A characterization of nilpotent Leibniz algebras

TL;DR

This paper investigates the conditions under which Leibniz algebras are nilpotent, extending known results from Lie algebras and clarifying the role of Leibniz-derivations in this context.

Contribution

It introduces a consistent definition of Leibniz-derivation for Leibniz algebras and proves that nilpotency is characterized by the existence of an invertible Leibniz-derivation.

Findings

A non-nilpotent Leibniz algebra can admit an invertible Leibniz-derivation.

A Leibniz algebra is nilpotent if and only if it admits an invertible Leibniz-derivation.

The invariance of the solvable radical under Leibniz-derivations extends from Lie to Leibniz algebras.

Abstract

W. A. Moens proved that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. In this paper we show that with the definition of Leibniz-derivation from W. A. Moens the similar result for non Lie Leibniz algebras is not true. Namely, we give an example of non nilpotent Leibniz algebra which admits an invertible Leibniz-derivation. In order to extend the results of paper W. A. Moens for Leibniz algebras we introduce a definition of Leibniz-derivation of Leibniz algebras which agrees with Leibniz-derivation of Lie algebras case. Further we prove that a Leibniz algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. Moreover, the result that solvable radical of a Lie algebra is invariant with respect to a Leibniz-derivation was extended to the case of Leibniz algebras.