Classical r-matrix structure for elliptic Ruijsenaars chain and 1+1 field analogue of Ruijsenaars-Schneider model

TL;DR

This paper establishes the classical r-matrix structure for the elliptic Ruijsenaars chain, demonstrates its integrability, and explores its non-relativistic limit leading to a field analogue of the elliptic Calogero-Moser model.

Contribution

It provides the first detailed classical r-matrix formulation for the elliptic Ruijsenaars chain and connects it to the field analogue of the elliptic Calogero-Moser model.

Findings

Liouville integrability of the elliptic Ruijsenaars chain established

Derived the Poisson brackets for the monodromy matrix

Reproduced the non-ultralocal r-matrix structure in the non-relativistic limit

Abstract

The classical dynamical $r$-matrix structure for the periodic elliptic Ruijsenaars chain is described. The Poisson brackets for the monodromy matrix are calculated as well, thus providing Liouville integrability of the model. Next, we study its continuous non-relativistic limit and reproduce the Maillet type non-ultralocal $r$-matrix structure for the field analogue of the elliptic Calogero-Moser model.