EMBEDDINGS OF A LIE-ALGEBRA INTO ITS UNIVERSAL ENVELOPING ALGEBRA

The problem of embedding of a semisimple Lie algebra into its universal enveloping algebra is solved. The solutions given by deformations of the trivial embedding are described. The problem is connected with calculation of cohomologies of Lie algebras with values in universal enveloping algebras. Solutions obtained lead to representations of Lie algebras by higher order differential operators. These new representations are realized in a direct sum of the fundamental representation and its dual one and are related to non-standard symplectic geometry calculus and Moyal bracket.