Reflection $K$ matrices associated with an Onsager coideal of $U_p(A^{(1)}_{n-1})$, $U_p(B^{(1)}_n)$, $U_p(D^{(1)}_n)$ and $U_p(D^{(2)}_{n+1})$

TL;DR

This paper explicitly constructs reflection $K$ matrices for certain quantum affine algebras, providing the first proof of the reflection equation for non-type A cases through intertwiners of Onsager coideal subalgebras.

Contribution

It introduces a novel approach using intertwiners of Onsager coideal subalgebras to derive reflection $K$ matrices for non-type A quantum affine algebras.

Findings

Derived reflection $K$ matrices for $U_p(A^{(1)}_{n-1})$, $U_p(B^{(1)}_n)$, $U_p(D^{(1)}_n)$, $U_p(D^{(2)}_{n+1})$

Provided the first proof of the reflection equation for non-type A cases

Connected the intertwiners to three-dimensional integrability and matrix product constructions.

Abstract

We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra $U_p(A^{(1)}_{n-1})$ in the fundamental representations and $U_p(B^{(1)}_{n}), U_p(D^{(1)}_{n}), U_p(D^{(2)}_{n+1})$ in the spin representations. They reproduce the reflection $K$ matrices obtained recently by the matrix product construction connected to the three dimensional integrability. In particular the present approach provides the first proof of the reflection equation for the non type $A$ cases.

Equations

$$b_i=f_i+{\beta }_i{k}_i^{-1}{e}_i+{\gamma }_i{k}_i^{-1}\in U_p(0\le i\le n^{\prime })$$

$$K(z,p):V_z\to V_{z^{-1}}^{\ast },K(z,p){\pi }_z(b)={\pi }_{z^{-1}}^{\ast }(b)K(z,p)(b\in \mathcal{B}).$$

$$(z;q{)}_m=\prod _{j=1}^m(1-z{q}^{j-1}),{\left(\genfrac{}{}{0ex}{}{m}{r}\right)}_q=\frac{(q;q{)}_m}{(q;q{)}_{m-r}(q;q{)}_r}.$$

$$n^{\prime }$$

$${\mathfrak{g}}^{\mathrm{tr}}$$

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

$$p_0=p_n=p\text{for }{D}_{n+1}^{(2)},p_n=p\text{for }{B}_n^{(1)},p_0=p\text{for }{\tilde{B}}_n^{(1)}.$$

$$\Delta k_i^{\pm 1}=k_i^{\pm 1}\otimes k_i^{\pm 1},\Delta e_i=1\otimes e_i+e_i\otimes k_i,\Delta f_i=f_i\otimes 1+k_i^{-1}\otimes f_i.$$

$$\mathrm{sp}$$

$${\mathrm{sp}}_m$$

$${\mathrm{sp}}^{\pm }$$

$${\pi }_z:U_p({\mathfrak{g}}^{\mathrm{tr}}),U_p({\mathfrak{g}}^{r,r^{\prime }})\to \mathrm{End}{V}_z,V_z=\underset{\alpha \in \mathrm{sp}}{⨁}\mathbb{C}(z)v_{\alpha }$$

$${\mathfrak{g}}^{\mathrm{tr}}=A_{n-1}^{(1)};e_j{v}_{\alpha }$$

$${\mathfrak{g}}^{1,r^{\prime }}=D_{n+1}^{(2)},{\tilde{B}}_n^{(1)};e_0{v}_{\alpha }$$

$${\mathfrak{g}}^{2,r^{\prime }}=B_n^{(1)},D_n^{(1)};e_0{v}_{\alpha }$$

$${\mathfrak{g}}^{r,r^{\prime }}(1\le r,r^{\prime }\le 2);e_j{v}_{\alpha }$$

$${\mathfrak{g}}^{r,1}=D_{n+1}^{(2)},B_n^{(1)};e_n{v}_{\alpha }$$

$${\mathfrak{g}}^{r,2}={\tilde{B}}_n^{(1)},D_n^{(1)};e_n{v}_{\alpha }$$

$${\pi }_z^m:U_p({\mathfrak{g}}^{\mathrm{tr}})$$

$${\pi }_z^{\pm }:U_p({\mathfrak{g}}^{2,2})$$

$$e_i^{\ast }=e_i,f_i^{\ast }=f_i,k_i^{\ast }=k_i^{-1}.$$

$${\pi }_z^{\ast }:U_p({\mathfrak{g}}^{\mathrm{tr}}),U_p({\mathfrak{g}}^{r,r^{\prime }})\to \mathrm{End}{V}_z^{\ast },V_z^{\ast }=\underset{\alpha \in \mathrm{sp}}{⨁}\mathbb{C}(z)v_{\alpha }^{\ast }$$

$$⟨{\pi }_z^{\ast }(g)v^{\ast },v^{\prime }⟩=⟨v^{\ast },{\pi }_z(g^{\ast })v^{\prime }⟩(g\in U_p,v^{\ast }\in V_z^{\ast },v^{\prime }\in V_z).$$

$${\pi }_z^{m\ast }:U_p({\mathfrak{g}}^{\mathrm{tr}})$$

$${\pi }_z^{\pm \ast }:U_p({\mathfrak{g}}^{2,2})$$

$$\mathcal{R}(x/y,p)$$

$${\mathcal{R}}^{\ast }(x/y,p)$$

$${\mathcal{R}}^{\ast \ast }(x/y,p)$$

$${\mathcal{R}}^{\ast }(z,p{)}_{\alpha ,\beta }^{\gamma ,\delta }=\mathcal{R}(z,p{)}_{\alpha ,1-\beta }^{\gamma ,1-\delta },{\mathcal{R}}^{\ast \ast }(z,p{)}_{\alpha ,\beta }^{\gamma ,\delta }=\mathcal{R}(z,p{)}_{1-\alpha ,1-\beta }^{1-\gamma ,1-\delta }.$$

$${\Delta }^{\mathrm{op}}(g)\mathcal{R}(x/y,p)$$

$${\Delta }^{\mathrm{op}}(g){\mathcal{R}}^{\ast }(x/y,p)$$

$${\Delta }^{\mathrm{op}}(g){\mathcal{R}}^{\ast \ast }(x/y,p)$$

$$\mathcal{R}(x/y,p)$$

$${\mathcal{R}}_{12}(z_{12},p){\mathcal{R}}_{13}(z_{13},p){\mathcal{R}}_{23}(z_{23},p)$$

$${\mathcal{R}}_{12}(z_{12},p){\mathcal{R}}_{13}^{\ast }(z_{13},p){\mathcal{R}}_{23}^{\ast }(z_{23},p)$$

$${\mathcal{R}}_{12}^{\ast }(z_{12},p){\mathcal{R}}_{13}^{\ast }(z_{13},p){\mathcal{R}}_{23}^{\ast \ast }(z_{23},p)$$

$${\mathcal{R}}_{12}^{\ast \ast }(z_{12},p){\mathcal{R}}_{13}^{\ast \ast }(z_{13},p){\mathcal{R}}_{23}^{\ast \ast }(z_{23},p)$$

$$s{s}^{\prime }=-\mathrm{i}ϵp^{-1},ϵ=\pm 1,$$

Full text

Reflection $\boldsymbol{K}$ matrices associated with

an Onsager coideal of $\boldsymbol{U_{p}(A^{(1)}_{n-1})},\boldsymbol{U_{p}(B^{(1)}_{n})},\boldsymbol{U_{p}(D^{(1)}_{n})}$ and $\boldsymbol{U_{p}(D^{(2)}_{n+1})}$

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 determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra $U_{p}(A^{(1)}_{n-1})$ in the fundamental representations and $U_{p}(B^{(1)}_{n}),U_{p}(D^{(1)}_{n}),U_{p}(D^{(2)}_{n+1})$ in the spin representations. They reproduce the reflection $K$ matrices obtained recently by the matrix product construction connected to the three dimensional integrability. In particular the present approach provides the first proof of the reflection equation for the non type $A$ cases.

