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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.

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Your terminology seems a little nonstandard. $X_f$ is usually called the Hamiltonian vector field associated to $f$. When one speaks of "the Hamiltionian", this refers the the function $f$. – Hans Lundmark Aug 13 '11 at 9:01
Thanks, Hans, for the precision. Yes, I should write hamiltonian vector field. – amine Aug 13 '11 at 11:38

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