On Levi-Malcev theorem for Leibniz algebras

TL;DR

This paper investigates the conditions under which Levi-Malcev theorem holds for finite-dimensional Leibniz algebras over characteristic zero fields, focusing on conjugacy of Levi subalgebras, especially over the complex numbers.

Contribution

It provides new criteria for the conjugacy of Levi subalgebras in Leibniz algebras, extending classical results from Lie algebra theory.

Findings

Levi subalgebras are conjugate under certain conditions.

Counterexamples where Levi subalgebras are not conjugate.

Complete classification over the complex numbers.

Abstract

The present paper is devoted to provide conditions for the Levi--Malcev theorem to hold or not to hold (i.e. for two Levi subalgebras to be or not conjugate by an inner automorphism) in the context of finite-dimensional Leibniz algebras over a field of characteristic zero. Particularly, in the case of the field $\mathbb{C}$ of complex numbers, we consider all possible cases in which Levi subalgebras are conjugate and not conjugate.