Quasiclassical Lie Algebras

In this paper we study finite dimensional non-semisimple Lie algebras that can be obtained as Lie algebras of skew-symmetric elements of associative algebras with involution. We call such algebras quasiclassical and characterize them in terms of existence of so-called “ * -plain” representations. We show that the theory of * -plain representations for quasiclassical Lie algebras is almost equivalent to representation theory of associative algebras with involution.