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Working paper

Baxter-Belavin R-matrices as non-abelian generalization of elliptic functions

A. Levin, Olshanetsky M., Zotov A.
It was shown in our previous paper that quantum ${\rm gl}_N$ $R$-matrices satisfy noncommutative analogues of the Fay identities in ${\rm gl}_N^{\otimes 3}$. In this paper we extend the list of $R$-matrix valued elliptic function identities. We propose counterparts of the Fay identities in ${\rm gl}_N^{\otimes 2}$, the symmetry between the Planck constant and the spectral parameter, quasi-periodicities with respect to these variables, the Kronecker double series representation of the R-matrix. As an application we construct $R$-matrix valued $2N^2\times 2N^2$ Lax pairs for the Painlev\'e VI equation (in the 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.