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...

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Bibliographic Details
Published in:arXiv.org 2023-01
Main Authors: Gutierrez-Sagredo, I, D Iglesias Ponte, Marrero, J C, Padrón, E, Ravanpak, Z
Format: Article
Language:English
Subjects:
Fields (mathematics)
Hamiltonian functions
Invariants
Lie groups
Online Access:Get full text
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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:2331-8422
DOI:10.48550/arxiv.2207.05511