Elliptic Ruijsenaars-Schneider and Calogero-Moser Models Represented by Sklyanin Algebra and sl(n) Gaudin Algebra

TL;DR

This paper explores the connections between elliptic integrable models (RS and CM) and algebraic structures (Sklyanin and Gaudin algebras), providing Lax pair representations and solutions for specific cases.

Contribution

It establishes a relationship between elliptic integrable models and algebraic structures, including explicit Lax pair representations and solutions for the n=2 case.

Findings

Lax pair representations for elliptic RS and CM models

Eigenvalues and eigenfunctions for the n=2 Lame equation

Connection between models and Sklyanin and Gaudin algebras

Abstract

The relationship between Elliptic Ruijsenaars-Schneider (RS) and Calogero-Moser (CM) models with Sklyanin algebra is presented. Lax pair representations of the Elliptic RS and CM are reviewed. For n=2 case, the eigenvalue and eigenfunction for Lame equation are found by using the result of the Bethe ansatz method.