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Статья

Quantum Baxter-Belavin R-matrices and multidimensional lax pairs for Painlevé VI

Theoretical and Mathematical Physics. 2015. Vol. 184. No. 1. P. 924-939.
Levin A., Olshanetsky M., Zotov A.

Quantum elliptic R-matrices satisfy the associative Yang-Baxter equation in Mat(N)⊗2, which can be regarded as a noncommutative analogue of the Fay identity for the scalar Kronecker function. We present a broader list of R-matrix-valued identities for elliptic functions. In particular, we propose an analogue of the Fay identities in Mat(N)⊗2. As an application, we use the ℤN×ℤN elliptic R-matrix to construct R-matrix-valued 2N2×2N2 Lax pairs for the Painlevé VI equation (in the elliptic form) with four free constants. More precisely, the case with four free constants corresponds to odd N, and even N corresponds to the case with a single constant in the equation. © 2015, Pleiades Publishing, Ltd.