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 unimodularity structure. particular, prove that vector fields a Lie group endowed with unimodular structure preserve multiple any left-invariant group. Conversely, also if there exists function such identity element is nondegenerate singularity associated field preserves form, then necessarily unimodular. Furthermore, illustrate our theory different interesting examples, both semisimple groups.