Geodesic
Quantum Baxter--Belavin $R$-matrices and multidimensional Lax pairs for Painlev\'e~VI
Teoretičeskaâ i matematičeskaâ fizika, Tome 184 (2015) no. 1, pp. 41-56

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Quantum elliptic $R$-matrices satisfy the associative Yang–Baxter equation in $\mathrm{Mat}(N)^{\otimes 2}$, 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 $\mathrm{Mat}(N)^{\otimes 2}$. As an application, we use the $\mathbb{Z}_N\times\mathbb{Z}_N$ elliptic $R$-matrix to construct $R$-matrix-valued $2N^2\times 2N^2$ Lax pairs for the Painlevé VI equation {(}in the elliptic form{\rm)} 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.
Mots-clés : quantum $R$-matrix, multidimensional Lax pair
Keywords: Painlevé equation. 
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     author = {A. M. Levin and M. A. Olshanetsky and A. V. Zotov},
     title = {Quantum {Baxter--Belavin} $R$-matrices and multidimensional {Lax} pairs for {Painlev\'e~VI}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     number = {1},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2015_184_1_a1/}
}
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A. M. Levin; M. A. Olshanetsky; A. V. Zotov. Quantum Baxter--Belavin $R$-matrices and multidimensional Lax pairs for Painlev\'e~VI. Teoretičeskaâ i matematičeskaâ fizika, Tome 184 (2015) no. 1, pp. 41-56. http://geodesic.mathdoc.fr/item/TMF_2015_184_1_a1/