On abelian subalgebras and ideals of maximal dimension in supersolvable lie algebras

TL;DR

This paper investigates the structure of abelian subalgebras and ideals in finite-dimensional supersolvable Lie algebras, providing characterizations and conditions for their maximal dimensions and relationships.

Contribution

It characterizes maximal abelian subalgebras in solvable Lie algebras and establishes the existence of abelian ideals of the same dimension under certain conditions.

Findings

Maximal abelian subalgebras characterized in solvable Lie algebras

Existence of abelian ideals in nilpotent Lie algebras with specific codimension

Examples illustrating the theoretical results

Abstract

In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two. Throughout the paper, we also give several examples to clarify some results.