;q{)}_{l-m}(q;q{)}_m},$$

$$\theta (\text{true})=1,\theta (\text{false})=0,{\mathbf{e}}_j=(0,\dots ,0,\stackrel{j}{1},0,\dots ,0)\in {\mathbb{Z}}^n(1\le j\le n).$$

$$\begin{array}{rl}& k_i{k}_i^{-1}=k_i^{-1}{k}_i=1,[k_i,k_j]=0,k_i{e}_j{k}_i^{-1}=q^{a_{ij}}{e}_j,k_i{f}_j{k}_i^{-1}=q^{-a_{ij}}{f}_j,[e_i,f_j]={\delta }_{ij}\frac{k_i-k_i^{-1}}{q-q^{-1}},\\ & \sum _{\nu =0}^{1-a_{ij}}(-1{)}^{\nu }{e}_i^{(1-a_{ij}-\nu )}{e}_j{e}_i^{(\nu )}=0,\sum _{\nu =0}^{1-a_{ij}}(-1{)}^{\nu }{f}_i^{(1-a_{ij}-\nu )}{f}_j{f}_i^{(\nu )}=0(i\ne j),\end{array}$$

$$\Delta k_i^{\pm 1}$$

$$S(k_i)$$

$$|\alpha |$$

$$\sigma (\alpha )$$

$${\pi }_{l,z}:$$

$${\pi }_{l,z}^{\ast }:$$

$$B_l=\{\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{Z}}_{+}^n\mid |\alpha |=l\}.$$

$$e_j{v}_{\alpha }$$

$$f_j{v}_{\alpha }$$

$$k_j{v}_{\alpha }$$

$$R(x/y):$$

$$R^{\ast }(x/y):$$

$$R^{\ast \ast }(x/y):$$

$$R(z)(v_{\alpha }\otimes v_{\beta })$$

$$R^{\ast }(z)(v_{\alpha }^{\ast }\otimes v_{\beta })$$

$$R^{\ast \ast }(z)(v_{\alpha }^{\ast }\otimes v_{\beta }^{\ast })$$

$$R(z{)}_{l{\mathbf{e}}_1,m{\mathbf{e}}_1}^{l{\mathbf{e}}_1,m{\mathbf{e}}_1}=R^{\ast }(z{)}_{l{\mathbf{e}}_1,m{\mathbf{e}}_1}^{l{\mathbf{e}}_1,m{\mathbf{e}}_1}=R^{\ast \ast }(z{)}_{l{\mathbf{e}}_1,m{\mathbf{e}}_1}^{l{\mathbf{e}}_1,m{\mathbf{e}}_1}=1.$$

$${\Phi }_q(\gamma |\beta ;\lambda ,\mu )$$

$${\overline{\Phi }}_q(\gamma |\beta ;\lambda ,\mu )$$

$$A(z{)}_{\alpha ,\beta }^{\gamma ,\delta }$$

$$A(z{)}_{\alpha ,\beta }^{\gamma ,\delta }=A(z{)}_{\rho (\gamma ),\rho (\delta )}^{\rho (\alpha ),\rho (\beta )}\prod _{i=1}^n\frac{(q^2;q^2{)}_{{\alpha }_i}(q^2;q^2{)}_{{\beta }_i}}{(q^2;q^2{)}_{{\gamma }_i}(q^2;q^2{)}_{{\delta }_i}}=z^{{\beta }_1-{\delta }_1}A(z{)}_{\sigma (\gamma ),\sigma (\delta )}^{\sigma (\alpha ),\sigma (\beta )}.$$

$$R(z{)}_{\alpha ,\beta }^{\gamma ,\delta }$$

$$R^{\ast }(z{)}_{\alpha ,\beta }^{\gamma ,\delta }$$

$$R^{\ast \ast }(z{)}_{\alpha ,\beta }^{\gamma ,\delta }$$

$$(1\otimes R(x))(R(xy)\otimes 1)(1\otimes R(y))$$

$$(1\otimes R^{\ast }(x))(R^{\ast }(xy)\otimes 1)(1\otimes R(y))$$

$$(1\otimes R^{\ast \ast }(x))(R^{\ast }(xy)\otimes 1)(1\otimes R^{\ast }(y))$$

$$(1\otimes R^{\ast \ast }(x))(R^{\ast \ast }(xy)\otimes 1)(1\otimes R^{\ast \ast }(y))$$

$$b_i=-e_i+q^2{k}_i{f}_i+\frac{q}{1-q}{k}_i\in U_q(i\in {\mathbb{Z}}_n)$$

$$K(z):V_{l,z}\to V_{l,z^{-1}}^{\ast },K(z)v_{\alpha }=\sum _{\gamma \in B_l}K(z{)}_{\alpha }^{\gamma }{v}_{\gamma }^{\ast },$$

$$K(z){\pi }_{l,z}(b)={\pi }_{l,z^{-1}}^{\ast }(b)K(z)(b\in {\mathcal{B}}_q).$$

Full text

Matrix product solution

to the reflection equation

associated with a coideal subalgebra of $\boldsymbol{U_{q}(A^{(1)}_{n-1})}$

Atsuo Kuniba

Atsuo Kuniba, Institute of Physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan

atsuo.s.kuniba@gmail.com

, 

Masato Okado

Masato Okado, Department of Mathematics, Osaka City University, Osaka, 558-8585, Japan

okado@sci.osaka-cu.ac.jp

 and 

Akihito Yoneyama

Akihito Yoneyama, Institute of Physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan

yoneyama@gokutan.c.u-tokyo.ac.jp

Abstract

We present a new solution to the reflection equation associated with a coideal subalgebra of $U_{q}(A^{(1)}_{n-1})$ in the symmetric tensor representations and their dual. Elements of the $K$ matrix are expressed by a matrix product formula involving terminating $q$-hypergeometric series in $q$-boson generators. At $q=0$, our result reproduces a known set-theoretical solution to the reflection equation connected to the crystal base theory.

