Lie Algebras with a finite number of ideals

TL;DR

This paper characterizes Lie algebras with finitely many ideals, providing explicit descriptions and graph representations, especially for those with trivial Frattini subalgebra, supported by numerous examples.

Contribution

It offers a detailed structural analysis and explicit classification of Lie algebras with finitely many ideals, including their graph representations and conditions.

Findings

Explicit description of Lie algebras with finitely many ideals

Graph representations via Hasse diagrams

Characterization of algebras with trivial Frattini subalgebra

Abstract

In this paper we focus on the structure of the variety of Lie algebras with a finite number of ideals and their graph representations using Hasse diagrams. The large number of necessary conditions on the algebraic structure of this type of algebras leads to the explicit description of those algebras in the variety with trivial Frattini subalgebra. To illustrate our results, we have included and discussed lots of examples throughout the paper.