On R-matrix identities related to elliptic anisotropic spin Ruijsenaars-Macdonald operators

TL;DR

This paper establishes new identities for elliptic R-matrices related to the GL_M group, advancing the understanding of solutions for the anisotropic spin Ruijsenaars model's quantum eigenvalue problem.

Contribution

It introduces and proves novel R-matrix identities for elliptic functions in the context of the GL_M group, extending scalar case results to matrix cases.

Findings

Derived elliptic function identities for GL_M R-matrices

Connected identities to the construction of quantum eigenstates

Paved the way for solving anisotropic spin Ruijsenaars models

Abstract

We propose and prove a set of identities for ${\rm GL}_M$ elliptic $R$-matrix (in the fundamental representation). In the scalar case ($M=1$) these are elliptic function identities derived by S.N.M. Ruijsenaars as necessary and sufficient conditions for his kernel identity underlying construction of integral solutions to quantum spinless Ruijsenaars-Schneider model. In this respect the result of the present paper can be considered as the first step towards constructing solutions of quantum eigenvalue problem for the anisotropic spin Ruijsenaars model.