1. Introduction

Reflection equation or boundary Yang-Baxter equation [6, 14, 25] plays a fundamental role in quantum integrable systems with boundaries. It is a quartic relation of the so called $R$ and $K$ matrices encoding the interactions in the bulk and at the boundary, respectively. In the recent work [17] new solutions to the reflection equation including a spectral parameter were obtained associated with the quantum affine algebra $U_{p}(A^{(1)}_{n-1})$ and conjectured for $U_{p}(D^{(2)}_{n+1}),U_{p}(B^{(1)}_{n})$ and $U_{p}(D^{(1)}_{n})$. They possess a number of distinct features. Firstly the $K$ matrices act on the vector space of the form $({\mathbb{C}}^{2})^{\otimes n}$, and as this structure may indicate, the companion $R$ matrices are those associated with the fundamental representations111The antisymmetric tensor representations of general degree between [math] and $n$ whose degree 1 case corresponds to the vector representation. of $U_{p}(A^{(1)}_{n-1})$ or the spin representation of $U_{p}(B^{(1)}_{n}),U_{p}(D^{(1)}_{n}),U_{p}(D^{(2)}_{n+1})$. This is in contrast to the most preceding works which are concerned with the vector representation or $U_{p}(A^{(1)}_{1})$. See for example [3, 19, 20, 21, 24]. Secondly the $K$ matrices are trigonometric and dense in that all the elements are nontrivial rational function of the (multiplicative) spectral parameter. Thirdly each element of the $K$ matrices admits a matrix product formula in terms of $q$-Bosons. Lastly the family of solutions labeled with $n$ are descendants of a single integrable structure in the three dimension called quantized reflection equation (Appendix A) which was extracted from the representation theory of the quantized coordinate ring $A_{q}(\mathrm{Sp}_{4})$ [15]222The relation between $q$ of $A_{q}$ and $p$ of $U_{p}$ is given by (96)..

In this paper we show that the $K$ matrices in [17] are nothing but the intertwiners of the Onsager coideal subalgebras inside $U_{p}$, and thereby establish the reflection equation in the non type $A$ cases $U_{p}(B^{(1)}_{n}),U_{p}(D^{(1)}_{n}),U_{p}(D^{(2)}_{n+1})$ for the first time. The result gives a characterization of the $K$ matrices in the framework of $U_{p}$, which is an essential complement to the construction by the matrix product method originating in $A_{q}(\mathrm{Sp}_{4})$.

Let us briefly recall the approach to the reflection equation based on the coideal subalgebras of $U_{p}$, which goes back to the affine Toda field theory with boundaries [8]. See also [8, 13, 24] and references therein. Let $e_{i},f_{i},k^{\pm 1}_{i}\,(i=0,\ldots,n^{\prime})$ be the generators of the Drinfeld-Jimbo quantum affine algebra $U_{p}$ [9, 11], where $n^{\prime}$ is defined in (3). The elements

[TABLE]

behave as $\Delta b_{i}=k_{i}^{-1}\otimes b_{i}+(f_{i}+\beta_{i}k^{-1}_{i}e_{i})\otimes 1$ under the coproduct $\Delta$ in (7) for any coefficients $\beta_{i},\gamma_{i}$. Thus the subalgebra $\mathcal{B}\subset U_{p}$ generated by $b_{0},\ldots,b_{n^{\prime}}$ satisfies $\Delta\mathcal{B}\subset U_{p}\otimes\mathcal{B}$ meaning that it is a left coideal333 Choosing the left and the right coideals is a matter of convention transformed by an automorphism of $U_{p}$. They fit the reflection at the right and the left boundary of the system, respectively. Here we adopt the left coideal to adjust to [17].. Regarding the boundary reflection as a linear map $K$ from a representation $(\pi_{z},V_{z})$ of $U_{p}$ to its dual $(\pi^{\ast}_{z^{-1}},V^{\ast}_{z^{-1}})$ with the reversed spectral parameter, one has

[TABLE]

Here the latter equation is postulating that the $U_{p}$-symmetry in the bulk is not preserved fully at the boundary but still survives as a smaller symmetry corresponding to $\mathcal{B}$. By composing the above $K$ and the $R$ matrices, one can construct a linear map $V_{x}\otimes V_{y}\rightarrow V^{\ast}_{x^{-1}}\otimes V^{\ast}_{y^{-1}}$ commuting with $\Delta\mathcal{B}$ in two ways along the two sides of the reflection equation. See (41). Therefore if a solution $K$ to (2) exists uniquely and $V_{x}\otimes V_{y}$ is irreducible as a $\mathcal{B}$ module, the reflection equation follows as a corollary. In other words, the intertwining relation (2) for such a coideal $\mathcal{B}$ achieves linearization of the reflection equation, eliminating the task of proving the original quartic relation “manually”. This is a boundary analogue of the classic idea that the cubic Yang-Baxter equation is attributed to the linear equation $[\check{R},\Delta U_{p}]$ =0 representing the $U_{p}$-symmetry [9, 11].

The coideal $\mathcal{B}$ must be small enough; otherwise the intertwining relation (2) may not allow a solution. Nonetheless it must also be large enough; otherwise $V_{x}\otimes V_{y}$ may not become irreducible as a $\mathcal{B}$ module. In this way one is led to a fundamental question; what is the right “size” or choice of the coideal $\mathcal{B}$ in order to make the linearization work legitimately for a given representation $V_{z}$? To our knowledge it is still an outstanding issue in general.

This paper presents such coideals $\mathcal{B}=\langle b_{0},\ldots,b_{n^{\prime}}\rangle$ that control the $K$ matrices for the prototypical setting; the fundamental representations of $U_{p}(A^{(1)}_{n-1})$ and the spin representation of $U_{p}(B^{(1)}_{n})$, $U_{p}(D^{(1)}_{n})$, $U_{p}(D^{(2)}_{n+1})$. They are realized by a specific choice of the coefficients in (1). It turns out in all the cases that the resulting generators $b_{i}$’s form a closed set of relations among themselves known as the generalized $p$-Onsager algebras [2]. This algebra has drawn considerable attention since its first appearance at $p=1$ [23] and has been generalized significantly. See for example [26, Rem. 9.1] for the early history, [13, Sec.1(1)] for an account of more recent studies and the references therein.

In this paper a left coideal subalgebra of $U_{p}$ isomorphic to a generalized $p$-Onsager algebra will just be called an Onsager coideal for short. In this terminology all the sought coideals in our examples are Onsager coideals. This is another main observation in this paper. Although it is yet to be understood conceptually, we remark that a quite parallel result has already been established in [16] for another prominent example; the symmetric tensor representations of $U_{p}(A^{(1)}_{n-1})$.

The paper is organized as follows. In Section 2 basic definitions of the relevant quantum affine algebras and their representations are recalled. Then the specific Onsager coideals are introduced. Their intertwiners are unique and automatically satisfy the reflection equation. In Section 3 matrix product construction of the $R$ matrices [5, 18] and $K$ matrices [17] are quoted in the form adapted to the present setting. In Section 4 the main result of the paper, Theorem 3, 4 and 5 are stated. The latter two establish for the first time the matrix product solutions in the non type $A$ cases. In Section 5 a proof of the main result is given. In Section 6 a multi-parameter generalization is formulated and the relation to the Onsager coideals is explained. Section 7 is a summary.

