Integrable systems from the classical reflection equation

TL;DR

This paper develops a method to construct integrable Hamiltonian systems on certain geometric spaces using classical reflection equations, linking classical and quantum integrable models.

Contribution

It introduces a new class of integrable systems on $G/K$ derived from the classical reflection equation, with explicit solutions via Lax pairs and factorization.

Findings

Systems are integrable on $G/K$ with classical reflection symmetry.

Time evolution described by Lax equations and factorization in $G$.

Connections established between classical systems and quantum spin chains with boundaries.

Abstract

We construct integrable Hamiltonian systems on $G/K$, where $G$ is a quasitriangular Poisson Lie group and $K$ is a Lie subgroup arising as the fixed point set of a group automorphism $\sigma$ of $G$ satisfying the classical reflection equation. In the case that $G$ is factorizable, we show that the time evolution of these systems is described by a Lax equation, and present its solution in terms of a factorization problem in $G$. Our construction is closely related to the semiclassical limit of Sklyanin's integrable quantum spin chains with reflecting boundaries.