Unimodularity and invariant volume forms for Hamiltonian dynamics on 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.