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arXiv · 1501.07351

Quantum Baxter-Belavin R-matrices and multidimensional Lax pairs for Painleve VI

A. Levin·M. Olshanetsky·A. Zotov
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Abstract

The quantum elliptic $R$-matrices of Baxter-Belavin type satisfy the associative Yang-Baxter equation in ${\rm Mat}(N,\mathbb C)^{\otimes 3}$. The latter can be considered as noncommutative analogue of the Fay identity for the scalar Kronecker function. In this paper we extend the list of $R$-matrix valued analogues of elliptic function identities. In particular, we propose counterparts of the Fay identities in ${\rm Mat}(N,\mathbb C)^{\otimes 2}$. As an application we construct $R$-matrix valued $2N^2\times 2N^2$ Lax pairs for the Painlevé VI equation (in elliptic form) with four free constants using ${\mathbb Z}_N\times {\mathbb Z}_N$ elliptic $R$-matrix. More precisely, the four free constants case appears for an odd $N$ while even $N$'s correspond to a single constant.

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A. Levin, M. Olshanetsky, A. Zotov. 2015-04-06. Quantum Baxter-Belavin R-matrices and multidimensional Lax pairs for Painleve VI. https://doi.org/10.1007/s11232-015-0306-y

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