Higher-order analogues of the unitarity condition for quantum R-matrices

Abstract

We prove a family of $n$-th order identities for quantum $R$-matrices of Baxter-Belavin type in fundamental representation. The set of identities includes the unitarity condition as the simplest one ($n=2$). Our study is inspired by the fact that the third order identity provides commutativity of the Knizhnik-Zamolodchikov-Bernard connections. On the other hand the same identity gives rise to $R$-matrix valued Lax pairs for the classical integrable systems of Calogero type. The latter construction uses interpretation of quantum $R$-matrix as matrix generalization of the Kronecker function. We present a proof of the higher order scalar identities for the Kronecker functions which is then naturally generalized to the $R$-matrix identities.