Appendix A recalls the quantized reflection equation [17] which is the basic ingredient in the three dimensional approach. This description is included since Section 6 requires a slight parametric generalization of [17]. Appendix B lists the explicit forms of the generators of the Onsager coideals. We use the notation:

[TABLE]

2. Quantum affine algebras

Let $U_{p}=U_{p}(A^{(1)}_{n-1})\,(n\geq 2),U_{p}(D^{(2)}_{n+1})\,(n\geq 2),U_{p}(B^{(1)}_{n})\,(n\geq 3),U_{p}(\tilde{B}^{(1)}_{n})\,(n\geq 3),U_{p}(D^{(1)}_{n})\,(n\geq 3)$ be the quantum affine algebras without derivation operator [9, 11]. We assume that $p$ is generic throughout. For convenience we use the notation

[TABLE]

$U_{p}$ is a Hopf algebra generated by $e_{i},f_{i},k^{\pm 1}_{i}\,(0\leq i\leq n^{\prime})$ satisfying the relations

[TABLE]

where $e^{(\nu)}_{i}=e^{\nu}_{i}/[\nu]_{p_{i}}!,\,f^{(\nu)}_{i}=f^{\nu}_{i}/[\nu]_{p_{i}}!$ and $[m]_{p}!=\prod_{k=1}^{m}[k]_{p}$ with $[m]_{p}=\frac{p^{m}-p^{-m}}{p-p^{-1}}$. The constants $p_{i}\,(0\leq i\leq n^{\prime})$ in (5) are all taken as $p_{i}=p^{2}$ except the following:

[TABLE]

Note that $p_{0}=p^{r},\,p_{n}=p^{r^{\prime}}$ for $\mathfrak{g}^{r,r^{\prime}}$. The Cartan matrix $(a_{ij})_{0\leq i,j\leq n^{\prime}}$ is determined from the Dynkin diagrams of the relevant affine Lie algebras according to the convention of [12]:

$\mathfrak{g}^{\mathrm{tr}}=A^{(1)}_{n-1}$\drawline(20,3)(67,30) \drawline(73,30)(120,3) 1$$2$$n\!\!-\!\!2$$n\!\!-\!\!10\mathfrak{g}^{1,1}=D^{(2)}_{n+1}$$<$$>[math]1$$n\!\!-\!\!1$$n$$\mathfrak{g}^{2,1}=B^{(1)}_{n}$$>[math]1$$2$$n\!\!-\!\!1$$n$$\mathfrak{g}^{1,2}=\tilde{B}^{(1)}_{n}$$<$$n\!\!-\!\!1[math]1$$n\!\!-\!\!2$$n$$\mathfrak{g}^{2,2}=D^{(1)}_{n}[math]1$$2$$n\!\!-\!\!2$$n\!\!-\!\!1$$n

Here the affine Lie algebra $\tilde{B}^{(1)}_{n}$ is just $B^{(1)}_{n}$ but only with different enumeration of the nodes as shown above. We keep it for uniformity of description. Thus for instance in $U_{p}(D^{(2)}_{n+1})$, one has $a_{01}=-2,a_{10}=-1$ and $k_{0}e_{1}=p^{-2}e_{1}k_{0}$, $k_{1}e_{0}=p^{-2}e_{0}k_{1}$ and $k_{1}e_{1}=p^{4}e_{1}k_{1}$. We also note that $A^{(1)}_{1}$ is exceptional in that $a_{00}=a_{11}=-a_{01}=-a_{10}=2$. We employ the coproduct $\Delta$ of the form

[TABLE]

The opposite coproduct is denoted by $\Delta^{\!\mathrm{op}}=P\circ\Delta$, where $P(u\otimes v)=v\otimes u$ is the exchange of the components.

2.1. Representations $\pi_{z},\pi^{m}_{z},\pi^{\pm}_{z}$

We introduce the labeling set of the bases of the relevant representations as

[TABLE]

Let ${\bf e}_{j}=(0,\ldots,\overset{j\mathrm{th}}{1},\ldots,0)\in{\mathbb{Z}}^{n}$ be the elementary vector whose unique non vanishing element 1 is located at the $j$th component from the left. It should not be confused with the generator $e_{j}$ of $U_{p}$. We consider the representation

[TABLE]

given as follows:

[TABLE]

where for example $f_{j}$ actually means $\pi_{z}(f_{j})$. The symbol $v_{\beta}$ on the RHS with $\beta\not\in\mathrm{sp}$ is to be understood as [math]. For $U_{p}(\mathfrak{g}^{\mathrm{tr}})$, the representation $(\pi_{z},V_{z})$ is decomposed as

[TABLE]

where each component $(\pi^{m}_{z},V^{m}_{z})$ is irreducible and called the $m$-th fundamental representation. For $U_{p}(\mathfrak{g}^{r,r^{\prime}})$, the representation $(\pi_{z},V_{z})$ is irreducible except for $(r,r^{\prime})=(2,2)$. In the latter case, it decomposes into $(\pi^{+}_{z},V^{+}_{z})$ and $(\pi^{-}_{z},V^{-}_{z})$ as

[TABLE]

The representations $(\pi_{z},V_{z})$ and $(\pi^{\pm}_{z},V^{\pm}_{z})$ of $U_{p}(\mathfrak{g}^{r,r^{\prime}})$ are called the spin representations.

2.2. $\ast$-dual representations

Let $\ast$ be the algebra anti-automorphism given by

[TABLE]

The $\ast$-dual representation $(\pi^{\ast}_{z},V^{\ast}_{z})$

[TABLE]

of $(\pi_{z},V)$ is defined by

[TABLE]

Here $\langle\;,\;\rangle$ denotes the dual pairing $\langle v^{\ast}_{\alpha},v_{\beta}\rangle=\delta_{\alpha,\beta}$. Practically in our case, the $\ast$-dual representations are obtained from (12) by formally identifying $v^{\ast}_{\alpha}=v_{{\bf 1}-\alpha}$, where ${\bf 1}=(1,\ldots,1)={\bf e}_{1}+\cdots+{\bf e}_{n}$.

According to (13) and (14), one has the decompositions:

[TABLE]

2.3. Intertwiner of $U_{p}$;

$R$ matrix

Intertwiners of $\Delta U_{p}$ are called $R$ matrices. We will be concerned with the three kinds of $R$ matrices as

[TABLE]

From the remark after (17), one may set

[TABLE]

The $R$ matrices are characterized up to normalization by the intertwining relations:

[TABLE]

for $g\in U_{p}$. According to (13) and (14), one has the decompositions:

[TABLE]

and similarly for $\mathscr{R}^{\ast}(x/y,p)$ and $\mathscr{R}^{\ast\ast}(x/y,p)$. The $R$ matrices satisfy the Yang-Baxter equations [4, 9, 11]

[TABLE]

where $z_{ij}=z_{i}/z_{j}$. Here the notation is standard; $\mathscr{R}_{ij}(z,p)$ for instance denotes the $\mathscr{R}(z,p)$ that acts on the $i$th and the $j$th components from the left in the three-fold tensor product and as the identity in the other one. For $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ and $U_{p}(\mathfrak{g}^{2,2})$, the Yang-Baxter equations actually hold in finer subspaces corresponding to the decompositions (27) and (28).

At this stage we do not specify the normalization of the $R$ matrices. One typical choice will be given in (68)–(70) via Theorem 2. The basic properties of these $R$ matrices like the spectral decomposition have been described in [7] for $U_{p}(A^{(1)}_{n-1})$, in [22] for $U_{p}(B^{(1)}_{n})$, $U_{p}(D^{(1)}_{n})$ and in [18] for $U_{p}(D^{(2)}_{n+1})$.

2.4. Left coideal subalgebras

Let $\epsilon,s,s^{\prime}$ be the parameters obeying

[TABLE]

where ${\mathrm{i}}=\sqrt{-1}$. For $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ we introduce the left coideal subalgebra $\mathcal{B}^{\mathrm{tr}}$ generated by $b_{0},\ldots,b_{n-1}$ given by

[TABLE]

For $U_{p}(\mathfrak{g}^{r,r^{\prime}})$ we introduce the left coideal subalgebra $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}$ labeled with $k,k^{\prime}\in\{1,2\}$ satisfying $r\leq k$ and $r^{\prime}\leq k^{\prime}$. So there are nine pairs $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}\subset U_{p}(\mathfrak{g}^{r,r^{\prime}})$ of such kind. They are generated by $b_{0},\ldots,b_{n}$ defined by

[TABLE]

where $d_{1,1}(u,v),d_{1,2}(u,v).d_{2,2}(u,v)$ are given by

[TABLE]

For convenience we list the elements $b_{i}$ in Appendix B. The coideals $\mathcal{B}^{\mathrm{tr}}$ and $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}$ are examples of generalized $p$-Onsager algebras [2]. This aspect will be explained in Section 6.2, in a more generalized setting including further parameters.