1. Introduction

Reflection equation [5, 20, 11] is a characteristic structure in quantum integrable systems in the presence of boundaries. It combines the $K$ matrix encoding the boundary interaction with the $R$ matrix, another fundamental object governing the integrability in the bulk [3]. A variety of solutions to the reflection equation have been constructed up to now. See for example [2, 16, 18, 19, 15] and references therein. In this Letter we present a new solution to the reflection equation having a number of outstanding features described below.

First, it is associated with the Drinfeld-Jimbo quantum affine algebra $U_{q}(A^{(1)}_{n-1})$ in the symmetric tensor representation $V_{l,z}$ and its dual $V^{\ast}_{l,z}$ with general degree $l\in{\mathbb{Z}}_{+}$. Here $z$ denotes the (multiplicative) spectral parameter and $q$ is assumed to be generic throughout. Both representations $V_{l,z},V^{\ast}_{l,z}$ have the bases $\{v_{\alpha}\}$, $\{v^{\ast}_{\alpha}\}$ labeled with an array $\alpha=(\alpha_{1},\ldots,\alpha_{n})\in{\mathbb{Z}}_{+}^{n}$ satisfying $\alpha_{1}+\cdots+\alpha_{n}=l$. They include the vector representation as the simplest case $V_{l=1,z}$. Our $K$ matrix $K(z)=K^{(l)}(z,q)$ is a linear operator reflecting the “particles” into their duals as $K(z):V_{l,z}\rightarrow V^{*}_{l,z^{-1}}$. As such, there are three kinds of $R$ matrices $R(z),R^{\ast}(z)$ and $R^{\ast\ast}(z)$ (12)–(14) coming naturally into the game. They are all well-understood conceptually, and admit explicit formulas owing to the recent works [13, 4, 12]. The reflection equation takes the form

[TABLE]

where $K_{1}(x)=K^{(l)}(x,q)\otimes 1$ and $K_{1}(y)=K^{(m)}(y,q)\otimes 1$. This is an equality of linear maps from $V_{l,x}\otimes V_{m,y}$ to $V^{\ast}_{l,x^{-1}}\otimes V^{\ast}_{m,y^{-1}}$, where the pair $(l,m)\in{\mathbb{Z}}_{+}^{2}$ is arbitrary. See (35) and (36) for a more concrete description.

Second, let us write the action of our $K$ matrix on the basis as $K(z)v_{\alpha}=\sum_{\beta}K(z)_{\alpha}^{\beta}v^{\ast}_{\beta}$. Then, it is dense in the sense that all the matrix elements $K(z)_{\alpha}^{\beta}$ are nontrivial rational function of $z$ and $q$. Put plainly, our $K(z)$ is trigonometric, dense, and of type $A$ with general rank $n$ and general “spin” $l$. These are distinct features from previous works for type $A$ which are mostly devoted to diagonal $K$’s or to the situation $\min(n-1,l)=1$111There are important exceptions [14, 15] related to this work although..

Third, our $K(z)$ is characterized, up to normalization, as the intertwiner of the coideal subalgebra $\mathcal{B}_{q}$ of $U_{q}(A^{(1)}_{n-1})$ generated by the elements

[TABLE]

Indeed, it is easy to check the right coideal nature $\Delta\mathcal{B}_{q}\subset\mathcal{B}_{q}\otimes U_{q}(A^{(1)}_{n-1})$ by applying the coproduct $\Delta$ in (2) to $b_{i}$. The idea to characterize the spectral parameter dependent $K$ matrices in terms of coideal subalgebras of quantum affine algebras was proposed long ago in the context of affine Toda field theory with boundaries. See for example [6], more recent [10, 19] and references therein. Our result may be viewed as a systematic implementation of it for the pair $\mathcal{B}_{q}\subset U_{q}(A^{(1)}_{n-1})$ and the representations $V_{l,z},V^{*}_{l,z}$. We note that the above $b_{i}$ has also appeared in the generalized $q$-Onsager algebra [1] up to convention.

