Quadratic algebras based on $SL(NM)$ elliptic quantum $R$-matrices

Abstract

We construct a quadratic quantum algebra based on the dynamical $RLL$-relation for the quantum $R$-matrix related to $SL(NM)$-bundles with a nontrivial characteristic class over an elliptic curve. This $R$-matrix simultaneously generalizes the elliptic nondynamical Baxter–Belavin and the dynamical Felder $R$-matrices, and the obtained quadratic relations generalize both the Sklyanin algebra and the relations in the Felder–Tarasov–Varchenko elliptic quantum group, which are reproduced in the respective particular cases $M=1$ and $N=1$.

Appendix A. Elliptic functions and their properties

In the definitions of $R$-matrices in this paper, we use the Kronecker elliptic functions

$$\begin{aligned} \, &\varphi_\alpha(u,x+\omega_\alpha) =\phi(u,x+\omega_\alpha)e^{(2\pi i/N)\alpha_2u},\qquad \omega_\alpha=\frac{\alpha_1+\alpha_2\tau}{N}, \\ &\phi(u,x)=\frac{\theta'(0)\theta(u+x)}{\theta(u)\theta(x)}, \end{aligned} $$

(A.1)

which are expressed through the odd theta function

$$\theta(u)=-\sum_{k\in\mathbb Z} \exp\biggl(\pi i\tau\biggl(k+\frac{1}{2}\biggr)^{\!2} +2\pi i\biggl(k+\frac{1}{2}\biggr) \biggl(u+\frac{1}{2}\biggr)\biggr). $$

(A.2)

Here, $\tau$ — a complex parameter with $\operatorname{Im}\tau>0$ — is the modular parameter of the elliptic curve underlying all elliptic functions

The main tool for the derivation of the quadratic relations is the addition formula (also known as the genus-one Fay identity) for the Kronecker functions

$$\phi(z,x)\phi(w,y)=\phi(z-w,x)\phi(w,x+y) +\phi(w-z,y)\phi(z,x+y) $$

(A.3)

and its degenerations corresponding to coincident values of variables

$$\begin{aligned} \, &\phi(z,x)\phi(z,y)=\phi(z,x+y)(E_1(z)+E_1(x)+E_1(y)-E_1(x+y+z)), \\ &\phi(z,x)\phi(z,-x)=E_2(z)-E_2(x), \end{aligned} $$

(A.4)

where the Eisenstein functions are used:

$$E_1(z)=\frac{\theta'(z)}{\theta(z)},\qquad E_2(z)=-E_1'(z) $$

(A.5)

In the definition of the Baxter–Belavin quantum $R$-matrix, the basis matrices $T_\alpha$ are used. They are defined as

$$\begin{aligned} \, &T_\alpha=T_{(\alpha_1, \alpha_2)} =e^{\pi i\alpha_1\alpha_2/N}Q^{\alpha_1}\Lambda^{\alpha_2}, \\ &Q_{jk}=\delta_{jk}e^{2\pi ik/N},\qquad \Lambda_{jk}=\begin{cases} 1, &j+1=k\mod N, \\ 0 &\text{otherwise}. \end{cases} \end{aligned} $$

(A.6)

Appendix B. An example of calculation verifying the $$RLL$$ -relation

We consider, for example, the $(E_{ij})_a(E_{ik})_b$-component of the $RLL$-relation:

$$\begin{aligned} \, R_{12}^{\mathrm{BB}}(\hbar,z_{12}) L_1^{ij}(z_1) L_2^{ik}(z_2) &=L_2^{ik}(z_2)L_1^{ij}(z_1)\phi(\hbar,-q_{jk}^{\{1\}})+L_2^{ij}(z_2) L_1^{ik}(z_1) R_{12}^{\mathrm{BB}}(q^{\{1\}}_{kj},z_{12}),\qquad j\ne k. \end{aligned}$$

(B.1)

This relation is given in $(N\times N)$-matrices. Decomposing it in the basis $T_\alpha$, we obtain the following scalar relations in components $(T_\alpha)_1(T_\beta)_2$ (after canceling all exponential factors):

$$\begin{aligned} \, &\theta(z_2+q_i^{\{2\}}-q_k^{\{1\}}+\omega_\beta)t_{ki}^\beta \cdot\theta(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_\alpha)t_{ji}^\alpha \cdot\phi(\hbar,-q_{jk}^{\{1\}})= \\ &=\sum_\gamma\varkappa_{\gamma \alpha} \varkappa_{\beta \gamma}[\phi(z_{12},\hbar+\omega_\gamma) \theta(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma})\times{} \\ &\hphantom{={}}\times t_{ji}^{\alpha-\gamma}\theta(z_2+q_i^{\{2\}} -q_k^{\{1\}}+\omega_{\beta+\gamma})t_{ki}^{\beta+\gamma} -\theta(z_2+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma})\times{} \\ &\hphantom{={}}\times t_{ji}^{\alpha-\gamma} \theta(z_1+q_i^{\{2\}}-q_k^{\{1\}}+\omega_{\beta+\gamma}) t_{ki}^{\beta+\gamma}\phi(z_{12},q_{kj}^{\{1\}} +\omega_{\alpha-\beta-\gamma})]. \end{aligned}$$

