Quantum Baxter–Belavin $R$-matrices and multidimensional Lax pairs for Painlevé VI

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.