Last but perhaps most intriguingly, our $K$ matrix has the elements that admit an explicit matrix product formula

[TABLE]

with a scalar $\varrho(z)$. The trace is taken over a $q$-boson Fock space on which ${\bf h}$ acts as the number operator. In terms of the creation ${\rm{\bf a}}^{\!+}$, the annihilation ${\rm{\bf a}}^{\!-}$ and the $q$-counting generator ${\rm{\bf k}}=q^{\bf h}$ of the $q$-boson, the matrix product operator is given as $\hat{G}^{j}_{i}=q^{-{\scriptstyle\frac{1}{2}}i^{2}}{\rm{\bf k}}^{-i}G_{i}^{j}$ with

[TABLE]

where ${}_{2}\phi_{1}$ denotes the $q$-hypergeometric function and $(m)_{+}=\max(m,0)$. A matrix product solution to the reflection equation of this kind was first obtained in [15]. It covered all the fundamental representations of $U_{q}(A^{(1)}_{n-1})$ whose simplest case goes back to [7]. According to [15], the matrix product structure is a signal of three dimensional (3D) integrability. It is an interesting open problem to elucidate such features for the solution in this Letter. In this regard we note that all the $R$ matrices appearing in the reflection equation (37) are known to admit a matrix product formula originating in the tetrahedron equation [12].

There are further notable properties in our $K$ matrix $K(z)$. At $z=q^{-l}$, elements of $K^{(l)}(z,q)$ exhibit a neat factorization (59). Combined with the similar property of the $R$ matrices [13, Th.2], it allows us to merge the spectral parameter to the spins $l,m\in{\mathbb{Z}}_{+}$ thereby upgrading the latter to generic parameters. Consequently we get a parametric generalization of the solution to the reflection equation. This achieves a boundary analogue of the result concerning the Yang-Baxter equation [13, sec.2.3]. Another feature of interest occurs at $q=0$, where our $K$ matrix and reflection equation (84) survive quite nontrivially. In fact they are frozen exactly to the set-theoretical (combinatorial) counterparts introduced in [14] to formulate the box-ball system with reflecting end.

The outline of the Letter is as follows. In the next section we recapitulate the relevant representations of $U_{q}(A^{(1)}_{n-1})$ and the three kinds of $R$ matrices. In Section 3 we introduce the coideal subalgebra $\mathcal{B}_{q}$ and characterize the $K$ matrix as the intertwiner. The reflection equation is formulated, which corresponds to a twisted one in the terminology of [19]. The proof of uniqueness of the intertwiner and the irreducibility of $V_{l,x}\otimes V_{m,y}$ as a $\mathcal{B}_{q}$ module will be given elsewhere. In Section 4 we present the matrix product solution to the intertwining relation. The proof becomes local in the direction of rank, and reduces to some quadratic relations of (nonterminating) $q$-hypergeometric series. In Section 5 a generalization of integer spins (degrees of symmetric tensors and their dual) to continuous parameters is described. In Section 6 we present the results in yet another gauge and elucidate the connection to the work [14] at $q=0$. Section 7 contains a brief summary and an outlook. The associated commuting double row transfer matrices (cf. [20]) are left for future study. We set ${\mathbb{Z}}_{+}={\mathbb{Z}}_{\geq 0}$ and use the following notations:

[TABLE]

2. $U_{q}(A^{(1)}_{n-1})$ and relevant $R$ matrices

2.1. $U_{q}(A^{(1)}_{n-1})$ and relevant representations

Let $U_{q}=U_{q}(A^{(1)}_{n-1})$ be the Drinfeld-Jimbo quantum affine algebra (without the derivation operator) generated by $e_{i},f_{i},k^{\pm 1}_{i}\,(i\in{\mathbb{Z}}_{n})$ obeying the relations

[TABLE]

where $\delta_{ij}=\theta(i=j),\,e^{(\nu)}_{i}=e^{\nu}_{i}/[\nu]!,\,f^{(\nu)}_{i}=f^{\nu}_{i}/[\nu]!$ and $[m]!=\prod_{j=1}^{m}[j]$. The Cartan matrix $(a_{ij})_{i,j\in{\mathbb{Z}}_{n}}$ is given by $a_{ij}=2\delta_{i,j}-\delta_{i,j+1}-\delta_{i,j-1}$222Note $a_{n-1,0}=a_{0,n-1}=-1$ because of $i,j\in{\mathbb{Z}}_{n}$.. We employ the coproduct $\Delta$ and the antipode $S$ of the form

[TABLE]

For integer arrays $\alpha=(\alpha_{1},\ldots,\alpha_{k}),\beta=(\beta_{1},\ldots,\beta_{k})\in{\mathbb{Z}}^{k}$ of any length $k$, we use the notation

[TABLE]

where $\sigma$ is a cyclic shift and $\rho$ is the reverse ordering. We will be concerned with the two irreducible representations of $U_{q}$ labeled with $l\in{\mathbb{Z}}_{+}$:

[TABLE]

where $B_{l}$ is a finite set of length $n$ arrays specified as

[TABLE]

The index $i$ of $\alpha=(\alpha_{i})\in B_{l}$ should always be understood as elements of ${\mathbb{Z}}_{n}$. Now the representations (6) and (7) are specified as

[TABLE]

where $\pi_{l,z}(g),\pi^{\ast}_{l,z}(g)$ with $g\in U_{q}$ are denoted by $g$ for simplicity. In the RHS, $v_{\beta},v^{\ast}_{\beta}$ with $\beta\not\in B_{l}$ should be understood as [math]. The representation $\pi_{l,z}$ is the (affinization of) degree $l$ symmetric tensor representation, and $\pi^{\ast}_{l,z}$ is its antipode dual. Namely, $(\pi^{\ast}_{l,z}(g)v^{\ast}_{\alpha},v_{\beta})=(v^{\ast}_{\alpha},\pi_{l,z}(S(g))v_{\beta})$ holds for any $\alpha,\beta\in B_{l}$ and $g\in U_{q}$ with respect to the bilinear pairing $(v^{\ast}_{\alpha},v_{\beta})=\delta_{\alpha,\beta}$. In terms of the classical part $U_{q}(A_{n-1})$, they are the irreducible representations labeled with the rectangular Young diagrams of shape $1\times l$ and $(n-1)\times l$, respectively.

