Spin generalization of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra

TL;DR

This paper develops a spin generalization of the elliptic Ruijsenaars-Schneider system, solves its equations using Riemann theta-functions, and links it to non-abelian 2D Toda chains and Sklyanin algebra representations.

Contribution

It introduces a new spin generalization of the elliptic Ruijsenaars-Schneider system and establishes its isomorphism with elliptic solutions of the non-abelian 2D Toda chain.

Findings

Constructed action-angle variables for the spin system.

Proved the isomorphism with elliptic solutions of the Toda chain.

Connected finite gap solutions with Sklyanin algebra representations.

Abstract

Action-angle type variables for spin generalizations of the elliptic Ruijsenaars-Schneider system are constructed. The equations of motion of these systems are solved in terms of Riemann theta-functions. It is proved that these systems are isomorphic to special elliptic solutions of the non-abelian 2D Toda chain. A connection between the finite gap solutions of solitonic equations and representations of the Sklyanin algebra is revealed and discrete analogs of the Lame operators are introduced. A simple way to construct representations of the Sklyanin algebra by difference operators is suggested.