The non-dynamical r-matrices of the degenerate Calogero-Moser models

TL;DR

This paper classifies and describes all non-dynamical r-matrices for degenerate Calogero-Moser models based on gl_n, connecting them to known solutions and clarifying their uniqueness.

Contribution

It provides a complete, elementary classification of non-dynamical r-matrices for these models, including gauge transformations and relations to classical r-matrices.

Findings

Hyperbolic/trigonometric case yields Cremmer-Gervais r-matrix.

Rational case relates to Frobenius subalgebra solutions.

Analysis clarifies the uniqueness of constant r-matrices in degenerate models.

Abstract

A complete description of the non-dynamical r-matrices of the degenerate Calogero-Moser models based on $gl_n$ is presented. First the most general momentum independent r-matrices are given for the standard Lax representation of these systems and those r-matrices whose coordinate dependence can be gauged away are selected. Then the constant r-matrices resulting from gauge transformation are determined and are related to well-known r-matrices. In the hyperbolic/trigonometric case a non-dynamical r-matrix equivalent to a real/imaginary multiple of the Cremmer-Gervais classical r-matrix is found. In the rational case the constant r-matrix corresponds to the antisymmetric solution of the classical Yang-Baxter equation associated with the Frobenius subalgebra of $gl_n$ consisting of the matrices with vanishing last row. These claims are consistent with previous results of Hasegawa and others, which imply that Belavin's elliptic r-matrix and its degenerations appear in the Calogero-Moser models. The advantages of our analysis are that it is elementary and also clarifies the extent to which the constant r-matrix is unique in the degenerate cases.