The Poisson bracket compatible with the classical reflection equation algebra

TL;DR

This paper introduces a new family of compatible Poisson brackets on polynomial matrices, enabling a multi-Hamiltonian framework for various integrable systems including spin chains, Toda lattices, and tops.

Contribution

It develops a novel family of compatible Poisson brackets that unify and extend the algebraic structures underlying several integrable models.

Findings

Derived a multi-Hamiltonian structure for integrable systems

Unified treatment of boundary conditions in integrable models

Extended the reflection equation algebra with new Poisson brackets

Abstract

We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the $XXX$ Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.