Poisson-Lie groups, bi-Hamiltonian systems and integrable deformations

TL;DR

This paper introduces a method to deform Lie-Poisson integrable systems into bi-Hamiltonian systems on non-abelian Poisson-Lie groups, enabling new integrable models and reductions, exemplified by the Lorenz and Euler top systems.

Contribution

It presents a novel construction of integrable bi-Hamiltonian deformations on Poisson-Lie groups from Lie-Poisson systems, expanding the framework for integrable deformations.

Findings

Constructed deformed bi-Hamiltonian systems on Poisson-Lie groups.

Derived two integrable Hamiltonian systems on the product group G_η × G_η.

Applied the method to Lorenz and Euler top systems, demonstrating its effectiveness.

Abstract

Given a Lie-Poisson completely integrable bi-Hamiltonian system on $\mathbb{R}^n$, we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson system on a non-abelian Poisson-Lie group $G_\eta$ of dimension $n$, where $\eta \in \mathbb{R}$ is the deformation parameter. Moreover, we show that from the two multiplicative (Poisson-Lie) Hamiltonian structures on $G_\eta$ that underly the dynamics of the deformed system and by making use of the group law on $G_\eta$, one may obtain two completely integrable Hamiltonian systems on $G_\eta \times G_\eta$. By construction, both systems admit reduction, via the multiplication in $G_\eta$, to the deformed bi-Hamiltonian system in $G_\eta$. The previous approach is applied to two relevant Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.