On Levi's Theorem for Leibniz algebras

TL;DR

This paper extends Levi's Theorem from Lie algebras to Leibniz algebras, showing the splitting property holds but conjugacy of complements does not, highlighting differences in structure between these algebraic systems.

Contribution

The paper generalizes Levi's Theorem to Leibniz algebras and demonstrates the failure of the conjugacy property in this broader context.

Findings

Levi's splitting theorem extends to Leibniz algebras.

Conjugacy of complements does not hold in Leibniz algebras.

Structural differences between Lie and Leibniz algebras are clarified.

Abstract

A Lie algebra over a field of characteristic 0 splits over its soluble radical and all complements are conjugate. I show that the splitting theorem extends to Leibniz algebras but that the conjugacy theorem does not.