GL(NM) quantum dynamical $R$-matrix based on solution of the associative Yang-Baxter equation

TL;DR

This paper constructs a new class of dynamical R-matrices for GL(NM) using solutions of the associative Yang-Baxter equation, unifying and extending known elliptic and vertex R-matrices.

Contribution

It introduces a novel GL(NM)-valued dynamical R-matrix derived from associative Yang-Baxter solutions, generalizing Felder's and Baxter-Belavin's R-matrices.

Findings

Reproduces Felder's R-matrix for N=1

Provides elliptic and degenerate R-matrices for M=1

Unifies different types of R-matrices under a common framework

Abstract

In this letter we construct ${\rm GL}_{NM}$-valued dynamical $R$-matrix by means of unitary skew-symmetric solution of the associative Yang-Baxter equation in the fundamental representation of ${\rm GL}_{N}$. In $N=1$ case the obtained answer reproduces the ${\rm GL}_{M}$-valued Felder's $R$-matrix, while in the $M=1$ case it provides the ${\rm GL}_{N}$ $R$-matrix of vertex type including the Baxter-Belavin's elliptic one and its degenerations.

Equations

$$R_{12}^{\hslash }(z_{12})R_{23}^{\eta }(z_{23})=R_{13}^{\eta }(z_{13})R_{12}^{\hslash -\eta }(z_{12})+R_{23}^{\eta -\hslash }(z_{23})R_{13}^{\hslash }(z_{13}),z_{ab}=z_a-z_b.$$

$$R_{13}^{\hslash }(z_{13})=\sum _{i,j,k,l=1}^N{R}_{ijkl}(\hslash ,z_{13})e_{ij}\otimes 1_N\otimes e_{kl},$$

$$\begin{array}[]{c}\displaystyle{R^{\hbar}_{12}(z)=-R_{21}^{-\hbar}(-z)=-P_{12}R_{12}^{-\hbar}(-z)P_{12}\,,\qquad P_{12}=\sum\limits_{i,j=1}^{N}e_{ij}\otimes e_{ji}}\end{array}$$

$$\begin{array}[]{c}\displaystyle{R^{\hbar}_{12}(z)R^{\hbar}_{21}(-z)=(\wp(\hbar)-\wp(z))\,1_{N}\otimes 1_{N}\,,}\end{array}$$

$$R_{12}^{\hslash }(z_{12})R_{13}^{\hslash }(z_{13})R_{23}^{\hslash }(z_{23})=R_{23}^{\hslash }(z_{23})R_{13}^{\hslash }(z_{13})R_{12}^{\hslash }(z_{12}).$$

$$\begin{array}[]{c}\displaystyle{\phi(\hbar,z_{12})\phi(\eta,z_{23})=\phi(\eta,z_{13})\phi(\hbar-\eta,z_{12})+\phi(\eta-\hbar,z_{23})\phi(\hbar,z_{13})\,,}\end{array}$$

$$\begin{array}[]{c}\displaystyle{\phi(\hbar,z)=\frac{\vartheta^{\prime}(0)\vartheta(\hbar+z)}{\vartheta(\hbar)\vartheta(z)}\,,\quad\vartheta(x)=\displaystyle{\sum_{k\in\mathbb{Z}}}\exp\left(\pi\imath\tau(k+\frac{1}{2})^{2}+2\pi\imath(x+\frac{1}{2})(k+\frac{1}{2})\right)\,,}\end{array}$$

$$\begin{array}[]{c}\displaystyle{R^{\hbox{\tiny{F}}}_{12}(\hbar,z_{1},z_{2}|\,q)=R^{\hbox{\tiny{F}}}_{12}(\hbar,z_{1}-z_{2}|\,q)=}\\ \\ \displaystyle{=\phi(\hbar,z_{1}-z_{2})\sum\limits_{i=1}^{M}E_{ii}\otimes E_{ii}+\sum\limits_{i\neq j}^{M}E_{ij}\otimes E_{ji}\,\phi(z_{1}-z_{2},q_{ij})+\sum\limits_{i\neq j}^{M}E_{ii}\otimes E_{jj}\,\phi(\hbar,-q_{ij})\,,}\end{array}$$

