Maximal subalgebras and chief factors of Lie algebras

Journal of Generalized Lie Theory and Applications

In group theory the chief factors allow a group to be studied by its representation theory on particularly natural irreducible modules. It is to be expected, therefore, that they will play an important role in the study of Lie algebras. In this article we survey a few of their properties.

2015

The number of Frattini chief factors or of chief factors which are complemented by a maximal subalgebra of a finite-dimensional Lie algebra L is the same in every chief series for L, by [Theorem 2.3][11]. However, this is not the case for the number of chief factors which are simply complemented in L. In this paper we determine the possible variation in that number.

2015

The purpose of this paper is to continue the study of chief factors of a Lie algebra and to prove a further strengthening of the Jordan-H\"older Theorem for chief series.

2018

The number of Frattini chief factors or of chief factors which are complemented by a maximal subalgebra of a nite-dimensional Lie algebra L is the same in every chief series for L, by [13, Theorem 2.3]. However, this is not the case for the number of chief factors which are simply complemented in L. In this paper we determine the possible variation in that number.

Proceedings of the American Mathematical Society

We call a subalgebra U of a Lie algebra L a CAP-subalgebra of L if for any chief factor H/K of L, we have H ∩ U = K ∩ U or H + U = K + U. In this paper we investigate some properties of such subalgebras and obtain some characterizations for a finite-dimensional Lie algebra L to be solvable under the assumption that some of its maximal subalgebras or 2-maximal subalgebras be CAP-subalgebras.

Proceedings of the American Mathematical Society

A chain S 0 < S 1 <. .. < S n = L is a maximal chain if each S i is a maximal subalgebra of S i+1. The subalgebra S 0 in such a series is called an n-maximal subalgebra. There are many interesting results concerning the question of what certain intrinsic properties of the maximal subalgebras of a Lie algebra L imply about the structure of L itself. Here we consider whether similar results can be obtained by imposing conditions on the n-maximal subalgebras of L, where n > 1.

Communications in Algebra, 2013

For a Lie algebra L and a subalgebra M of L we say that a subalgebra U of L is a supplement to M in L if L = M + U. We investigate those Lie algebras all of whose maximal subalgebras have abelian supplements, those that have nilpotent supplements, those that have nil supplements, and those that have supplements with the property that their derived algebra is inside the maximal subalgebra being supplemented. For the algebras over an algebraically closed field of characteristic zero in the last three of these classes we find complete descriptions; for those in the first class partial results are obtained.

2007

In this paper we study the class ${\cal F}$ of Lie algebras having a flag of subalgebras, and the class ${\cal Ch}_{lm}$ of Lie algebras having a maximal chain of lower modular subalgebras. We show that ${\cal F} \subseteq {\cal Ch}_{lm}$ and that both are extensible formations that are subalgebra closed. We derive a number of properties relating to these two classes, including a classification of the algebras in each class over a field of characteristic zero.

Proceedings of the Edinburgh Mathematical Society, 2011

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 characterizations of solvable and supersolvable Lie algebras.

Journal of Pure and Applied Algebra, 2012

Relationships between certain properties of maximal subalgebras of a Lie algebra L and the structure of L itself have been studied by a number of authors. Amongst the maximal subalgebras, however, some exert a greater influence on particular results than others. Here we study properties of those maximal subalgebras that contain Engel subalgebras, and of those that also have codimension greater than one in L.