On the construction of reductive Lie-admissible algebras

Abstract

We discuss a method to construct reductive Lie-admissible algebras which is based on the construction of nonassociative algebras with a specified simple Lie algebra D of derivations. As special cases, we construct two classes of reductive Lie-admissible algebras ( A,∗) of dimensions 7 and 8 with D=sl(3) and D= G 2, and determine their associated reductive Lie algebras g −= A −⊕sl(3) and g −= A −⊕ G 2. The split octonion, para-octonion, 7-dimensional simple Malcev algebra and simple Lie algebras of type A 3, G 2, B 3 arise from this construction. Representations of simple Lie algebras play a main role.