$$\begin{array}[]{c}\displaystyle{q_{ij}=q_{i}-q_{j}\,,}\end{array}$$

$$\begin{array}[]{c}\displaystyle{R^{\hbar}_{12}(z_{1},z_{2}|\,q)R^{\hbar}_{13}(z_{1},z_{3}|\,q-\hbar^{(2)})R^{\hbar}_{23}(z_{2},z_{3}|\,q)=}\\ \\ \displaystyle{\hskip 113.81102pt=R^{\hbar}_{23}(z_{2},z_{3}|\,q-\hbar^{(1)})R^{\hbar}_{13}(z_{1},z_{3}|\,q)R^{\hbar}_{12}(z_{1},z_{2}|\,q-\hbar^{(3)})\,,}\end{array}$$

$$\begin{array}[]{c}\displaystyle{R^{\hbar}_{12}(z_{1},z_{2}|\,q+\hbar^{(3)})=P_{3}^{\hbar}\,R^{\hbar}_{12}(z_{1},z_{2}|\,q)\,P_{3}^{-\hbar}\,,\quad P_{3}^{\hbar}=\sum\limits_{k=1}^{M}1_{M}\otimes 1_{M}\otimes E_{kk}\exp\Big{(}\hbar\frac{\partial}{\partial q_{k}}\Big{)}\,.}\end{array}$$

$$\begin{array}[]{c}\displaystyle{{\bf R}^{\hbar}_{1^{\prime}2^{\prime}12}(z,w)=\sum\limits_{i=1}^{M}\stackrel{{\scriptstyle 1^{\prime}}}{{E}}_{ii}\otimes\stackrel{{\scriptstyle 2^{\prime}}}{{E}}_{ii}\otimes\,R^{\hbar}_{12}(z-w)+\sum\limits_{i\neq j}^{M}\stackrel{{\scriptstyle 1^{\prime}}}{{E}}_{ij}\otimes\stackrel{{\scriptstyle 2^{\prime}}}{{E}}_{ji}\otimes R_{12}^{\,q_{ij}}(z-w)+}\\ \displaystyle{+\sum\limits_{i\neq j}^{M}\stackrel{{\scriptstyle 1^{\prime}}}{{E}}_{ii}\otimes\stackrel{{\scriptstyle 2^{\prime}}}{{E}}_{jj}\otimes\stackrel{{\scriptstyle 1}}{{1}}_{N}\otimes\stackrel{{\scriptstyle 2}}{{1}}_{N}\phi(\hbar,-q_{ij})\,,}\end{array}$$

$$\begin{array}[]{c}\displaystyle{\mathbf{R}^{\hbar}_{1^{\prime}2^{\prime}12}(z_{1},z_{2}\mid q)\mathbf{R}^{\hbar}_{1^{\prime}3^{\prime}13}(z_{1},z_{3}\mid q-\hbar^{(2)})\mathbf{R}^{\hbar}_{2^{\prime}3^{\prime}23}(z_{2},z_{3}\mid q)=}\\ \\ \displaystyle{=\mathbf{R}^{\hbar}_{2^{\prime}3^{\prime}23}(z_{2},z_{3}\mid q-\hbar^{(1)})\mathbf{R}^{\hbar}_{1^{\prime}3^{\prime}13}(z_{1},z_{3}\mid q)\mathbf{R}^{\hbar}_{1^{\prime}2^{\prime}12}(z_{1},z_{2}\mid q-\hbar^{(3)})\,,}\end{array}$$

$$\begin{array}[]{c}\displaystyle{\mathbf{R}^{\hbar}_{1^{\prime}2^{\prime}12}(z_{1},z_{2}\mid q+\hbar^{(3)})={\bf P}_{3^{\prime}}^{\hbar}\,\,\mathbf{R}^{\hbar}_{1^{\prime}2^{\prime}12}(z_{1},z_{2}\mid q)\,{\bf P}_{3^{\prime}}^{-\hbar}\,,}\\ \\ \displaystyle{{\bf P}_{3^{\prime}}^{\hbar}=\sum\limits_{k=1}^{M}\stackrel{{\scriptstyle 1^{\prime}}}{{1}}_{M}\otimes\stackrel{{\scriptstyle 2^{\prime}}}{{1}}_{M}\otimes\stackrel{{\scriptstyle 3^{\prime}}}{{E}}_{kk}\otimes\stackrel{{\scriptstyle 1}}{{1}}_{N}\otimes\stackrel{{\scriptstyle 2}}{{1}}_{N}\otimes\stackrel{{\scriptstyle 3}}{{1}}_{N}\exp\Big{(}\hbar\frac{\partial}{\partial q_{k}}\Big{)}\,.}\end{array}$$

