Lie-central derivations, Lie-centroids and Lie-stem Leibniz algebras

摘要

In this paper, we introduce the notion of a Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Lie-derivations in the case where their image is contained in the Lie-center, and call them Lie-central derivations. We provide a characterization of Lie-stem Leibniz algebras by their Liecentral derivations, and prove several properties of the Lie algebra of Lie-central derivations for Lie-nilpotent Leibniz algebras of class 2. We also introduce ID*-Lie-derivations. An ID*-Lie-derivation of a Leibniz algebra g is a Lie-derivation of g in which the image is contained in the second term of the lower Lie-central series of g, and which vanishes on Lie-central elements. We provide an upper bound for the dimension of the Lie algebra ID_*~(Lie)(g) of ID_*-Lie-derivation of g, and prove that the sets ID_*~(Lie)(g) and ID_*~(Lie)(q) are isomorphic for any two Lie-isoclinic Leibniz algebras g and q.