On complemented non-abelian chief factors of a Lie algebra

TL;DR

This paper investigates the variability in the number of complemented chief factors in a Lie algebra, extending understanding of their structure and how it differs from Frattini chief factors.

Contribution

It characterizes the possible variations in the number of simply complemented chief factors across different chief series of a Lie algebra.

Findings

Number of simply complemented chief factors can vary between chief series.

The paper determines the possible range of this variation.

Provides structural insights into Lie algebra chief factors.

Abstract

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 \cite[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.