Dynamical $r$-matrices for the Elliptic Calogero-Moser Model

TL;DR

This paper demonstrates that the Lax operator for the elliptic Calogero-Moser system has a classical r-matrix structure, generalizing previous trigonometric results, and satisfies a generalized Yang-Baxter equation.

Contribution

It introduces a new elliptic r-matrix structure for the Calogero-Moser system, extending known trigonometric cases and involving a dynamical matrix.

Findings

The Lax operator possesses a classical r-matrix structure.

The r-matrix generalizes previous trigonometric results.

It satisfies a generalized Yang-Baxter equation.

Abstract

For the integrable $N$-particle Calogero-Moser system with elliptic potential it is shown that the Lax operator found by Krichever possesses a classical $r$-matrix structure. The $r$-matrix is a natural generalisation of the matrix found recently by Avan and Talon (hep-th/9210128) for the trigonometric potential. The $r$-matrix depends on the spectral parameter and only half of the dynamical variables (particles' coordinates). It satisfies a generalized Yang-Baxter equation involving another dynamical matrix.