Yang-Baxter equations with two Planck constants

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 $\mathrm{GL}(N)$ Sklyanin elliptic algebras. Then we proceed to a natural generalization of the Baxter-Belavin quantum $R$-matrix to the case $\mathrm{Mat}(N,\mathbb{C}{)}^{\otimes 2}\otimes \mathrm{Mat}(M,\mathbb{C}{)}^{\otimes 2}$. It can be viewed as symmetric form of $\mathrm{GL}(NM)$ $R$-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.

20 pages, minor corrections