On n-maximal subalgebras of Lie algebras

TL;DR

This paper investigates the properties of n-maximal subalgebras in Lie algebras and explores how conditions on these subalgebras influence the overall structure of the algebra, extending known results about maximal subalgebras.

Contribution

It introduces the concept of n-maximal subalgebras and analyzes their properties to derive structural insights about Lie algebras, expanding the understanding beyond maximal subalgebras.

Findings

Characterization of n-maximal subalgebras in Lie algebras

Conditions on n-maximal subalgebras imply specific structural properties

Extension of classical results from maximal to n-maximal subalgebras

Abstract

A chain $S_0 < S_1 < \ldots < S_n = L$ is a {\em 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 {\em $n$-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 $n>1$.