Yang-Baxter equations with two Planck constants

TL;DR

This paper explores Yang-Baxter equations related to associative analogs, leading to finite-dimensional quantum algebras, and introduces a symmetric generalization of the Baxter-Belavin R-matrix involving two Planck constants.

Contribution

It generalizes the Baxter-Belavin R-matrix to a symmetric form with two Planck constants and studies the resulting algebraic structures and equations.

Findings

Finite-dimensional quantum algebras from Yang-Baxter equations

Symmetric form of GL(NM) R-matrix with two Planck constants

R-matrices satisfy quadratic and cubic Yang-Baxter-like equations

Abstract

We consider Yang-Baxter equations arising from its associative analog and study corresponding exchange relations. They generate finite-dimensional quantum algebras which have form of coupled ${\rm GL}(N)$ Sklyanin elliptic algebras. Then we proceed to a natural generalization of the Baxter-Belavin quantum $R$-matrix to the case ${\rm Mat}(N,\mathbb C)^{\otimes 2}\otimes {\rm Mat}(M,\mathbb C)^{\otimes 2}$. It can be viewed as symmetric form of ${\rm GL}(NM)$ $R$-matrix in the sense that the Planck constant and the spectral parameter enter (almost) symmetrically. Such type (symmetric) $R$-matrices are also shown to satisfy the Yang-Baxter like quadratic and cubic equations.