FINITE-DIMENSIONAL NON-COMMUTATIVE POISSON ALGEBRAS. II

Here is a structure theorem of a finite-dimensional non-commutative Poisson algebra A. A nice element ε of A will be found, so that the Lie module action of an element of a large Poisson subalgebra of A on A is described in terms of ε and the ordinary associative commutator. Consequently, we can figure out a structure of A when the Jacobson radical rad A satisfies (rad A)² = 0. This structure theorem leads us to a classification of the finite-dimensional simple Poisson A-modules.