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

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 st...

Full description

Saved in:
Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2023-01, Vol.56 (1), p.15203
Main Authors: Gutierrez-Sagredo, I, Iglesias Ponte, D, Marrero, J C, Padrón, E, Ravanpak, Z
Format: Article
Language:English
Subjects:
Hamiltonian systems
invariant volume forms
modular class
modular vector fields
Poisson–Lie groups
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8121/acb116