2.5. Intertwiner of coideal; $K$ matrix

Intertwiners of the coideals are called $K$ matrices. It is a linear map

[TABLE]

satisfying the intertwining relation

[TABLE]

where $b\in\mathcal{B}^{\mathrm{tr}}$ for $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ and $b\in\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}$ for $U_{p}(\mathfrak{g}^{r,r^{\prime}})$. If $V_{z}$ is irreducible as a module over $\mathcal{B}^{\mathrm{tr}}$ or $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}$, then (40) characterizes the $K$ matrix up to normalization.

Consider the two maps going from $V_{x}\otimes V_{y}$ to $V^{\ast}_{x^{-1}}\otimes V^{\ast}_{y^{-1}}$ composed of $R$ and $K$ matrices as

[TABLE]

Here $P$ is the transposition introduced after (7), $\mathscr{R}(z),\mathscr{R}^{\ast}(z),\mathscr{R}^{\ast\ast}(z)$ are abbreviation of $\mathscr{R}(z,p)$, $\mathscr{R}^{\ast}(z,p)$, $\mathscr{R}^{\ast\ast}(z,p)$, respectively and $\mathscr{K}_{2}(z)=1\otimes\mathscr{K}(z,p)$. Since the $K$ matrices act only on the right component, the both maps commute with $\Delta\mathcal{B}\subset U_{p}\otimes\mathcal{B}$ ($\mathcal{B}=\mathcal{B}^{\mathrm{tr}}$ or $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}$) owing to the intertwining relations (24)–(26) and (40). Therefore if $V_{x}\otimes V_{y}$ with generic $x,y$ is irreducible as a $\mathcal{B}$ module and the $R$ and the $K$ matrices are properly normalized, the above diagram implies the reflection equation [8]

[TABLE]

where $\mathscr{R}_{12}(z)=\mathscr{R}(z,p)$, $\mathscr{R}^{\ast}_{21}(z)=P\mathscr{R}^{\ast}(z,p)P$, $\mathscr{R}^{\ast}_{12}(z)=\mathscr{R}^{\ast}(z,p)$, $\mathscr{R}^{\ast\ast}_{21}(z)=P\mathscr{R}^{\ast\ast}(z,p)P$ and $\mathscr{K}_{1}(z)=\mathscr{K}(z,p)\otimes 1$. In terms of the matrix elements for the transition $v_{\alpha}\otimes v_{\beta}\mapsto v^{\ast}_{\alpha^{\prime\prime\prime}}\otimes v^{\ast}_{\beta^{\prime\prime\prime}}$, it reads

[TABLE]

As the set of equations this is equivalent, due to (23), to

[TABLE]

where $\mathscr{K}^{\ast}(z,p)^{\beta}_{\alpha}:=\mathscr{K}(z,p)^{{\bf 1}-\beta}_{\alpha}$. This is the form which essentially agrees with [17, eqs.(73), (82)]. The $K(z)$ in [17] corresponds to $\mathscr{K}^{\ast}(z,p)$ here having the matrix elements $\mathscr{K}^{\ast}(z,p)^{\beta}_{\alpha}$. As seen in (44), using $\mathscr{K}^{\ast}(z,p)$ enables one to formulate the reflection equation without involving $\mathscr{R}^{\ast}(z,p)$ and $\mathscr{R}^{\ast\ast}(z,p)$.

When the intertwining relations (24)–(26) and (40) are decomposed into the ones in finer irreducible submodules over the coideals, the same argument shows that the reflection equation holds individually in those subspaces.

At this stage we do not specify the normalization of the $K$ matrices. One typical example will be given in (85)–(87) via Theorem 3, 4 and 5. The reflection equations (43) and (44) are diagrammatically expressed as follows:

\beta^{\prime\prime\prime},y^{-1}$$\beta,y$$\alpha^{\prime\prime\prime},x^{-1}$$\alpha,x$$\beta^{\prime\prime}$$\beta^{\prime}$$\alpha^{\prime\prime}$$\alpha^{\prime}$$=$$\beta^{\prime\prime\prime},y^{-1}$$\alpha,x$$\alpha^{\prime\prime\prime},x^{-1}$$\beta,y$$\alpha^{\prime\prime}$$\alpha^{\prime}$$\beta^{\prime}$$\beta^{\prime\prime}

The boundary reflection changes the spectral parameters $x$ and $y$ into $x^{-1}$ and $y^{-1}$.

3. Matrix product construction

3.1. $q$-Bosons

Let $F_{q}=\bigoplus_{m\geq 0}{\mathbb{C}}|m\rangle$ be the Fock space equipped with the $q$-Boson operators ${\rm{\bf a}}^{\!\pm},{\rm{\bf k}}$. Similarly let $F_{q^{2}}$ be the one with ${\rm{\bf A}}^{\!\pm},{\rm{\bf K}}$ as follows:

[TABLE]

We use the same notation for the base vectors either for $F_{q}$ or $F_{q^{2}}$, which will not cause a confusion. The following $q$-Boson relations hold444The operators ${\rm{\bf k}},{\rm{\bf K}}$ in [17] differ from the ones here by the factors $q^{\scriptstyle\frac{1}{2}},q$ corresponding to the zero point energy.:

[TABLE]

Denote the dual space of $F_{q^{r}}\,(r=1,2)$ by $F^{\ast}_{q^{r}}=\bigoplus_{m\geq 0}{\mathbb{C}}\langle m|$ such that $\langle m|m^{\prime}\rangle=\delta_{m,m^{\prime}}(q^{2r};q^{2r})_{m}$. We endow them with the $q$-Boson action by

[TABLE]

These definitions satisfy $(\langle m|X)|m^{\prime}\rangle=\langle m|(X|m^{\prime}\rangle)$.

3.2. Boundary vectors

For $r=1,2$, we introduce the elements called boundary vectors:

[TABLE]

They will be utilized in the matrix product constructions (62) and (79). They are characterized by

[TABLE]

3.3. Matrix product construction from $L$

Define an operator $L$ by [5, 17]

[TABLE]

We introduce the linear operators $R(z,q),R^{\ast}(z,q),R^{\ast\ast}(z,q)$ with $R=R^{\mathrm{tr}},R^{r,r^{\prime}}$ with $r,r^{\prime}\in\{1,2\}$ by

[TABLE]

where the elements are given by the matrix product formulas:

[TABLE]

The normalization factors appearing here are taken as

[TABLE]

For the definition of the symbol $|\alpha|$, see (9). These choices make all the matrix elements of $R^{\mathrm{tr}}(z,q)$ and $R^{r,r^{\prime}}(z,q)$ rational functions in $z$ and $q$. For instance we have

[TABLE]

From the construction it is easy to see the properties:

[TABLE]

Comparing (57) and [17, eq.(5)], one can show that these matrix elements are related to $S(z)^{\gamma,\delta}_{\alpha,\beta}$ in [17, eqs.(57), (58)] by

[TABLE]

with a scalar $\varphi(z)$ that only depends on the normalization. Examples of $S^{\mathrm{tr}}(z),S^{r,r^{\prime}}(z)$ are available in [17, App.C]. The factor $({\mathrm{i}}\epsilon)^{|\beta-\gamma|}$ in (74) does not influence the reflection equation on account of (71).

3.4. Matrix product construction from $G$

Define an operator $G$ by

[TABLE]

where $s$ and $s^{\prime}$ are parameters satisfying

[TABLE]

The definition (75) is a slight modification of [17, eq.(6)]. We introduce the linear operators $K(z,q)=K^{\mathrm{tr}}(z,q)$ and $K^{r,r^{\prime}}(z,q)$ by

[TABLE]

Note that the dependence on $s,s^{\prime}$ is suppressed in the notation. The elements are given by the matrix product formulas:

[TABLE]

The operator $z^{\bf h}$ can be moved anywhere by means of

[TABLE]

The normalization factors are taken as

[TABLE]

These choices make all the matrix elements of $K^{\mathrm{tr}}(z,q)$ and $K^{r,r^{\prime}}(z,q)$ rational in $z$ and $q$. For instance we have

[TABLE]

By the construction they have the properties:

[TABLE]

where the latter two relations can be derived from [17, eq.(1)] and $\kappa^{k,k^{\prime}}(z)=\kappa^{k^{\prime},k}(z)$555 Eq.(91) corresponds to [17, eq.(84)]..

