Abstract:
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.
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