A Characterization of Nilpotent Leibniz Algebras

Abstract

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 [17] 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 [17] 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.

On the Kantor product

“…In [10] it was proved that a finite-dimensional Leibniz algebra over a field of characteristic zero which admitting an invertible derivation is nilpotent. The Lemma is proved.…”

A characterization of nilpotent nonassociative algebras by invertible Leibniz-derivations

“…In paper [34] a generalization of derivations and pre-derivations of Lie algebras is defined as a Leibnizderivation of order k. Moens proved that a finite-dimensional Lie algebra over a field of characteristic zero is nilpotent if and only if it admits an invertible Leibniz-derivation. After that, Fialowski, Khudoyberdiyev and Omirov [13] showed that with the definition of Leibniz-derivations from [34] the similar result for non-Lie Leibniz algebras is not true. Namely, they gave an example of non-nilpotent Leibniz algebra which admits an invertible Leibniz-derivation.…”

Degenerations of Zinbiel and nilpotent Leibniz algebras

“…The study of the structure theory and other properties of Leibniz algebras was initiated by Loday in [23]. Leibniz algebras were also studied in [12,15,20,28]. Zinbiel algebras were introduced by Loday in [22].…”