Comparing (75) and [17, eq.(6)], one can show that these matrix elements are related to [17, eqs.(74), (83)] by

[TABLE]

with some scalar $\phi(z)$ that only depends on the normalization. Examples of $K^{\mathrm{tr}}(z),K^{r,r^{\prime}}(z)$ on the RHS are available in [17, App.C]. Setting $(s,s^{\prime})=({\mathrm{i}}t,-{\mathrm{i}}q/t)$ so as to satisfy (76), one can stay within “real” coefficients $K^{\mathrm{tr}}(z,q)^{\beta}_{\alpha},K^{k,k^{\prime}}(z,q)^{\beta}_{\alpha}\in{\mathbb{Q}}(q,t,z)$. This feature will also be observed explicitly in the examples in Section 3.5.

Remark 1**.**

Comparing (57) and (75), we have

[TABLE]

From the similar relation between the boundary vectors (54), we find a curious fact that the elements of the $K$ matrices (78)–(80) are equal to the special elements of the $R$ matrices (61)–(64) with $q$ replaced by $q^{\scriptstyle\frac{1}{2}}$:

[TABLE]

where $\rho(z)$ is a scalar depending on the normalization only, and the LHS actually means the case $s=s^{\prime}=\epsilon q^{\scriptstyle\frac{1}{2}}$.

3.5. Examples

The quantities (78)–(80) are evaluated by the following formulas [17, eq.(81)]666Quoted by taking it into account that ${\rm{\bf k}}$ in [17] is equal to $q^{\scriptstyle\frac{1}{2}}{\rm{\bf k}}$ in this paper.:

[TABLE]

where $\theta(\text{true})=1,\theta(\text{false})=0$.

As already mentioned, matrix elements have been given explictly in [17, App.C] for $K^{\mathrm{tr}}(z,q)$ up to $n=3$ and $K^{k,k^{\prime}}(z,q)$ up to $n=2$, which are connected to the present paper by (92). So we illustrate here the calculation of $K^{1,1}(z,q)$ with $n=3$, which corresponds to $U_{p}(\mathfrak{g}^{1,1})=U_{p}(D^{(2)}_{4})$.

From (91) and (81) with $z^{\bf h}$ replaced by the special case ${\rm{\bf k}}=q^{\bf h}$, the elements777For simplicity, $K^{1,1}(z,q)^{\beta}_{\alpha}$ with $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3}),\beta=(\beta_{1},\beta_{2},\beta_{3})$ is denoted by $K^{1,1}(z,q)^{\beta_{1}\beta_{2}\beta_{3}}_{\alpha_{1}\alpha_{2}\alpha_{3}}$ rather than $K^{1,1}(z,q)^{(\beta_{1},\beta_{2},\beta_{3})}_{(\alpha_{1},\alpha_{2},\alpha_{3})}$. are reduced to $K^{1,1}(z,q)^{000}_{000}=1$ in (86) and the following up to powers of $q,s$ and $z$:

[TABLE]

Let us derive $K^{1,1}(z,q)_{100}^{011}$ in the last line. From (79) and $s^{\prime}=q/s$ it is calculated as

[TABLE]

which leads to the sought result.

4. Main result

In the rest of the paper we will always assume that $\epsilon,q,p,s,s^{\prime}$ are related by

[TABLE]

This unifies (76) and (33).

4.1. Summary of known results

Let us first recall the known result on the $R$ matrices.

Theorem 2**.**

The $R$ matrices possess the matrix product formulas as follows:

[TABLE]

The results (97) and (98) are essentially due to [5] and [18], respectively.

From (23) and (64), Theorem 2 implies

[TABLE]

Next we summarize the results on the $K$ matrices in [17]. For $U_{p}(\mathfrak{g}^{\mathrm{tr}})$, the reflection equation involving $R^{\mathrm{tr}}(z,q),K^{\mathrm{tr}}(z,q)$ was proved in [17, eq.(73)]. For $U_{p}(\mathfrak{g}^{r,r^{\prime}})$, the reflection equation involving $R^{r,r^{\prime}}(z,q)$ and $K^{k,k^{\prime}}(z,q)$ with $r\leq k,r^{\prime}\leq k^{\prime}$ was shown to follow from the property [17, eq.(78)] of the boundary vectors. Proving it is an interesting open problem. In this paper we will achieve a proof of the reflection equation for $U_{p}(\mathfrak{g}^{r,r^{\prime}})$ via a different route of resorting to the representation theory of the coideal subalgebras of $U_{p}$.

4.2. $K$ matrices

Now we state the main results of the paper. They characterize the $K$ matrices [17] by the matrix product construction (recalled in Section 3.4 with a slight parametric generalization) as the intertwiner of the coideal subalgebras described in Section 2.4–2.5.

Theorem 3**.**

The $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ modules $V^{m}_{x}$ and $V^{m}_{x}\otimes V^{m^{\prime}}_{y}$ with generic $x,y$ are irreducible as $\mathcal{B}^{\mathrm{tr}}$ modules for any $0\leq m,m^{\prime}\leq n$. There is a $K$ matrix having the decomposition

[TABLE]

such that each component is characterized up to normalization by a finer version of the intertwining relation (40):

[TABLE]

It is given by the matrix product formula

[TABLE]

Theorem 4**.**

For $(r,r^{\prime})\neq(2,2)$, the $U_{p}(\mathfrak{g}^{r,r^{\prime}})$ modules $V_{x}$ and $V_{x}\otimes V_{y}$ with generic $x,y$ are irreducible as $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}$ modules if $r\leq k$ and $r^{\prime}\leq k^{\prime}$. The $K$ matrix characterized up to normalization by the intertwining relation (40) is given by the matrix product formula:

[TABLE]

Theorem 5**.**

The $U_{p}(\mathfrak{g}^{2,2})$ modules $V^{\pm}_{x}$, $V^{\pm}_{x}\otimes V^{\pm}_{y}$ and $V^{\pm}_{x}\otimes V^{\mp}_{y}$ with generic $x,y$ are irreducible as $\mathcal{B}^{2,2}_{2,2}$ modules. There is a $K$ matrix having the decomposition

[TABLE]

such that each component is characterized up to normalization by a finer version of the intertwining relation (40):

[TABLE]

It is given by the matrix product formula

[TABLE]

4.3. Reflection equation

Write ${R}(z,q),{R}^{\ast}(z,q),{R}^{\ast\ast}(z,q),{K}(z,q)$ simply as ${R}(z),{R}^{\ast}(z),{R}^{\ast\ast}(z),{K}(z)$, respectively. From the argument that led to (42) and Theorem 3–5, we have

Corollary 6**.**

The following reflection equation holds:

[TABLE]

where $r\leq k$ and $r^{\prime}\leq k^{\prime}$ in the latter. The equality (108) splits into the one in $\mathrm{Hom}(V^{m}_{x}\otimes V^{m^{\prime}}_{y},V^{m\ast}_{x^{-1}}\otimes V^{m^{\prime}\ast}_{y^{-1}})$ for each pair $(m,m^{\prime})\in[0,n]^{2}$. The equality (109) with $(r,r^{\prime})=(2,2)$ splits into the one in $\mathrm{Hom}(V^{\sigma}_{x}\otimes V^{\sigma^{\prime}}_{y},V^{\sigma\ast}_{x^{-1}}\otimes V^{\sigma^{\prime}\ast}_{y^{-1}})$ for each pair $(\sigma,\sigma^{\prime})\in\{+,-\}^{2}$.

The result (108) reconfirms [17, eq.(73)]. The result (109) yields the first proof of [17, eq.(82)]. As for the special components $\mathscr{K}^{1}(z,q)$ and $\mathscr{K}^{n-1}(z,q)$ in (101), an explicit list of the matrix elements was given in [10].

5. Proof

5.1. Existence

Let us prove the existence of the intertwiner of the coideal $\mathscr{K}(z,p)$ in (40). Our strategy is to show that the matrix product constructed $K$ matrices in Section 3.4 indeed fulfill the intertwining relation. We assume the relation (96) among the parameters throughout.

