On ad-semisimple Lie algebras

Abstract

The question, whether any ad-semisimple Lie algebra is commutative, is related to the existence of simple-semiabelian Lie algebras. These algebras determine the Brauer group of a field and are, in addition to that, significant for the study of Lie algebras all whose proper subalgebras are nilpotent. In this paper we consider a class of Lie algebras related to several topics of Lie algebra theory and field theory. We begin with the study of finite dimensional Lie algebras G having the property that the adjoint representation sends every element of G to a semisimple derivation. We call such a Lie algebra ad-semisimple. Our study of these may be regarded as a complement to the classical investigation of ad-nilpotent Lie algebras due to Engel. However, the results of the first section show that, in contrast with Engels theory, the theory of ad-semisimple Lie algebras is highly sensitive to the choice of the base field. This feature becomes more evident if we confine our studies to the subclass of simple-semiabelian Lie algebras. Sections 3-5 are devoted to a brief discussion of some aspects of the structure of this interesting class; a more thorough investigation can be found in 15 1. Theorem 5.2 and Section 6 are closely related to a paper by Towers [ 11 I. After discussing the relationship between simple-semiabelian Lie algebras and restricted Lie algebras having a nonsingular p-map (i.e., a p-map whose only zero is 0), we illustrate, in the final section, some aspects of the linkage between the Brauer group of the underlying field and the isomorphism classes of simple-semiabelian Lie algebras: The Brauer group of the field admitting no simple-semiabelian Lie algebras is trivial. It is conceivable that the Brauer group determines the isomorphism classes of simple-semiabelian Lie algebras. The cases of IR and 6, for example, indicate the existence of a more general, yet discerning relationship.