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

Abstract

We propose and prove a set of identities for ${\mathrm{GL}}_M$ 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.

19 pages

References in corpus (4)

• Relativistic Classical Integrable Tops and Quantum R-matrices
• Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications
• Relativistic interacting integrable elliptic tops
• Wave function for $GL(n,\mathbb{R})$ hyperbolic Sutherland model II. Dual Hamiltonians