$$R_{ab}^{\hslash }(z_{ab})R_{bc}^{\eta }(z_{bc})=R_{bc}^{\eta -\hslash }(z_{bc})R_{ac}^{\hslash }(z_{ac})+R_{ac}^{\eta }(z_{ac})R_{ab}^{\hslash -\eta }(z_{ab}),$$

$$R_{ab}^{\hslash }(z_{ab})R_{ac}^{\eta }(z_{ac})R_{bc}^{\hslash }(z_{bc})-R_{bc}^{\eta }(z_{bc})R_{ac}^{\hslash }(z_{ac})R_{ab}^{\eta }(z_{ab})=R_{ac}^{\hslash +\eta }(z_{ac})(\wp (\hslash )-\wp (\eta )),$$

$$\begin{array}[]{c}\displaystyle{R^{\eta}_{ab}R^{\hbar}_{ac}R^{\eta}_{bc}+R^{\hbar}_{ab}R^{\eta}_{ac}R^{\hbar}_{bc}=R^{\eta}_{bc}R^{\hbar}_{ac}R^{\eta}_{ab}+R^{\hbar}_{bc}R^{\eta}_{ac}R^{\hbar}_{ab}\,,\quad R_{ab}^{\hbar}=R^{\hbar}_{ab}(z_{a}-z_{b})\,,}\end{array}$$

$$\begin{array}[]{c}\displaystyle{R^{q_{ik}}_{12}(z_{12})R^{q_{kj}}_{13}(z_{13})R^{q_{ik}}_{23}(z_{23})+\phi(\hbar,q_{ik})\phi(\hbar,q_{ki})R^{q_{ij}}_{13}(z_{13})=}\\ \\ \displaystyle{=R^{q_{kj}}_{23}(z_{23})R^{q_{ik}}_{13}(z_{13})R^{q_{kj}}_{12}(z_{12})+\phi(\hbar,q_{kj})\phi(\hbar,q_{jk})R^{q_{ij}}_{13}(z_{13})\,.}\end{array}$$

$$\begin{array}[]{c}\displaystyle{R^{q_{ik}}_{12}(z_{12})R^{q_{kj}}_{13}(z_{13})R^{q_{ik}}_{23}(z_{23})-R^{q_{kj}}_{23}(z_{23})R^{q_{ik}}_{13}(z_{13})R^{q_{kj}}_{12}(z_{12})=}\\ \\ \displaystyle{=(\wp(q_{ik})-\wp(q_{kj}))R^{q_{ik}+q_{kj}}_{13}(z_{13})=(\wp(q_{ik})-\wp(q_{kj}))R^{q_{ij}}_{13}(z_{13})}\end{array}$$

$$\begin{array}[]{c}\displaystyle{\phi(\hbar,q_{ik})\phi(\hbar,q_{ki})=\wp(\hbar)-\wp(q_{ik})\,,\quad\quad\phi(\hbar,q_{kj})\phi(\hbar,q_{jk})=\wp(\hbar)-\wp(q_{kj})\,.}\end{array}$$

$$\begin{array}[]{c}\displaystyle{R_{12}^{\hbar}(z)=\hbar^{-1}1_{N}\otimes 1_{N}+r_{12}(z)+O(\hbar)\,.}\end{array}$$

$$\begin{array}[]{c}\displaystyle{[r_{12}(z_{12}),r_{13}(z_{13})]+[r_{12}(z_{12}),r_{23}(z_{23})]+[r_{13}(z_{13}),r_{23}(z_{23})]=0\,.}\end{array}$$

