Classical R-Matrices and the Feigin-Odesskii Algebra via Hamiltonian and Poisson Reductions

TL;DR

This paper develops a formula for classical r-matrices in integrable systems via Hamiltonian reduction, applying it to derive new results for elliptic Calogero-Moser systems and Feigin-Odesskii algebras.

Contribution

It introduces a new formula for classical r-matrices using Dirac brackets in Hamiltonian reductions and applies it to elliptic integrable models and Feigin-Odesskii algebra construction.

Findings

Derived a classical r-matrix for elliptic Calogero-Moser system with spin.

Constructed a classical Feigin-Odesskii algebra via Poisson reduction.

Extended integrable lattice models within a Hitchin-type framework.

Abstract

We present a formula for a classical $r$-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the respective gauge transformations are Lie group ones. Our formula is in terms of Dirac brackets, and some new observations on these brackets are made. We apply our method to derive a classical $r$-matrix for the elliptic Calogero-Moser system with spin starting from the Higgs bundle over an elliptic curve with marked points. In the paper we also derive a classical Feigin-Odesskii algebra by a Poisson reduction of some modification of the Higgs bundle over an elliptic curve. This allows us to include integrable lattice models in a Hitchin type construction.