Lie algebras with given properties of subalgebras and elements

TL;DR

This paper classifies finite-dimensional Lie algebras over characteristic zero fields based on properties of their subalgebras and elements, including anisotropic, regular, minimal nonabelian, and depth 2 algebras.

Contribution

It provides new results characterizing Lie algebras with specific subalgebra and element properties, expanding understanding of their structure.

Findings

Classification of anisotropic Lie algebras.

Characterization of regular Lie algebras.

Analysis of minimal nonabelian and depth 2 Lie algebras.

Abstract

Results about the following classes of finite-dimensional Lie algebras over a field of characteristic zero are presented: anisotropic (i.e., Lie algebras for which each adjoint operator is semisimple), regular (i.e., Lie algebras in which each nonzero element is regular in the sense of Bourbaki), minimal nonabelian (i.e., nonabelian Lie algebras all whose proper subalgebras are abelian), and algebras of depth 2 (i.e., Lie algebras all whose proper subalgebras are abelian or minimal nonabelian).