The index complex of a maximal subalgebra of a Lie algebra

TL;DR

This paper introduces the index complex of a maximal subalgebra in a Lie algebra, using it to analyze how maximal subalgebras influence the algebra's structure and to characterize solvable and supersolvable Lie algebras.

Contribution

It defines the index complex of a maximal subalgebra and applies it to derive new characterizations of solvable and supersolvable Lie algebras.

Findings

New characterizations of solvable Lie algebras

New characterizations of supersolvable Lie algebras

Insight into the influence of maximal subalgebras on Lie algebra structure

Abstract

Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is said to be a completion for M if C is not contained in M but every proper subalgebra of C that is an ideal of L is contained in M. The set I(M) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal subalgebras on the structure of a Lie algebra, in particular finding new characterisations of solvable and supersolvable Lie algebras.