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

References in corpus (4)

• Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations
• Classical integrable systems and soliton equations related to eleven-vertex R-matrix
• Quantum Baxter-Belavin R-matrices and multidimensional Lax pairs for Painleve VI
• Parameter-dependent associative Yang-Baxter equations and Poisson brackets

Cited by in corpus (7)

• Trigonometric integrable tops from solutions of associative Yang-Baxter equation
• Higher rank 1+1 integrable Landau-Lifshitz field theories from associative Yang-Baxter equation
• Relativistic elliptic matrix tops and finite Fourier transformations
• Higher order analogues of unitarity condition for quantum R-matrices
• Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices
• Odd supersymmetrization of elliptic R-matrices
• R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\mathrm{BC}}_n$ type