Maximal subalgebras and chief factors of Lie algebras

TL;DR

This paper explores the structure of Lie algebras through their chief factors, introducing concepts like L-connectedness and crowns, and extends the Jordan-Hölder theorem to this context.

Contribution

It introduces the concept of crowns in Lie algebras and establishes a strengthened Jordan-Hölder theorem relating Frattini chief factors.

Findings

Characterization of Lie algebras with core-free maximal subalgebras

Definition of L-connectedness among chief factors

Extension of Jordan-Hölder theorem for Lie algebras

Abstract

This paper is a continued investigation of the structure of Lie algebras in relation to their chief factors, using concepts that are analogous to corresponding ones in group theory. The first section investigates the structure of Lie algebras with a core-free maximal subalgebra. The results obtained are then used in section two to consider the relationship of two chief factors of L being L-connected, a weaker equivalence relation on the set of chief factors than that of being isomorphic as L-modules. A strengthened form of the Jordan-Holder Theorem in which Frattini chief factors correspond is also established for every Lie algebra. The final section introduces the concept of a crown, a notion introduced in group theory by Gaschutz, and shows that it gives much information about the chief factors