An approach to the integrability of a dynamical system $X\in \mathfrak{X}(M)$ on a manifold $M$ of dimension $n$, according to Euler and Jacobi, is to look for $n-2$ first integrals which are functionally independent, and an invariant volume form. Under these conditions, the system can be integrated by quadratures, that is, to determine the trajectories of $X$ by means of a finite number of algebraic operations and quadratures of some functions.
For those dynamical systems which are Hamiltonian with respect to a Poisson structure, there are two possible approaches in order to find invariant volume forms: the first one is to describe the symplectic leaves of the Poisson structure and apply Liouville's theorem to obtain invariant volume forms on the leaves; the second is to look for an invariant volume form on the whole manifold. A disadvantage that shows up for the first approach is that the symplectic leaves for some types of Poisson manifolds are hard to compute. Therefore, in these cases it seems more natural to use the second approach, finding invariant forms on the whole manifold. Our goal is the second approach for Hamiltonian systems on Poisson-Lie groups.
We discuss several relations between the existence of an invariant volume form for a Hamiltonian system on a Poisson-Lie group $G$ and the unimodularity of the Poisson-Lie structure. We illustrate our main theory and our results with different interesting examples.
This talk is based on:
[1] I. Gutierrez-Sagredo, D. Iglesias Ponte, J. C. Marrero, E. Padr\'on, Z. Ravanpak, Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups, (2022) Preprint: arXiv:2207.05511.
[2] A. Ballesteros, J. C. Marrero, Z. Ravanpak, Poisson-Lie groups, bi-Hamiltonian systems, and integrable deformations,
J. Phys. A: Math. Theor. Vol. 50 (2017), 145204.