Given a Poisson tensor $\pi$ on a Lie group $G$. The hamiltonian $X_f$ associated to a smooth function $f\in C^\infty(G)$ is definied by $$X_f=-[\pi,f]$$ where $[\,,\,]$ is the Schouten bracket
I would like to ask if the Poisson tensors for which all hamiltonians are left invariant are already known and studied?
Another related question:
What are the Poisson tensors that map left invariant forms on left invariant vector fields, apart from those associtaed to the classical Yang-Baxter equation?
Thanks for your help.