2.2. $R$ matrices

For simplicity denote the tensor product representation $(\pi^{\ast}_{l,x}\otimes\pi_{m,y})\circ\Delta$ just by $\pi^{\ast}_{l,x}\otimes\pi_{m,y}$, etc. Consider the three types of quantum $R$ matrices which are characterized, up to normalization, by the commutativity with $U_{q}$ as

[TABLE]

Note that dependence on $l,m,q$ is suppressed in the $R$ matrices. We specify the matrix elements by

[TABLE]

and the normalization

[TABLE]

In order to provide explicit formulas for the $R$ matrices, we prepare their building blocks. For complex parameters $\lambda,\mu$ and arrays $\beta=(\beta_{1},\ldots,\beta_{k}),\gamma=(\gamma_{1},\ldots,\gamma_{k})\in{\mathbb{Z}}_{+}^{k}$ with any length $k$, define

[TABLE]

where $\theta(\gamma\leq\beta)$ stands for $\prod_{i=1}^{k}\theta(\gamma_{i}\leq\beta_{i})$. The function $\Phi_{q}(\gamma|\beta;\lambda,\mu)$ was introduced in [13, eq.(19)] in the study of a stochastic $R$ matrix for $U_{q}$. Following [4] we define a quadratic combination of (19) as

[TABLE]

where $\alpha,\gamma\in B_{l}$ and $\beta,\delta\in B_{m}$ and $\overline{\alpha}=(\alpha_{1},\ldots,\alpha_{n-1})$ stands for the truncation of $\alpha=(\alpha_{1},\ldots,\alpha_{n})$. The sum in (21) extends over $\overline{\xi},\overline{\eta}\in{\mathbb{Z}}_{+}^{n-1}$ satisfying $\overline{\xi}+\overline{\eta}=\overline{\gamma}+\overline{\delta}$. There are finitely many such $\overline{\xi}$ and $\overline{\eta}$. The function $A(z)^{\gamma,\delta}_{\alpha,\beta}$ satisfies

[TABLE]

Now the elements of $R$ matrices are expressed as follows ($\delta_{\alpha}^{\beta}=\theta(\alpha=\beta)$):

[TABLE]

See the comments after (79) for the origin of these formulas. The $R$ matrices satisfy the Yang-Baxter equations [3] reversing the components of the tensor products $V_{l_{1},z_{1}}\otimes V_{l_{2},z_{2}}\otimes V_{l_{3},z_{3}},V^{\ast}_{l_{1},z_{1}}\otimes V_{l_{2},z_{2}}\otimes V_{l_{3},z_{3}},V^{\ast}_{l_{1},z_{1}}\otimes V^{\ast}_{l_{2},z_{2}}\otimes V_{l_{3},z_{3}},V^{\ast}_{l_{1},z_{1}}\otimes V^{\ast}_{l_{2},z_{2}}\otimes V^{\ast}_{l_{3},z_{3}}$. In terms of $x=z_{1}/z_{2},y=z_{2}/z_{3}$, they read

[TABLE]

3. A coideal subalgebra and $K$ matrix

Consider the element

[TABLE]

and let $\mathcal{B}_{q}$ be the subalgebra of $U_{q}$ generated by $\{b_{i}\mid i\in{\mathbb{Z}}_{n}\}$. From $\Delta(b_{i})=b_{i}\otimes k_{i}+1\otimes(-e_{i}+q^{2}k_{i}f_{i})$, we see $\Delta\mathcal{B}_{q}\subset\mathcal{B}_{q}\otimes U_{q}$, meaning that $\mathcal{B}_{q}$ is a right coideal subalgebra of $U_{q}$. Consider the operator $K(z)=K^{(l)}(z,q)$

[TABLE]

which satisfies the intertwining relation

[TABLE]

It suffices to impose (32) for the generators $b=b_{i}\,(i\in{\mathbb{Z}}_{n})$. From (9)–(11), it reads explicitly as

[TABLE]

where $|\alpha|=|\gamma|=l$ and $K(z)^{\gamma}_{\alpha}=0$ unless $\alpha,\gamma\in B_{l}$.

The essentials for our construction are the following claim.

Theorem 1**.**

The solution $K(z)$ to the intertwining relation (32) or equivalently (33) $(\forall i\in{\mathbb{Z}}_{n})$ is unique up to normalization. Moreover, $V_{l,x}\otimes V_{m,y}$ is irreducible as a $\mathcal{B}_{q}$ module for generic $x$ and $y$.