Moving all $t_{ab}$ to the right yields

$$\begin{aligned} \, &\theta(z_2+q_i^{\{2\}}-q_k^{\{1\}}+\omega_\beta) \theta(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\hbar+\omega_\alpha) \phi(\hbar,-q_{jk}^{\{1\}})t_{ki}^\beta t_{ji}^\alpha= \\ &=\sum_\gamma\varkappa_{\gamma\alpha}\varkappa_{\beta\gamma} [\phi(z_{12},\hbar+\omega_\gamma) \theta(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma})\times{} \\ &\hphantom{={}}\times\theta(z_2+q_i^{\{2\}}-q_k^{\{1\}}+\hbar+\omega_{\beta+\gamma})-{} \\ &\hphantom{={}}-\theta(z_2+q_i^{\{2\}} -q_j^{\{1\}}+\omega_{\alpha-\gamma}) \theta(z_1+q_i^{\{2\}}-q_k^{\{1\}}+\hbar+\omega_{\beta+\gamma})\times{} \\ &\hphantom{={}}\times\phi(z_{12},q_{kj}^{\{1\}}+\omega_{\alpha-\beta-\gamma})] t_{ji}^{\alpha-\gamma}t_{ki}^{\beta+\gamma}. \end{aligned}$$

We divide both sides by $\theta(z_2+q_i^{\{2\}}-q_k^{\{1\}}+\omega_\beta) \theta(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\hbar+\omega_\alpha)$ and consider the expression in the brackets in the right-hand side. It can be simplified:

$$\begin{aligned} \, &\phi(z_{12},\hbar+\omega_\gamma) \frac{\theta(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}) \theta(z_2+q_i^{\{2\}}-q_k^{\{1\}}+\hbar+\omega_{\beta+\gamma})} {\theta(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\hbar+\omega_\alpha) \theta(z_2+q_i^{\{2\}}-q_k^{\{1\}}+\omega_\beta)}-{} \\ &\hphantom{={}}-\phi(z_{12},q_{kj}^{\{1\}}+\omega_{\alpha-\beta-\gamma}) \frac{\theta(z_1+q_i^{\{2\}}-q_k^{\{1\}}+\hbar+\omega_{\beta+\gamma}) \theta(z_2+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma})} {\theta(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\hbar+\omega_\alpha) \theta(z_2+q_i^{\{2\}}-q_k^{\{1\}}+\omega_\beta)}= \\ &=\phi(z_{12},\hbar+\omega_\gamma) \frac{\phi(z_2+q_i^{\{2\}}-q_k^{\{1\}}+\omega_\beta,\hbar+\omega_\gamma)} {\phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma},\hbar+\omega_\gamma)}-{} \\ &\hphantom{={}}-\phi(z_{12}, q_{kj}^{\{1\}}+\omega_{\alpha-\beta-\gamma}) \frac{\phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\hbar+\omega_\alpha, q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha})} {\phi(z_2+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha})}= \\ &=\frac{\phi(z_{12},\hbar+\omega_\gamma) \phi(z_2+q_i^{\{2\}}-q_k^{\{1\}}+\omega_\beta,\hbar+\omega_\gamma) } {\phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma},\hbar+\omega_\gamma) }\times{} \\ &\hphantom{={}}\qquad\qquad\times\frac{\phi(z_2+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha})} {\phi(z_2+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha})}-{} \\ &\hphantom{={}}-\frac{\phi(z_{12},q_{kj}^{\{1\}}+\omega_{\alpha-\beta-\gamma}) \phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\hbar+\omega_\alpha, q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha}) } {\phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma},\hbar+\omega_\gamma) }\times{}\\ &\hphantom{={}}\qquad\qquad\times\frac{\phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma},\hbar+\omega_\gamma)} {\phi(z_2+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha})}. \end{aligned}$$

Applying the Fay identity to $\phi$ and using the property $\phi(x,-x)=0$, we obtain

$$\begin{aligned} \, &\phi(z_2+q_i^{\{2\}}-q_k^{\{1\}}+\omega_\beta,\hbar+\omega_\gamma) \phi(z_2+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha})= \\ &\qquad=\phi(q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha},\hbar+\omega_\gamma) \phi(z_2+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, q_{jk}^{\{1\}}+\hbar+\omega_{\beta+2 \gamma-\alpha}), \\ &\phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\hbar+\omega_\alpha, q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha}) \phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma},\hbar+\omega_\gamma)= \\ &\qquad=\phi(\hbar+\omega_\gamma,q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha}) \phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, q_{jk}^{\{1\}}+\hbar+\omega_{\beta+2 \gamma-\alpha}). \end{aligned}$$

We can pull the factor $\phi(\hbar+\omega_\gamma,q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha})$ out of the numerator; by Fay’s identity, the remaining part is then exactly equal to the denominator:

$$\begin{aligned} \, &\phi(z_{12}, \hbar+\omega_\gamma) \phi(z_2+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, q_{jk}^{\{1\}}+\hbar+\omega_{\beta+2 \gamma-\alpha})-{} \\ &\qquad\qquad{}-\phi(z_{12},q_{kj}^{\{1\}}+\omega_{\alpha-\beta-\gamma}) \phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, q_{jk}^{\{1\}}+\hbar+\omega_{\beta+2 \gamma-\alpha})= \\ &\qquad=\phi(z_1+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, \hbar+\omega_\gamma) \phi(z_2+q_i^{\{2\}}-q_j^{\{1\}}+\omega_{\alpha-\gamma}, q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha}). \end{aligned}$$

Using this simplification, we obtain the required relation without spectral parameters:

$$\sum_\gamma\varkappa_{\gamma\alpha}\varkappa_{\beta\gamma} \phi(\hbar+\omega_\gamma,q_{jk}^{\{1\}}+\omega_{\beta+\gamma-\alpha}) t_{ji}^{\alpha-\gamma}t_{ki}^{\beta+\gamma} =\phi(\hbar,-q_{jk}^{\{1\}})t_{ki}^\beta t_{ji}^\alpha.$$

All other relations can be verified similarly by considering the other components of the $RLL$-relation.