Supplements to maximal subalgebras of Lie algebras

TL;DR

This paper studies Lie algebras where maximal subalgebras have specific types of supplements, providing complete classifications over algebraically closed fields of characteristic zero for some cases.

Contribution

It offers a comprehensive classification of Lie algebras with maximal subalgebras having abelian, nilpotent, or nil supplements, and partial results for those with supplements whose derived algebra lies inside the subalgebra.

Findings

Complete descriptions for algebras with nilpotent, nil, and derived algebra properties over algebraically closed fields of characteristic zero.

Partial results for algebras with abelian supplements.

Characterization of supplements in various classes of Lie algebras.

Abstract

For a Lie algebra $L$ and a subalgebra $M$ of $L$ we say that a subalgebra $U$ of $L$ is a {\em supplement} to $M$ in $L$ if $L = M + U$. We investigate those Lie algebras all of whose maximal subalgebras have abelian supplements, those that have nilpotent supplements, those that have nil supplements, and those that have supplements with the property that their derived algebra is inside the maximal subalgebra being supplemented. For the algebras over an algebraically closed field of characteristic zero in the last three of these classes we find complete descriptions; for those in the first class partial results are obtained.