C-Supplemented Subalgebras of Lie Algebras

TL;DR

This paper characterizes c-supplemented Lie algebras over any field, extending the concept from group theory to Lie algebra substructures and providing a complete classification.

Contribution

It offers a comprehensive classification of c-supplemented Lie algebras, a novel extension of the subgroup concept to Lie algebra subalgebras.

Findings

Complete classification of c-supplemented Lie algebras

Extension of subgroup c-supplementation to Lie algebras

Characterization valid over any field

Abstract

A subalgebra $B$ of a Lie algebra $L$ is {\em c-supplemented} in $L$ if there is a subalgebra $C$ of $L$ with $L = B + C$ and $B \cap C \leq B_L$, where $B_L$ is the core of $B$ in $L$. This is analogous to the corresponding concept of a c-supplemented subgroup in a finite group. We say that $L$ is {\em c-supplemented} if every subalgebra of $L$ is c-supplemented in $L$. We give here a complete characterisation of c-supplemented Lie algebras over a general field.