$$\begin{array}[]{c}\displaystyle{[r_{12}(z_{12}),r_{13}(z_{13})]+[r_{12}(z_{12}),r_{23}(z_{23})]+[r_{13}(z_{13}),r_{23}(z_{23})]+}\\ \\ \displaystyle{[\hat{\partial}_{1},r_{23}(z_{23})]-[\hat{\partial}_{2},r_{13}(z_{13})]+[\hat{\partial}_{3},r_{12}(z_{12})]=0\,,}\end{array}$$

$$\begin{array}[]{c}\displaystyle{\hat{\partial}_{3}=\sum\limits_{k=1}^{M}1_{M}\otimes 1_{M}\otimes E_{kk}\,\partial_{q_{k}}\,,\quad\quad P_{3}^{\hbar}\stackrel{{\scriptstyle(\ref{a09})}}{{=}}{1_{M}}^{\otimes 3}+\hbar\,\hat{\partial}_{3}+O(\hbar^{2})\,.}\end{array}$$

$$\begin{array}[]{c}\displaystyle{{\bf r}_{1^{\prime}2^{\prime}12}(z)=\sum\limits_{i=1}^{M}\stackrel{{\scriptstyle 1^{\prime}}}{{E}}_{ii}\otimes\stackrel{{\scriptstyle 2^{\prime}}}{{E}}_{ii}\otimes\,r_{12}(z)+\sum\limits_{i\neq j}^{M}\stackrel{{\scriptstyle 1^{\prime}}}{{E}}_{ij}\otimes\stackrel{{\scriptstyle 2^{\prime}}}{{E}}_{ji}\otimes R_{12}^{\,q_{ij}}(z)\,,}\end{array}$$

$$\begin{array}[]{c}\displaystyle{[\mathbf{r}_{1^{\prime}2^{\prime}12}(z_{12}),\mathbf{r}_{1^{\prime}3^{\prime}13}(z_{13})]+[\mathbf{r}_{1^{\prime}2^{\prime}12}(z_{12}),\mathbf{r}_{2^{\prime}3^{\prime}23}(z_{23})]+[\mathbf{r}_{1^{\prime}3^{\prime}13}(z_{13}),\mathbf{r}_{2^{\prime}3^{\prime}23}(z_{23})]+}\\ \\ \displaystyle{+[\hat{\partial}_{1^{\prime}},\mathbf{r}_{2^{\prime}3^{\prime}23}(z_{23})]-[\hat{\partial}_{2^{\prime}},\mathbf{r}_{1^{\prime}3^{\prime}13}(z_{13})]+[\hat{\partial}_{3^{\prime}},\mathbf{r}_{1^{\prime}2^{\prime}12}(z_{12})]=0\,.}\end{array}$$

$$\begin{array}[]{c}\displaystyle{\hat{\partial}_{3^{\prime}}=\sum\limits_{k=1}^{M}\stackrel{{\scriptstyle 1^{\prime}}}{{1}}_{M}\otimes\stackrel{{\scriptstyle 2^{\prime}}}{{1}}_{M}\otimes\stackrel{{\scriptstyle 3^{\prime}}}{{E}}_{kk}\otimes\stackrel{{\scriptstyle 1}}{{1}}_{N}\otimes\stackrel{{\scriptstyle 2}}{{1}}_{N}\otimes\stackrel{{\scriptstyle 3}}{{1}}_{N}\partial_{q_{k}}\,,\quad{\bf P}_{3^{\prime}}^{\hbar}\stackrel{{\scriptstyle(\ref{a201})}}{{=}}{1_{MN}}^{\otimes 3}+\hbar\,\hat{\partial}_{3^{\prime}}+O(\hbar^{2})\,.}\end{array}$$

Full text

Yang-Baxter equations.

