On the Lengths of Certain Chains of Subalgebras in Lie Algebras

TL;DR

This paper investigates the lengths of various chains of subalgebras in Lie algebras, characterizing conditions under which these chains have equal length, especially over fields of characteristic zero.

Contribution

It provides a characterization of Lie algebras with equal-length chains of subalgebras, linking algebra structure to chain properties over characteristic zero fields.

Findings

Maximal chains of subalgebras can have equal length only if L=R or specific field conditions hold.

In characteristic zero, the structure of L determines the chain lengths.

The paper identifies when chains of modular subalgebras and quasi-ideals coincide in length.

Abstract

In this paper we study the lengths of certain chains of subalgebras of a Lie algebra L: namely, a chief series, a maximal chain of minimal length, a chain of maximal length in which each subalgebra is modular in L, and a chain of maximal length in which each subalgebra is a quasi-ideal of L. In particular we show that, over a field of characteristic zero, a Lie algebra L with radical R has a maximal chain of subalgebras and a chain of subalgebras all of which are modular in L of the same length if and only if L = R, or $\sqrt{F} \not \subseteq F$ and L/R is a direct sum of isomorphic three-dimensional simple Lie algebras.