Proposition 7**.**

The $K$ matrices given by the matrix product construction (78)–(80) satisfy the intertwining relation

[TABLE]

where $(K,\mathcal{B})=(K^{\mathrm{tr}},\mathcal{B}^{\mathrm{tr}})$ for $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ and $(K,\mathcal{B})=(K^{k,k^{\prime}},\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}})$ for $U_{p}(\mathfrak{g}^{r,r^{\prime}})$ with $r\leq k$ and $r^{\prime}\leq k^{\prime}$.

Proof.

It suffices to show (110) for $b=b_{0},\ldots,b_{n^{\prime}}$ given by (34)–(38). There are seven cases to consider; (i) $b=b_{j}$ (34) for $j\in{\mathbb{Z}}_{n}$, (ii) $b=b_{0}$ (35) for $(r,k)=(1,2)$, (iii) $b=b_{0}$ (35) for $(r,k)=(1,1)$, (iv) $b=b_{0}$ (35) for $(r,k)=(2,2)$, (v) $b=b_{n}$ (37) for $(r,k)=(1,2)$, (vi) $b=b_{n}$ (37) for $(r,k)=(1,1)$, (vii) $b=b_{n}$ (37) for $(r,k)=(2,2)$. It is easy to show that (v), (vi), (vii) are attributed respectively to (ii), (iii), (iv) with $z=1$ by the exchange $s\leftrightarrow s^{\prime}$, $G^{j}_{i}\leftrightarrow G^{i}_{j}$ and $\langle m|\leftrightarrow|m\rangle$. See also [17, eq.(1)] for a hint. Thus we shall only treat the cases (i)–(iv) in the sequel.

(i) $b=b_{j}$ (34) for $j\in{\mathbb{Z}}_{n}$. This covers $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ whose sub-case $0<j<n$ also does the other $U_{p}(\mathfrak{g}^{r,r^{\prime}})$. In terms of the matrix element for the transition $v_{\alpha}\mapsto v^{\ast}_{\beta}$, the equation (110) is stated as

[TABLE]

where $K=K^{\mathrm{tr}}\,(j\in{\mathbb{Z}}_{n})$ or $K=K^{k,k^{\prime}}\,(0<j<n)$. In view of the matrix product formulas (78)–(80), this follows from the quadratic relation (we set $(\alpha_{j},\alpha_{j+1},\beta_{j},\beta_{j+1})=(a,a^{\prime},b,b^{\prime})$)

[TABLE]

where we regard $G^{b}_{a}=0$ unless $a,b\in\{0,1\}$. The factor $z^{\pm\delta_{j0}}$ has disappeared owing to (81). It is elementary to check (112) by using (48).

(ii) $b=b_{0}$ (35) for $(r,k)=(1,2)$. We are to check

[TABLE]

Due to (79) and (80) this follows from

[TABLE]

Thanks to (81) this further reduces to $z=1$, which can be checked easily by (56).

(iii) $b=b_{0}$ (35) for $(r,k)=(1,1)$. We are to check

[TABLE]

Due to (79) this follows from

[TABLE]

Due to (81) this reduces to $z=1$, which can be checked by (55).

(iv) $b=b_{0}$ (35) for $(r,k)=(2,2)$. We are to check

[TABLE]

Due to (79) and (80) this follows from

[TABLE]

Again this reduces to $z=1$ case, which can be verified by (56) and (48). ∎

5.2. Uniqueness and the proof of the reflection equation

In this subsection we give a proof of the irreducibility of $V_{x}\otimes V^{\prime}_{y}$ as a module over $\mathcal{B}$ for various $\mathcal{B}$ in Section 2.4 and $V,V^{\prime}$ in Section 2.1. Once this irreducibility is proven, the proof of the reflection equation (42) is complete. If we consider $V^{\prime}_{y}$ to be a trivial module, the uniqueness of the $K$ matrix in Section 2.5 is also confirmed.

To show $V_{x}\otimes V^{\prime}_{y}$ is irreducible for generic values of $x,y$, it suffices to show it is so at a special value $y=x$. We consider $U_{p}$ or $\mathcal{B}$ as algebras over ${\mathbb{C}}(p)$ and $V_{x},V^{\prime}_{x}$ as modules over ${\mathbb{C}}(p,x)$. Define $A=\{f(p,x)\in{\mathbb{C}}(p,x)\mid f(p,x)\text{ is regular at }x=0\}$. $A$ is a local ring with the maximal ideal $xA$ and $A/xA\simeq{\mathbb{C}}(p)$.

Let $W$ be a nonzero subspace of $V_{x}\otimes V^{\prime}_{x}$ invariant under $\mathcal{B}$ and set

[TABLE]

Apparently, $L$ is a finitely generated module over $A$ and $L^{\prime}$ is a sub $A$-module of $L$. Set $\tilde{b}_{i}=x_{0}^{\delta_{i0}}b_{i}$, where $x_{0}=x^{2}$ for $U_{p}(\mathfrak{g}^{2,1})$, $U_{p}(\mathfrak{g}^{2,2})$ and $x_{0}=x$ for the other $U_{p}$. Note that the action of $\Delta(\tilde{b}_{i})$ preserves $L$.

Let $U_{p,0}$ be the subalgebra of $U_{p}$ generated by $e_{i},f_{i},k_{i}$ with $i\neq 0$. Suppose that as a $U_{p,0}$-module we have the following decomposition.

[TABLE]

Here we number $\{U_{j}\}_{0\leq j\leq l}$ in such a way that $j<k$ holds whenever $\lambda_{j}>\lambda_{k}$ occurs888 $\lambda>\mu$ is defined by $\lambda\neq\mu$ and $\lambda-\mu$ is a linear combination of simple roots with nonnegative coefficients. where $\lambda_{j}$ is the highest weight of $U_{j}$. Take any nonzero vector $u$ from $L^{\prime}$. According to the decomposition (121), $u$ can be expressed as

[TABLE]

where $u_{j,\gamma}$ is a weight vector of weight $\gamma$ which belongs to $U_{j}$. One can assume there exists $(j,\gamma)$ such that $u_{j,\gamma}\not\equiv 0$ mod $xL$. Here and in what follows, $\equiv$ means an equality mod $xA$. Take the minimal $j_{0}$ such that $u_{j_{0},\gamma}\not\equiv 0$ for some $\gamma$. Applying $\Delta(\tilde{b}_{i})$ ($i\neq 0$) if necessary, one can assume $u_{j_{0},\gamma_{0}}\not\equiv 0$ where $\gamma_{0}$ is the lowest weight of $U_{j_{0}}$.

Lemma 8**.**

Suppose $j>0$. Then there exists a sequence $i_{1},\ldots,i_{m}\in\{0,1,\ldots,n^{\prime}\}$ such that $\Delta(\tilde{b}_{i_{1}}\cdots\tilde{b}_{i_{m}})u$ has a nonzero component in $U_{j^{\prime}}$ ($j^{\prime}<j$).

Proof.

Since the proof depends on the cases of $U_{p}$ and $\mathcal{B}$, we illustrate it for (i) $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ with $n=4$ and $V^{m}_{x}\otimes V^{m^{\prime}}_{x}$ with $m=m^{\prime}=2$, and (ii) $U_{p}(\mathfrak{g}^{1,1})$ with $n=2$ and $V_{x}\otimes V_{x}$ (spin representation).

(i) In this case the decomposition (121) reads as

[TABLE]

Suppose $j=1$. One can assume (122) is of the following form:

[TABLE]

Here $\cdots$ contains a linear combination of weight vectors of weight strictly higher than that of $v_{(1100)}\otimes v_{(1010)}$. Noting that $\tilde{b}_{0}=x_{0}f_{0}$ mod $xL^{\prime}$. We have

[TABLE]

It may happen that $\cdots$, terms with higher weights, contain a nonzero component in $U_{0}$. In this case the proof is over. Suppose they do not. Applying $\tilde{b}_{3}\tilde{b}_{1}$ further, we get

[TABLE]

Note that $\cdots$ contain terms produced by applying $k_{i}^{-1}e_{i}$ or $k_{i}^{-1}$ in $\tilde{b}_{i}$ for $i=1,3$. However the weights remain higher than that of $v_{(1010)}\otimes v_{(0011)}$. Finally, applying $\tilde{b}_{0}$, we obtain

[TABLE]

The first term apparently belongs to $U_{0}$.

