Finite-dimensional non-commutative Poisson algebras Original Research Article

Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie algebra structure together with the Leibniz law. For finite-dimensional ones we show that if they are semisimple 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 finite-dimensional non-commutative Poisson algebras whose Lie structures are reductive.