We will prove this for a more general setting elsewhere based partly on the existence of the crystal base [8]. In what follows, $K(z)$ denotes the unique intertwiner normalized as

[TABLE]

Consider the intertwiner $V_{l,x}\otimes V_{m,y}\rightarrow V^{\ast}_{l,x^{-1}}\otimes V^{\ast}_{m,y^{-1}}$ of the $\mathcal{B}_{q}$ modules constructed in two ways as

[TABLE]

where $K_{1}(x)=K^{(l)}(x,q)\otimes 1$ and $K_{1}(y)=K^{(m)}(y,q)\otimes 1$. The dependence of each $R$ matrix on $l,m$ should be understood appropriately. The composition of (35) and the inverse of (36) gives a map on $V_{l,x}\otimes V_{m,y}$ commuting with $\Delta\mathcal{B}_{q}$. Then the second assertion in Theorem 1 tells that it must be a scalar multiple of the identity operator. The scalar is $1$ due to the normalization (18) and (34). In this way, we obtain the reflection equation

[TABLE]

of the linear operators $V_{l,x}\otimes V_{m,y}\rightarrow V^{\ast}_{l,x^{-1}}\otimes V^{\ast}_{m,y^{-1}}$ for the intertwiner $K(z)$ characterized by the first assertion in Theorem 1. In short, Theorem 1 achieves linearization; the reflection equation which is quadratic in $K(z)$ becomes a corollary of the linear intertwining relation (32). In terms of matrix elements (37) reads

[TABLE]

where $a_{0},a_{3}\in B_{l},b_{0},b_{3}\in B_{m}$ and the sums range over $a_{1},a_{2}\in B_{l},b_{1},b_{2}\in B_{m}$ on the both sides. On the LHS (resp. RHS), they are to obey the weight conservation $a_{1}+b_{1}=a_{0}+b_{0},a_{1}-b_{2}=a_{2}-b_{3}$ (resp. $a_{1}-b_{0}=a_{2}-b_{1},a_{2}+b_{2}=a_{3}+b_{3})$.

a_{3},x^{-1}$$a_{0},x$$b_{3},y^{-1}$$b_{0},y$$a_{2}$$a_{1}$$b_{2}$$b_{1}$$=$$a_{3},x^{-1}$$b_{0},y$$b_{3},y^{-1}$$a_{0},x$$b_{2}$$b_{1}$$a_{1}$$a_{2}

Remark 2**.**

For the coideal subalgebra generated by $-e_{i}+c_{i}k_{i}f_{i}+d_{i}k_{i}$ with $c_{i}d_{i}\neq 0\,(\forall i\in{\mathbb{Z}}_{n})$, a necessary condition for the existence of $K(z):\;V_{l,z}\rightarrow V^{\ast}_{l,w^{-1}}$ with $n\geq 3$ is

[TABLE]

Such cases can always be reduced to (30) by applying an algebra automorphism $\omega:e_{i}\mapsto\mu_{i}e_{i},f_{i}\mapsto\mu^{-1}_{i}f_{i},k^{\pm 1}_{i}\mapsto k_{i}^{\pm 1}$ of $U_{q}$ for appropriate constants $\mu_{i}$. For $n=2$, the intertwiner exists without assuming the right constraint in (39). However a matrix product formula for such a case is not known in general.

4. Matrix product construction

Let $\mathcal{A}_{q}$ be the algebra generated by ${\rm{\bf a}}^{\!+},{\rm{\bf a}}^{\!-},{\rm{\bf k}}$ obeying the relations

[TABLE]

The algebra $\mathcal{A}_{q}$ will be called $q$-boson. It is equipped with an anti-algebra automorphism

[TABLE]

Let $F_{q}=\bigoplus_{m\geq 0}{\mathbb{C}}|m\rangle$ and $F^{\ast}_{q}=\bigoplus_{m\geq 0}{\mathbb{C}}\langle m|$ be the Fock space and its dual equipped with the bilinear pairing $\langle m|m^{\prime}\rangle=(q;q)_{m}\delta_{m,m^{\prime}}$. They can be endowed with an $\mathcal{A}_{q}$ module structure by

[TABLE]

It satisfies $(\langle m|X)|m^{\prime}\rangle=\langle m|(X|m^{\prime}\rangle)$. We also use ${\bf h}$ acting on the Fock spaces as ${\bf h}|m\rangle=m|m\rangle$ and $\langle m|{\bf h}=\langle m|m$. Thus one may set ${\rm{\bf k}}=q^{\bf h}$. By definition, the trace on $F_{q}$ means $\mathrm{Tr}(w^{\bf h}X)=\sum_{m\geq 0}w^{m}\frac{\langle m|X|m\rangle}{(q;q)_{m}}$ when convergent. The traces appearing in the sequel are always reduced to and evaluated by $\mathrm{Tr}(w^{\bf h}{\rm{\bf k}}^{r})=\frac{1}{1-q^{r}w}$ for some $w$ and $r\in{\mathbb{Z}}$ by relation (40).

