A Characterisation of Nilpotent Lie Algebras by Invertible Leibniz-Derivations

Abstract Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques. We only consider finite-dimensional Lie algebras over a fields of characteristic zero. Key Words: DerivationInvertibleLie algebraLeibniz algebraNilpotentPre-derivation2010 Mathematics Subject Classification: 17B4017B30 Notes Communicated by I. Shestakov.

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