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

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.

arXiv comment23 pages, 2 figures. Revised version

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Cited by in corpus (4)

• Hamiltonian structure of compartmental epidemiological models
• Invariant Poisson-Nijenhuis structures on Lie groups and classification
• Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups
• Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures