Invariant Operators in Simple Lie Algebras and Flexible Malcev-Admissible Algebras with A⁻ Simple

Abstract

We have determined in Chapter 2 (Corollary 2.34 and Theorem 2.38) the structure of flexible Malcev-admissible algebras A over a field F of characteristic 0, when A⁻ is split semisimple over F. In this chapter we determine the structure of such algebras A, when A⁻ is simple over F. The result for semisimple A⁻ immediately follows from the case for simple A⁻. Since the derivation algebra Der M of a semisimple Malcev algebra M is semisimple, we regard A as a Der A⁻-module. Representation theory of simple Lie algebras plays a main role in our investigation. Specifically, we first determine the so-called adjoint operators of a split simple Lie algebra in its irreducible modules, which in turn characterize the Der A⁻-module actions imposed on the multiplication of A. An adjoint operator of a Lie algebra L into an L-module V is an L-module homomorphism of L into the tensor product module V ⊗ V*, where V* is the dual module of V. The results in this chapter are based on work of Okubo and Myung [3], Benkart and Osborn [1], and Myung [5].