Higher order analogues of unitarity condition for quantum R-matrices

TL;DR

This paper establishes higher order identities for quantum R-matrices of Baxter-Belavin type, extending the unitarity condition and linking these identities to integrable systems and connections in mathematical physics.

Contribution

It introduces a family of n-th order identities for quantum R-matrices, generalizing the unitarity condition and connecting to integrable systems and KZB connections.

Findings

Proves higher order identities for quantum R-matrices.

Shows these identities include unitarity as the second order case.

Links identities to integrable systems and KZB connections.

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.