Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups

TL;DR

This paper explores the relationship between unimodularity of Poisson-Lie structures and the existence of invariant volume forms in Hamiltonian systems, establishing conditions under which volume preservation implies unimodularity.

Contribution

It proves the equivalence between unimodularity and volume-preserving Hamiltonian vector fields on Poisson-Lie groups, with illustrative examples on semisimple and unimodular groups.

Findings

Hamiltonian vector fields preserve volume forms on unimodular Poisson-Lie groups.

Existence of volume-preserving Hamiltonian vector fields implies unimodularity.

Examples demonstrate the theory on various Poisson-Lie groups.

Abstract

In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and the unimodularity of the Poisson-Lie structure. In particular, we prove that Hamiltonian vector fields on a Lie group endowed with a unimodular Poisson-Lie structure preserve a multiple of any left-invariant volume on the group. Conversely, we also prove that if there exists a Hamiltonian function such that the identity element of the Lie group is a nondegenerate singularity and the associated Hamiltonian vector field preserves a volume form, then the Poisson-Lie structure is necessarily unimodular. Furthermore, we illustrate our theory with different interesting examples, both on semisimple and unimodular Poisson-Lie groups.