Consider a matrix-valued function $R_{12}^{\hbar}(z)\in{\rm Mat}(N,\mathbb{C})^{\otimes 2}$, which solves the $\@@underline{\hbox{associative Yang-Baxter equation}}$ [9, 19]:

[TABLE]

Here, following notations of the Quantum Inverse Scattering Method [21], an operator $R_{ab}^{\hbar}(z)$ in (1) is considered as ${\rm Mat}(N,\mathbb{C})^{\otimes 3}$-valued. It acts non-trivially in the $a$-th and $b$-th tensor components only. For example, $R_{13}^{\hbar}(z)$ is of the form

[TABLE]

where the set $\{e_{ij}\}$ is the standard basis in ${\rm Mat}(N,\mathbb{C})$, $1_{N}$ – is the identity matrix in ${\rm Mat}(N,\mathbb{C})$ and $R_{ijkl}(\hbar,z_{12})$ are functions of complex variables $\hbar$ (the Planck constant) and $z$ (the spectral parameter).

Let the solution of (1) satisfies also the properties of the $\@@underline{\hbox{skew-symmetry}}$

[TABLE]

and $\@@underline{\hbox{unitarity}}$

[TABLE]

where $\wp(x)$ – is the Weierstrass $\wp$-function. We assume that it is equal to $1/\sinh^{2}(x)$ or $1/x^{2}$ for trigonometric (hyperbolic) or rational $R$-matrices respectively. Notice that solution of (1) with the properties (3)-(4) is a true quantum $R$-matrix of vertex type, i.e. it satisfies the $\@@underline{\hbox{quantum (non-dynamical) Yang-Baxter equation}}$111The latter statement is easily verified. See e.g. [15].:

[TABLE]

Equation (1) can be view as matrix extension of the $\@@underline{\hbox{genus one Fay trisecant identity}}$:

[TABLE]

which coincides with (1) in scalar ($N=1$) case. It plays a crucial role in the theory of classical and quantum integrable systems [13, 22, 5]. Solution of (6) satisfying the (scalar versions of) properties (3)-(4) is the $\@@underline{\hbox{Kronecker function}}$:

[TABLE]

where $\hbox{Im}(\tau)>0$. Its trigonometric and rational limits are given by $\coth(\hbar)+\coth(z)$ and $\hbar^{-1}+z^{-1}$ respectively. Similarly, the elliptic solution of (1) with properties (3)-(4) is known [19] to be given by the Baxter-Belavin’s $R$-matrix [4]. The trigonometric solutions were classified in [20]. They include the XXZ $R$-matrix, its 7-vertex deformation [6] and their ${\rm GL}_{N}$ generalizations [2] (see a brief review in [12]). The rational solutions consist of the XXX $R$-matrix, its 11-vertex deformation [6] and their ${\rm GL}_{N}$ generalizations [23, 14] – deformations of the ${\rm GL}_{N}$ Yang’s $R$-matrix $R_{12}^{\hbar}(z)=\hbar^{-1}1_{N}\otimes 1_{N}+z^{-1}P_{12}$. Summarizing, we deal with the $R$-matrices considered as matrix generalizations of the Kronecker function (including its trigonometric and rational versions).

To formulate the main result we also need the \@@underline{\hbox{Felder's dynamical {\rm GL}_{M}$$R-matrix}} [8]:

[TABLE]

where $q_{1},...,q_{M}$ – are (free) dynamical parameters,

[TABLE]

and the set $\{E_{ij}\}$ is the standard basis in ${\rm Mat}(M,\mathbb{C})$.

The $R$-matrix (8) is a solution of the $\@@underline{\hbox{quantum dynamical Yang-Baxter equation}}$:

[TABLE]

where the shifts of the dynamical arguments $\{q_{i}\}$ are performed as follows:

[TABLE]

Quantum dynamical ${\rm GL}_{NM}$ $R$-matrix.

Consider the following ${\rm Mat}(NM,\mathbb{C})$-valued expression:

[TABLE]

where the ${\rm Mat}(NM,\mathbb{C})$ indices are represented in a way that the ${\rm Mat}(M,\mathbb{C})$-valued tensor components are numbered by the primed numbers, and the ${\rm Mat}(N,\mathbb{C})$-valued components are those without primes (as previously). Put it differently, the indices are arranged through ${\rm Mat}(NM,\mathbb{C})^{\otimes 2}\cong{\rm Mat}(M,\mathbb{C})^{\otimes 2}\otimes{\rm Mat}(N,\mathbb{C})^{\otimes 2}$. The order of tensor components is, in fact, not important. It is chosen as in (12) just to emphasize its similarity with the Felder’s $R$-matrix (8). The latter is reproduced from (12) in the $N=1$ case, when the ${\rm GL}_{N}$ $R$-matrix entering (12) turns into the Kronecker function (7).

