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

I. Gutierrez-Sagredo; D. Iglesias Ponte; J. C. Marrero; E. Padrón; Z. Ravanpak

doi:10.1088/1751-8121/acb116

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arXiv · 2207.05511

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

I. Gutierrez-Sagredo·D. Iglesias Ponte·J. C. Marrero·E. Padrón·Z. Ravanpak

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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.

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BibTeX RIS

I. Gutierrez-Sagredo, D. Iglesias Ponte, J. C. Marrero, E. Padrón, Z. Ravanpak. 2022-07-12. Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups. https://doi.org/10.1088/1751-8121/acb116

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