A Note on Derivations of Lie Algebras

TL;DR

This paper proves that finite-dimensional Lie algebras over characteristic zero fields with certain derivation properties are necessarily solvable, extending understanding of the structure of such algebras.

Contribution

It establishes that specific conditions on derivations imply the solvability of finite-dimensional Lie algebras over characteristic zero fields.

Findings

Lie algebra with abelian derivation algebra is solvable

Existence of a derivation with certain properties implies solvability

Provides conditions linking derivations to algebra structure

Abstract

In this note, we will prove that a finite dimensional Lie algebra $L$ of characteristic zero, admitting an abelian algebra of derivations $D\leq Der(L)$ with the property $$ L^n\subseteq \sum_{d\in D}d(L) $$ for some $n\geq 1$, is necessarily solvable. As a result, if $L$ has a derivation $d:L\to L$, such that $L^n\subseteq d(L)$, for some $n\geq 1$, then $L$ is solvable.