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

Abstract

In this paper, we introduce the notion Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Liederivations in the case where their image is contained in the Lie-center, call them Lie-central derivations. We provide a characterization of Lie-stem Leibniz algebras by their Lie-central 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. A 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 that vanishes on Lie-central elements. We provide an upperbound 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.