The results of the paper are summarized in the following

Theorem Let $R_{12}^{\hbar}(z)$ be some ${\rm GL}_{N}$ quantum non-dynamical $R$-matrix satisfying the associative Yang-Baxter equation (1) and the properties (3)-(4). Then the expression (12) is a quantum dynamical $R$-matrix, i.e. it satisfies the quantum dynamical Yang-Baxter equation:

[TABLE]

where the shifts of arguments $\{q_{i}\}$ are performed similarly to (11):

[TABLE]

$\@@underline{\hbox{\em{Proof:}}}$ It is useful to write (1) as

[TABLE]

where $a,b,c$ are distinct numbers from the set $\{1,2,3\}$. Besides (15) and the properties (3)-(4) the proof of (13) uses the Yang-Baxter equation (5) for the $\mathrm{GL}_{N}$ $R$-matrix and the following cubic relation:

[TABLE]

which is true under hypothesis of the theorem. If $\hbar=\eta$ it reduces to (5). In the general case (16) leads (due to skew-symmetry of its r.h.s.) to

[TABLE]

known as the Yang-Baxter equation with two Planck constants [16]. The verification of (13) is a straightforward but cumbersome calculation. Consider, for example, the equation arising in the tensor component $\stackrel{{\scriptstyle 1^{\prime}}}{{E}}_{ij}\otimes\stackrel{{\scriptstyle 2^{\prime}}}{{E}}_{kk}\otimes\stackrel{{\scriptstyle 3^{\prime}}}{{E}}_{ji}$ with $i\neq j\neq k\neq i$:

[TABLE]

To prove it one should use (16) written in the form

[TABLE]

and the well-known property of the Kronecker function (scalar version of the unitarity)

[TABLE]

The rest of the tensor components are verified similarly. $\blacksquare$

In the elliptic case, when $R_{12}^{\hbar}(z)$ is the Baxter-Belavin’s $R$-matrix, the result of the theorem is known [18]. Similar results for the classical $r$-matrices were obtained previously by P. Etingof and O. Schiffmann [7] and later in [17, 24], where the Hitchin type systems were described on the Higgs bundles with non-trivial characteristic classes. Recently, these type models appeared in the context of $R$-matrix valued Lax pairs and quantum long-range spin chains [10, 11]. In [18] the answer (12) was verified explicitly in the elliptic case without use of the associative Yang-Baxter equation. In this respect the approach of this paper provides much simpler proof. What is more important, the answer (12) is also valid for all trigonometric and rational degenerations of the elliptic $R$-matrix (satisfying the properties required in the Theorem). In the light of results of [11] the $R$-matrix (12) is the one necessary for quantization of the (generalized) model of interacting tops.

Classical ${\rm gl}_{NM}$ $r$-matrix.

As a by-product of the Theorem we also get the classical dynamical Yang-Baxter equation for the classical $r$-matrix of the generalized interacting tops [11]. Consider the classical limit of the ${\rm GL}_{N}$ $R$-matrix from the Theorem:

[TABLE]

The coefficient $r_{12}(z)$ is the classical $r$-matrix, and the quantum Yang-Baxter equation (5) reduces in the limit (21) to the $\@@underline{\hbox{classical (non-dynamical) Yang-Baxter equation}}$:

[TABLE]

Similarly, the classical dynamical $r$-matrix appears from (8) through (21). It satisfies the $\@@underline{\hbox{classical dynamical Yang-Baxter equation}}$:

[TABLE]

which underlies the Poisson structure of the spin Calogero-Moser model [3]. Here

[TABLE]

In the same way, starting from the quantum $R$-matrix (12) one gets the classical $r$-matrix

[TABLE]

and the classical dynamical Yang-Baxter equation follows from (13):

[TABLE]

with

[TABLE]

Acknowledgments.

The second author is a Young Russian Mathematics award winner. The work was performed at the Steklov Mathematical Institute of Russian Academy of Sciences, Moscow. This work is supported by the Russian Science Foundation under grant 19-11-00062.