Suppose $j=2$. One can assume (122) is of the following form:

[TABLE]

Since $\dim U_{0}=1$, There are no other terms. Applying $\tilde{b}_{0}$ we have

[TABLE]

Since it is not proportional to (5.2), it should contain a nonzero component in $U_{0}$ or $U_{1}$.

(ii) Since the situation is similar, we only list formulas. For $j=1$ (122) is of the form:

[TABLE]

and we have

[TABLE]

For $j=2$ (122) is of the form:

[TABLE]

and we have

[TABLE]

∎

From this lemma, one can assume $j_{0}=0$ and $\gamma_{0}$ is the lowest weight of $U_{0}$. We can also show the following lemma.

Lemma 9**.**

For a vector $u=\sum_{j,\gamma}u_{j,\gamma}$ such that $u_{0,\gamma_{0}}\not\equiv 0$ where $\gamma_{0}$ is the lowest weight of $U_{0}$, there exists a sequence $i_{1},\ldots,i_{m}\in\{0,1,\ldots,n^{\prime}\}$ such that

[TABLE]

where $u_{0}$ is a nonzero highest weight vector of $U_{0}$.

Proof.

As in the proof of Lemma 8, we illustrate the proof for (i) $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ with $n=4$ and $V^{m}_{x}\otimes V^{m^{\prime}}_{x}$ with $m=m^{\prime}=2$, and (ii) $U_{p}(\mathfrak{g}^{1,1})$ with $n=2$ and $V_{x}\otimes V_{x}$ (spin representation).

(i) The vector $u$ is of the form:

[TABLE]

where $\cdots$ contains a linear combination of weight vectors of weight strictly higher than that of $v_{(1100)}\otimes v_{(1100)}$. Applying $\tilde{b}_{0}^{2}\tilde{b}_{3}^{2}\tilde{b}_{1}^{2}\tilde{b}_{0}^{2}$, we have

[TABLE]

Since $v_{(0011)}\otimes v_{(0011)}$ is a highest weight vector, the part $\cdots$ should vanish.

(ii) The vector $u$ is of the form:

[TABLE]

where $\cdots$ contains a linear combination of weight vectors of weight strictly higher than that of $v_{(11)}\otimes v_{(11)}$. Applying $\tilde{b}_{0}^{2}\tilde{b}_{1}^{2}\tilde{b}_{0}^{2}$, we have

[TABLE]

Since $v_{(00)}\otimes v_{(00)}$ is a highest weight vector, the part $\cdots$ should vanish. ∎

Hence, one finds that $u_{0}$ is contained in $L^{\prime}$ mod $xL^{\prime}$. Now apply $\Delta(\tilde{b}_{i})$ successively to obtain new vectors in $L^{\prime}$. Since new vectors obtained are those by applying $\Delta(x_{0}^{\delta_{i0}}f_{i})$, $u_{0}$ generates all vectors in $L$ mod $xL$, so we have $L=L^{\prime}+xL$. By Nakayama’s lemma (cf. [1]):

Lemma 10**.**

Suppose that $R$ is a ring and $M$ is an $R$-module. Let $N$ be a submodule of $M$. If $M/N$ is finitely generated over $R$ and $M=N+J(R)M$, then $N=M$. Here $J(R)$ is the Jacobson radical of $R$.

with $R=A$ ($J(R)=xA$) and $M=L,N=L^{\prime}$, one obtains $L^{\prime}=L$, and hence $W=V_{x}\otimes V^{\prime}_{x}$.

The proof of the irreducibility of $V_{x}\otimes V^{\prime}_{y}$ is complete.

6. Generalization

6.1. Inhomogeneous matrix product

The parameters $s,s^{\prime}$ entering (75) may be taken independently in each $G$ in the matrix products (78)–(80) as long as the relation (76) is kept. To be concrete, we write (75) obeying the constraint (76) as

[TABLE]

Introduce the parameters $s_{1},\ldots,s_{n}$ and consider the inhomogeneous generalization of (78)–(80):

[TABLE]

where the normalization factors have been put aside for simplicity. Using them as the matrix elements define $K^{\mathrm{tr}}(z,q;s_{1},\ldots,s_{n}),K^{k,k^{\prime}}(z,q;s_{1},\ldots,s_{n})\in\mathrm{Hom}(V_{z},V^{\ast}_{z^{-1}})$ similarly to (77). The homogeneous case (77)–(80) corresponds to $K(z,q,s,\ldots,s)$. The general case is reduced to it as

[TABLE]

where $S$ and $S^{\ast}$ are the diagonal matrices defined by

[TABLE]

As for the $R$ matrices, the weight conservation (71) and the relation (64) imply

[TABLE]

From (141) and (143) it follows that $K(z,q;s_{1},\ldots,s_{n})$ also satisfies the same reflection equations as (108) and (109) without affecting the companion $R$ matrices. In this sense $s_{1},\ldots,s_{n}$ in (139) and (140) are just a sort of gauge parameters as far as the reflection equation is concerned.

On the other hand their influence on the intertwining relation (110) is less trivial. In fact it is transformed to

[TABLE]

One may interpret this as the intertwining relation for the representation connected to $\pi_{z}$ by the conjugation by $S$. However there is another option to stay within the same representation saying that it is the coideal $\mathcal{B}=\langle b_{0},\ldots,b_{n^{\prime}}\rangle$ that has been deformed by the parameters $s_{1},\ldots,s_{n}$. The latter is presented as $\mathcal{B}(s_{1},\ldots,s_{n})=\langle\hat{b}_{0},\ldots,\hat{b}_{n^{\prime}}\rangle$ in terms of the new generators $\hat{b}_{i}$ satisfying

[TABLE]

Below we describe such $\hat{b}_{i}$’s generating $\mathcal{B}(s_{1},\ldots,s_{n})$ concretely.

$\mathcal{B}^{\mathrm{tr}}(s_{1},\ldots,s_{n})$ is the left coideal subalgebra of $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ generated by

[TABLE]

Of course it actually depends on the ratio of $s_{i}$’s, hence forms an $(n-1)$-parameter family.

$\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}(s_{1},\ldots,s_{n})$ with $r\leq k$ and $r^{\prime}\leq k^{\prime}$ is the left coideal subalgebra of $U_{p}(\mathfrak{g}^{r,r^{\prime}})$ generated by

[TABLE]

where $s^{\prime}_{j}=q/s_{j}=-{\mathrm{i}}\epsilon/(ps_{j})$ in all the cases. The definitions (146)–(153) reduce to (34)–(37) in the homogeneous case $s_{1}=\cdots=s_{n}=s=q/s^{\prime}$.

The argument so far is summarized as a multi-parameter generalization of Proposition 7:

Proposition 11**.**

Under the condition $s_{j}s^{\prime}_{j}=q=-{\mathrm{i}}\epsilon p^{-1}\,(1\leq j\leq n)$, the $K$ matrices defined by the inhomogeneous matrix products (139) and (140) satisfy the intertwining relation

[TABLE]

where $(K,\mathcal{B})=(K^{\mathrm{tr}},\mathcal{B}^{\mathrm{tr}})$ for $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ and $(K,\mathcal{B})=(K^{k,k^{\prime}},\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}})$ for $U_{p}(\mathfrak{g}^{r,r^{\prime}})$ with $r\leq k$ and $r^{\prime}\leq k^{\prime}$.

One can check the claim directly as the proof of Proposition 7.

6.2. Relation to generalized Onsager algebras

Consider the element

[TABLE]

where $c_{i},\bar{c}_{i},d_{i}$ are coefficients. In view of

[TABLE]

the subalgebra $\mathcal{A}\subset U_{p}$ generated by $A_{0},\ldots,A_{n}$ forms a left coideal $\Delta\mathcal{A}\subset U_{p}\otimes\mathcal{A}$.