For each pair $(i,j)\in{\mathbb{Z}}_{+}^{2}$, define an element $G^{j}_{i}\in\mathcal{A}_{q}$ by

[TABLE]

where $\phi$ is a shorthand for the $q$-hypergeometric series

[TABLE]

The RHS of (42) is terminating and actually involves finitely many terms. Note the properties

[TABLE]

Theorem 3**.**

The $K$ matrix characterized by (32) and (34) has the elements expressed by the matrix product formula:

[TABLE]

Due to the right property in (44) and $l^{2}=|\alpha|^{2}=\sum_{i=1}^{n}\alpha_{i}^{2}+2\langle\alpha,\alpha\rangle$ for $\alpha\in B_{l}$, formula (45) is also written as:

[TABLE]

where the prefactor of the trace is independent of $\alpha$ and $\gamma$. Let us sketch a (rather brute force) proof. Substitute (45) into (33). Applying the right relation in (44) and $\langle\gamma\pm{\bf e}_{i}\mp{\bf e}_{i+1},\alpha\rangle-\langle\gamma,\alpha\rangle=\pm(\alpha_{i+1}-l\delta_{i0})$, $\langle\gamma,\alpha\pm{\bf e}_{i}\mp{\bf e}_{i+1}\rangle-\langle\gamma,\alpha\rangle=\pm(-\gamma_{i}+l\delta_{i0})$, we find that (33) follows from the $\delta_{i0}$-free relation:

[TABLE]

Substitute (42) into (47) and remove a common factor after applying the $q$-commutation relations in (40). Regarding integer powers of $q$ as generic variables, one is left to show quadratic relations of the $q$-hypergeometric series. Below we illustrate a typical case $\alpha_{1}>\gamma_{1}$ and $\alpha_{2}<\gamma_{2}$. (The invariance of (47) by $\iota$ in (41) reduces the task in the proof to some extent.) The relevant quadratic relation reads

[TABLE]

with $y=u_{1}^{2}v_{1}^{-1}u_{2}^{-2}v_{2}w$. Applying Heine’s contiguous relations to the factors \phi\Bigl{(}{\bullet,\bullet\atop\bullet};w\Bigr{)}, one can rewrite the RHS as A\phi\Bigl{(}{q^{-1}u_{1},-u_{1}\atop-q^{-1}v_{1}};w\Bigr{)}+B\phi\Bigl{(}{u_{1},-q^{-1}u_{1}\atop-q^{-1}v_{1}};w\Bigr{)} with $A,B$ being linear combinations in \phi\Bigl{(}{\bullet,\bullet\atop\bullet};y\Bigr{)}. Then it is straightforward, though tedious, to check $A=0,B=0$ by (43). We remark that all the relations like (48) hold for generic $u_{i},v_{i}$, hence for nonterminating $q$-hypergeometric series.

4.1. Basic properties and examples

From the matrix product formula (45) it is easy to derive

[TABLE]

The array $\alpha^{(i)}\in{\mathbb{Z}}_{+}^{n-1}$ is obtained from $\alpha\in{\mathbb{Z}}_{+}^{n}$ by dropping the $i$th component $\alpha_{i}$. The equality (50) is due to $G^{0}_{0}=1$ and $\langle\gamma^{(i)},\alpha^{(i)}\rangle=\langle\gamma,\alpha\rangle$ when $\alpha_{i}=\gamma_{i}=0$. It implies a reduction with respect to rank $n$ when some components are simultaneously [math]. In what follows we present the result of an explicit evaluation of (45) for a few typical cases.

Example 4**.**

Consider $U_{q}(A^{(1)}_{1})$ for general $l$. Due to (49), $K(z)^{\gamma_{1},\gamma_{2}}_{\alpha_{1},\alpha_{2}}=z^{\gamma_{2}-\alpha_{2}}K(z)^{\alpha_{1},\alpha_{2}}_{\gamma_{1},\gamma_{2}}$ holds. Thus, we present the result assuming $s:=\gamma_{1}-\alpha_{1}=\alpha_{2}-\gamma_{2}\geq 0$ without loss of generality.

[TABLE]

Example 5**.**

Consider $U_{q}(A^{(1)}_{n-1})$ with $l=1$. The relevant matrix product operators are

[TABLE]

Thus, formula (45) yields

[TABLE]

In fact this is the $l=1$ case of more general

[TABLE]

Example 6**.**

Consider $U_{q}(A^{(1)}_{n-1})$ with $l=2$. In view of (49) and (50), the matrix elements that are not covered by Examples 4 and 5 are reduced to the following cases of $n=3$:

[TABLE]

Let us close the section with the conjecture

[TABLE]

where $Q_{0}(\gamma,\alpha)$ is defined after (87). This indicates that the present gauge as well as the one treated in Section 6 also has a curious connection to the crystal theory [8, 9, 17, 14].

5. Parametric generalization

5.1. Factorization at special point

The function (19) has two simplifying points:

[TABLE]

Applying it to (21) and (23)–(25), we get

[TABLE]

where we assume $\alpha,\gamma\in B_{l}$ and $\beta,\delta\in B_{m}$ in all the cases. Up to an overall factor, (58) is due to [13, Th.2]. By the argument similar to the proof of it there, one can show that the $K$ matrix also has the factorization

[TABLE]

