Back to Search
Start Over
Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups
- Publication Year :
- 2022
-
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.<br />Comment: 17 pages
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2207.05511
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1088/1751-8121/acb116