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A characterization of nilpotent Leibniz algebras. (English) Zbl 1292.17002
W. A. Moens [Commun. Algebra 41, No. 7, 2427–2440 (2013; Zbl 1300.17015)] proved that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. In this paper the authors show, that if one takes the Moens’ definition of Leibniz-derivations, then a similar result is not true for non-Lie Leibniz algebras. Namely, they give an example of non-nilpotent Leibniz algebra that admits an invertible Leibniz-derivation. In order to extend the above criteria of nilpotency for Leibniz algebras, the authors introduce another definition of a Leibniz-derivation for Leibniz algebras that agrees with the Moens’ definition in the Lie algebras case. They prove that a Leibniz algebra is nilpotent if and only if it admits an invertible Leibniz-derivation in the new sense. Moreover, similar to the the Lie algebras case, it is proved that a solvable radical of a Leibniz algebra is invariant with respect to a Leibniz-derivation.

MSC:
17A32 Leibniz algebras
17B30 Solvable, nilpotent (super)algebras
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