5.2. Upgrading

$\lambda=q^{-l},\mu=q^{-m}$ to generic parameters

In the reflection equation (37), specialize the spectral parameters to $x=q^{-l},y=q^{-m}$. Assuming $l\geq m$, one finds that all the $R$ and $K$ matrices have the factorized elements given in the previous subsection. (Note that (58) should be applied after the exchange $l\leftrightarrow m$.) Apart from the powers of $q$, (56)–(58) consist of the $q^{2}$-multinomial $(q^{2};q^{2})_{l}/\prod_{i=1}^{n}(q^{2};q^{2})_{\alpha_{i}}=(-q^{2l-|\overline{\alpha}|+1})^{|\overline{\alpha}|}(q^{-2l};q^{2})_{|\overline{\alpha}|}/\prod_{i=1}^{n-1}(q^{2};q^{2})_{\overline{\alpha}_{i}}$ for $\alpha\in B_{l}$. Here $\overline{\alpha}$ is a truncation of $\alpha$ explained after (21). Similar rewriting is possible also for (59). The powers of $q$ are handled by $\langle\alpha,\beta\rangle=\langle\overline{\alpha},\overline{\beta}\rangle+|\overline{\alpha}|(m-|\overline{\beta}|)$ for $\beta\in B_{m}$. Then from the argument similar to [13, Sec.2.3], it follows that the reflection equation, as well as the Yang-Baxter equation, holds as an identity of a rational function in which $\lambda=q^{-l}$ and $\mu=q^{-m}$ are regarded as generic parameters independent of $q$. Local spin variables in such a setting range over $\overline{\alpha}\in{\mathbb{Z}}^{n-1}_{+}$ rather than $\alpha\in B_{l}$. Below we describe the resulting $R$ and $K$ matrices resetting $\overline{\alpha}\in{\mathbb{Z}}^{n-1}_{+}$ to a simpler notation $\alpha\in{\mathbb{Z}}_{+}^{k}$.

For $k\geq 1$, introduce the infinite dimensional space

[TABLE]

Consider the linear operators depending on the continuous parameters $\lambda,\mu$ as

[TABLE]

where $\mathscr{Q}=\mathscr{R},\mathscr{R}^{\ast},\mathscr{R}^{\ast\ast}$. The matrix elements are defined by

[TABLE]

where $\overline{\Phi}_{q^{2}}$ is given by (20). Then, the Yang-Baxter equations and the reflection equation are valid:

[TABLE]

The Yang-Baxter (resp. reflection) equations hold as identities of the operators on $W^{\otimes 3}$ (resp. $W^{\otimes 2}$). The result (69) was obtained in [13, Sec.2.3] up to a gauge of $\mathscr{R}^{\ast\ast}(\lambda,\mu)$. Two remarks are in order.

(i) $\mathscr{K}(\lambda)$ and $\mathscr{R}^{\ast}(\lambda,\mu)$ are not locally finite in that the corresponding RHS of (62) contains infinitely many terms. However, the Yang-Baxter and the reflection equations make sense as the identities of matrix elements which are finite for any prescribed transitions $u_{\alpha}\otimes u_{\beta}\otimes u_{\gamma}\mapsto u_{\alpha^{\prime}}\otimes u_{\beta^{\prime}}\otimes u_{\gamma^{\prime}}$ and $u_{\alpha}\otimes u_{\beta}\mapsto u_{\alpha^{\prime}}\otimes u_{\beta^{\prime}}$.

(ii) The Yang-Baxter equations (66) – (69) remain valid under the replacement

[TABLE]

where $\varphi_{1}$ (resp. $\varphi_{2},\varphi_{3}$) is any bilinear (resp. linear) function. This can be utilized to simplify (64) and (65) to some extent. However there is no bilinear function $\varphi^{\prime}(\cdot,\cdot)$ such that the transformation $\mathscr{K}(\lambda)_{\alpha}^{\gamma}\mapsto q^{\varphi^{\prime}(\gamma,\alpha)}\mathscr{K}(\lambda)_{\alpha}^{\gamma}$ combined with (71)–(73) preserves the reflection equation.

6. Another gauge

The results in Sections 2 and 3 can also be stated in another gauge which suits the study of the limit $q\rightarrow 0$ in relation to the crystal theory [8].

6.1. Representation

$\pi^{\vee}_{l,z}$ and associated $R$ matrices

Consider the representation ([13, eq.(2)], [12, eq.(3.14)])

[TABLE]

where again $\pi^{\vee}_{l,z}(g)$ are abbreviated to $g$. It is easy to see the equivalence

[TABLE]

by means of $\{\alpha\pm({\bf e}_{i}-{\bf e}_{i+1})\}-\{\alpha\}=\pm(-1+n\delta_{i0})$. See (4) for the definition of the symbol $\{\alpha\}$. Denote the counterparts of the $R$ matrices in (13) and (14) by

[TABLE]

Under the normalization $R^{\vee}(z)^{l{\bf e}_{1},m{\bf e}_{1}}_{l{\bf e}_{1},m{\bf e}_{1}}=R^{\vee\vee}(z)^{l{\bf e}_{1},m{\bf e}_{1}}_{l{\bf e}_{1},m{\bf e}_{1}}=1$ as in (18), their matrix elements are given by

[TABLE]

The above formula for $R^{\vee\vee}(z)_{\alpha,\beta}^{\gamma,\delta}$ was obtained in [4] extending the result of [13]. The one for $R^{\vee}(z)_{\alpha,\beta}^{\gamma,\delta}$ and (23)–(25) can be deduced from it by applying the crossing symmetry and the results in [12] especially eqs.(2.7), (2.42) and Th.3.1 therein. The Yang-Baxter equations (26)–(29) with $\ast$ replaced by $\vee$ are valid.

