Non-abelian Poisson brackets: algebraic aspects

This is a continuation of the previous Seminar's talk given by prof. V. Sokolov.
I suppose to discuss algebraic questions related to  so-called "non-abelian"
Poisson brackets introduced in the previous talk. I hope to clarify their relations
with brackets of W. Crawley- Boevey and double brackets of M. Van den Bergh.
My talk is based on joint papers (in press and in progress) with A. Odesskii and
V. Sokolov.