$R$-matrices for Elliptic Calogero-Moser Models

TL;DR

This paper investigates the classical $R$-matrix structures in elliptic Calogero-Moser models, revealing the absence of momentum-independent $R$-matrices for four or more particles and constructing spectral parameter-dependent $R$-matrices for general elliptic potentials.

Contribution

It demonstrates the non-existence of momentum-independent $R$-matrices for $n ext{ } ext{particles}$ with $n ext{ } ext{at least} ext{ } 4$, and constructs $R$-matrices with spectral parameters for general elliptic potentials.

Findings

No momentum-independent $R$-matrix for $n ext{ } ext{particles}$ when $n ext{ } ext{ge} 4$

Reproduces known dynamical $R$-matrices for degenerations of elliptic potentials

Constructs $R$-matrices with spectral parameters for general elliptic potentials

Abstract

The classical $R$-matrix structure for the $n$-particle Calogero-Moser models with (type IV) elliptic potentials is investigated. We show there is no momentum independent $R$-matrix (without spectral parameter) when $n\ge4$. The assumption of momentum independence is sufficient to reproduce the dynamical $R$-matrices of Avan and Talon for the type I,II,III degenerations of the elliptic potential. The inclusion of a spectral parameter enables us to find $R$-matrices for the general elliptic potential.