FINITE-DIMENSIONAL NONCOMMUTATIVE POISSON ALGEBRAS

Non-commutative Poisson algebras are the algebras having an associativ e algebra structure and a Lie algebra structure together with the Leib niz law. For finite-dimensional ones we show that if they are semisimp le as associative algebras then they are standard, on the other hand, if they are semisimple as Lie algebras then their associative products are trivial. We also give the descriptions of the structures of finit e-dimensional non-commutative Poisson algebras whose Lie structures ar e reductive.