The classical r-matrix method for nonlinear sigma-model

TL;DR

This paper presents a Lie-Poisson r-matrix framework for the canonical Poisson structure of nonlinear sigma-models, revealing that hidden singularities of the Lax matrix determine the Poisson structure.

Contribution

It introduces a novel Lie-Poisson r-matrix approach to describe the Poisson structure of nonlinear sigma-models, linking it to Lax matrix singularities.

Findings

Poisson structure characterized by Lie-Poisson r-matrix bracket

Hidden singularities of Lax matrix influence the Poisson structure

Framework connects algebraic structures with sigma-model dynamics

Abstract

The canonical Poisson structure of nonlinear sigma-model is presented as a Lie-Poisson r-matrix bracket on coadjoint orbits. It is shown that the Poisson structure of this model is determined by some `hidden singularities' of the Lax matrix.