In [2] the condition on the coefficients $c_{i},\bar{c}_{i},d_{i}$ are determined so that there are a closed set of relations among $A_{i}$’s that make $\mathcal{A}\subset U_{p}$ isomorphic to the generalized $p$-Onsager algebra. Let us concentrate on $U_{p}(\mathfrak{g}^{\mathrm{tr}})$ and $U_{p}(\mathfrak{g}^{r,r^{\prime}})$ treated in this paper. Then except for $U_{p}(A^{(1)}_{1})$ which is the smallest example from the former, the condition is described as follows:

[TABLE]

where $(a_{ij})_{0\leq i,j\leq n^{\prime}}$ is the Cartan matrix appearing in (5). (The case $U_{p}(A^{(1)}_{1})$ is exceptional in that there is no constraint on the parameters.) In the resulting generalized $p$-Onsager algebra, the structure constants themselves are dependent on $c_{i},\bar{c}_{i}$ obeying the above constraint. We call such a coideal subalgebra of $U_{p}$ an Onsager coideal for simplicity.

It turns out that $\mathcal{B}^{\mathrm{tr}}(s_{1},\ldots,s_{n})$ and $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}(s_{1},\ldots,s_{n})$ defined via the generators $\hat{b}_{i}$’s in (146)–(153) satisfy (157) and (158). This fact can be checked case by case. In fact for the generic situation where the adjacent nodes $i,j$ are apart from boundaries of a Dynkin diagram, (157) follows from (146) or (150) by setting $c_{i}=\frac{s_{i+1}}{s_{i}},\bar{c}_{i}=\frac{s_{i}}{s_{i+1}}$ and $d_{i}=\frac{{\mathrm{i}}\epsilon}{1-p^{2}}$. As an example of non generic cases, consider $U_{p}(D^{(2)}_{n+1})$ for which (158) matters for $(i,j)=(1,0)$ and $(n-1,n)$. It indeed holds because $\hat{b}_{i}$ in either case is given by (150) hence $c_{i}\bar{c}_{i}+d_{i}^{2}(1-p^{2})^{2}=0$ is valid. By such verification one can establish

Proposition 12**.**

$\mathcal{B}^{\mathrm{tr}}(s_{1},\ldots,s_{n})$* and $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}(s_{1},\ldots,s_{n})\,(r\leq k,r^{\prime}\leq k^{\prime})$ are Onsager coideals.*

Synthesizing Proposition 11 and Proposition 12 we arrive at the conclusion of this section:

Theorem 13**.**

For $(K,R,\mathcal{B})=(K^{\mathrm{tr}},R^{\mathrm{tr}},\mathcal{B}^{\mathrm{tr}})$ and $(K^{k,k^{\prime}},R^{r,r^{\prime}},\mathcal{B}^{k,k^{\prime}})$ with $r\leq k,r^{\prime}\leq k^{\prime}$, the matrix product construction (139)–(140) leads to the intertwiner $K(z,q;s_{1},\ldots,s_{n})$ of the Onsager coideal $\mathcal{B}(s_{1},\ldots,s_{n})$ which satisfies the reflection equation

[TABLE]

We note that the constraint like (157) and (158) has also emerged in [16, Rem.2] as a necessary condition for the existence of the intertwining $K$ matrix for the symmetric tensor representations of $U_{p}(A^{(1)}_{n-1})$.

7. Summary

We have introduced the Onsager coideals $\mathcal{B}(s_{1},\ldots,s_{n})$ with generators in (146)–(153) and determined their intertwiner, the $K$ matrices, characterized by (154) for the fundamental representations of $U_{p}(A^{(1)}_{n-1})$ and the spin representations of $U_{p}(B^{(1)}_{n})$, $U_{p}(D^{(1)}_{n})$ and $U_{p}(D^{(2)}_{n+1})$ specified in (12). The results are expressed in the matrix product form as in (139) and (140), and satisfy the reflection equation (159). (The companion $R$ matrices are also given by the matrix product formulas in (61)–(64).)

This paper elucidates a prominent role of Onsager coideals in the study of reflection $K$ matrices. It also sheds new light on the entirely different approach in [17] based on the three dimensional integrability and the quantized reflection equation digested in Appendix A. In fact the conjectural results there for $U_{p}(B^{(1)}_{n})$, $U_{p}(D^{(1)}_{n})$, $U_{p}(D^{(2)}_{n+1})$ have been established here for the first time. We have stressed two essential aspects; connection to the Onsager coideals and matrix product construction by $q$-Bosons. Their interrelation deserves a further investigation.

Appendix A Quantized reflection equation

Let us recall the basic ingredients that led to the matrix product solutions (78), (79) to the reflection equation. We regard $L$ (57) as an operator in $\in\mathrm{End}({\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2}\otimes F_{q^{2}})$ defined by

[TABLE]

where $u_{0},u_{1}$ are bases of ${\mathbb{C}}^{2}={\mathbb{C}}u_{0}\oplus{\mathbb{C}}u_{1}$. We also introduce its variants corresponding to (58)–(60):

[TABLE]

Similarly we regard $G$ (75) as an operator in $\mathrm{End}({\mathbb{C}}^{2}\otimes F_{q})$ defined by

[TABLE]

The quantized reflection equation [17] is the reflection equation up to conjugation:

[TABLE]

where the conjugation operator $\mathcal{K}\in\mathrm{End}(F_{q^{2}}\otimes F_{q}\otimes F_{q^{2}}\otimes F_{q})$ is called 3D $K$. It is depicted as follows:

[TABLE]

The equality (164) is to hold in $\mathrm{End}(\overset{1}{{\mathbb{C}}^{2}}\otimes\overset{2}{{\mathbb{C}}^{2}}\otimes\overset{3}{F_{q^{2}}}\otimes\overset{4}{F_{q}}\otimes\overset{5}{F_{q^{2}}}\otimes\overset{6}{F_{q}})$, where the labels $1,\ldots,6$ are assigned just for explanation. The vertical straight line on each side is the reflecting boundary. The other arrows labeled with $1,2$ signify ${\mathbb{C}}^{2}$. At each vertex one should imagine an extra arrow penetrating it perpendicularly to the sheet from the back to the front. Those going through $4,6$ (resp. $3,5$) are assigned with $F_{q}$ (resp. $F_{q^{2}}$). One may regard $\mathcal{K}_{3456}$ in the left (resp. right) hand side as a point in the back (resp. front) of the diagram where the four arrows going toward (resp. coming from) the vertices $3,4,5,6$ intersect. If the labels of these spaces are exhibited, (164) becomes

[TABLE]

In terms of the operators in (57), (75), (161) and (162), this is expressed as

[TABLE]

for any $i_{0},j_{0},i_{3},j_{3}\in\{0,1\}$, where the sums are taken over $i_{1},i_{2},j_{1},j_{2}\in\{0,1\}$. The explicit form of the sixteen equations (166) is available in [15, App. A]999The operators ${\rm{\bf A}}^{\!\pm},{\rm{\bf K}},{\rm{\bf a}}^{\!\pm},{\rm{\bf k}}$ in [15] has the same meaning as the ones in this paper, although the 3D $K$ is denoted by $\mathscr{K}$ rather than $\mathcal{K}$ here.. We remark that (166) becomes actually independent of the parameters $\epsilon,s,s^{\prime}$ entering $L$ (57) and $G$ (75) as long as the constraint (96) is satisfied. The 3D $K$ $\mathcal{K}$ is uniquely determined from the quantized reflection equation up to normalization as in [15, eqs.(3.27)-(3.28)]. Then the matrix product solutions (78)–(80) to the ordinary reflection equation are constructed following the reduction method explained in [17, Sec.5] assuming [17, eq.(78)] for $U_{p}(\mathfrak{g}^{r,r^{\prime}})$.

Appendix B Generators of $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}$

Let us write down the generators $b_{0},\ldots,b_{n}$ of $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}$ specified in (35)–(38) explicitly. For any $\mathcal{B}^{r,r^{\prime}}_{k,k^{\prime}}$ with $r\leq k$ and $r^{\prime}\leq k^{\prime}$ we have

[TABLE]

In what follows we list $b_{0},b_{n}$ only.

[TABLE]

Acknowledgments

A.K. thanks Vladimir Mangazeev for sending the preprint [20]. He 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.