6.2. $K$ matrix and reflection equation

From now on, we set

[TABLE]

but allow coexistence of $q$ and $p$ when it eases the presentation. Let $\mathcal{B}^{\prime}_{q}$ be the right coideal subalgebra of $U_{q}$ generated by

[TABLE]

This is related to $b_{i}$ in (30) via $b^{\prime}_{i}=-p^{-1}\omega(b_{i})$ where $\omega$ denotes the automorphism mentioned in Remark 2 with $\forall\mu_{i}=p$. Let

[TABLE]

be the unique map satisfying the intertwining relation

[TABLE]

and the normalization $K^{\prime}(z)^{l{\bf e}_{1}}_{l{\bf e}_{1}}=1$. From the construction so far we find that its matrix elements are related to those of $K(z)$ as

[TABLE]

Similarly to (37), it satisfies the reflection equation

[TABLE]

as linear operators $V_{l,x}\otimes V_{m,y}\rightarrow V^{\vee}_{l,x^{-1}}\otimes V^{\vee}_{m,y^{-1}}$.

6.3. Combinatorial $R$ and $K$ at $q=0$

At $q=0$, the $R$ matrices survive nontrivially as

[TABLE]

where $P_{i}(\alpha,\beta)=\min(\alpha_{i+1},\beta_{i+1})$, Q_{i}(\alpha,\beta)=\min_{1\leq k\leq n}\Bigl{\{}\sum_{1\leq j<k}\alpha_{i+j}+\sum_{k<j\leq n}\beta_{i+j}\Bigr{\}}. The denominator in the second formula is given by $R^{\vee}(z)^{l{\bf e}_{1},m{\bf e}_{2}}_{l{\bf e}_{1},m{\bf e}_{2}}=((-q)^{1-n}z)^{m}\frac{(q^{l-m+n}z^{-1};q^{2})_{m}}{(q^{l-m-n+2}z;q^{2})_{m}}$ from (79). In the RHS, we regard $\alpha,\gamma\in B_{l},\beta,\delta\in B_{m}$ as elements of crystals [8], and $\mathrm{R},\mathrm{R}^{\vee},\mathrm{R}^{\!\vee\vee}$ denote the classical part of the combinatorial $R$’s defined in eqs.(2.1), (2.2) and (2.4) in [14], respectively. They are nontrivial bijections $B_{m}\times B_{l}\rightarrow B_{l}\times B_{m}$ obeying the Yang-Baxter equations [14, eq.(2.7)]. The quantities $P_{i}(\alpha,\beta),Q_{i}(\alpha,\beta)$ are versions of energy functions and known to play an important role [9, 17, 14].

As for the $K$ matrix (83), it has the following behavior at $q=-p^{2}=0$:

[TABLE]

The denominator here can be written down explicitly from (53) and (83). The transformation $\alpha\mapsto\gamma=\sigma(\alpha)$ viewed as a bijection on $B_{l}$ essentially reproduces the combinatorial $K$ matrix introduced in [14, eq.(2.8)] to formulate the box-ball system with reflecting end. Together with the combinatorial $R$’s in the above, it forms a set-theoretical solution to the reflection equation. The latter is known to admit a further generalization to the birational maps [14, App.A]. We conclude that the reflection equation (84), after exchange of the two components, achieves a $q$-melting of the combinatorial reflection equation [14, eq.(2.13)].

Example 7**.**

Let $n=5$. We denote $v_{(2,1,0,2,0)}\in V_{5,z}$ by one-row semistandard tableau $11244$ and similarly $v^{\vee}_{(0,1,0,3,1)}\in V^{\vee}_{5,z}$ by $\bar{2}\bar{4}\bar{4}\bar{4}\bar{5}$, etc. With a proper normalization at $q=0$, the action of the two sides of $(\ref{sin2})_{x=y=1}$ on a base vector $12235\otimes 124\in V_{5,1}\otimes V_{3,1}$ proceed, according to (85)–(88), as follows:

[TABLE]

The agreement of the output is an example of the set-theoretical reflection equation [14].

7. Summary and outlook

In Theorem 1 we have characterized a $K$ matrix as the intertwiner of the coideal subalgebra $\mathcal{B}_{q}$ of $U_{q}(A^{(1)}_{n-1})$ generated by (30). By construction, it satisfies the reflection equation (37). In Theorem 3 we have constructed it in a matrix product form in terms of terminating $q$-hypergeometric series of $q$-boson generators.

At $q=0$, the $K$ matrix here reproduces one of the set-theoretical $K$ matrices called “Rotateleft” in [14, eq.(2.10)]. When $n$ is even, there are further solutions known as “Switch1n” and “Switch12” [14, eqs.(2.11), (2.12)] which also admit decent generalizations into geometric versions [14, app.A]. To incorporate them into the framework of this Letter, possibly with some other coideal subalgebra, is a natural problem to be addressed. Another important theme is to explore the 3D aspects of the matrix product (Theorem 3) from the viewpoint of [15]. It amounts to embedding the relations among the operators $G^{j}_{i}$ (42) into some sort of quantized reflection equation. We hope to report on these issues elsewhere.

Acknowledgments

The authors thank Vladimir Mangazeev, Zengo Tsuboi and Bart Vlaar for comments. A.K. is supported by Grants-in-Aid for Scientific Research No. 18H01141 from JSPS. M.O. is supported by Grants-in-Aid for Scientific Research No. 15K13429 and No. 16H03922 from JSPS.