Abstract
W. A. Moens proved that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. In this paper, using the definition of a Leibniz-derivation from Moens (2010), we show that a similar result for non-Lie Leibniz algebras is not true. Namely, we give an example of non-nilpotent Leibniz algebra that admits an invertible Leibniz-derivation. In order to extend the results of the paper by Moens (2010) for Leibniz algebras, we introduce a definition of a Leibniz-derivation of Leibniz algebras that agrees with Leibniz-derivation of the Lie algebra case. Further, we prove that a Leibniz algebra is nilpotent if and only if it admits an invertible Leibniz-derivation of Definition 3.4. Moreover, the result that a solvable radical of a Lie algebra is invariant with respect to a Leibniz-derivation was extended to the case of Leibniz algebras.
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The research of the first author was partially supported by the grant OTKA K77757.
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Fialowski, A., Khudoyberdiyev, A.K. & Omirov, B.A. A Characterization of Nilpotent Leibniz Algebras. Algebr Represent Theor 16, 1489–1505 (2013). https://doi.org/10.1007/s10468-012-9373-z
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DOI: https://doi.org/10.1007/s10468-012-9373-z