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

Abstract

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.

On some properties of $\mathsf{Lie}$-centroids of Leibniz algebras

“…In this section we recall some notions and results from [4] and we provide some new results concerning Lie-central derivations and Lie-centroids. Definition 3.1.…”

“…In the recent manuscript [4], it was introduced the concept of centroid of a Leibniz algebra with respect to the Liesation functor, named Lie-centroid, together with the study of the interplay between Lie-central derivations, Lie-centroids and Lie-stem Leibniz algebras.…”

“…Our goal in this paper is to continue with the analysis of properties of the Liecentroid introduced in [4] and study its interaction with Lie-central derivations, generalised Lie-derivations, quasi-Lie-centroids and almost inner Lie-derivations, establishing a parallelism between the absolute results (classical properties for Lie algebras) with the results obtained. We note that not all classical results are immediately translated into the relative case, so new requirements are needed.…”

(TLie)-Leibniz algebras and related properties

“…Recently, several results in studies of derivations of Lie algebras have been extended on Leibniz algebras [4][5][6]13]. Our aim in this paper is to study properties of a subclass of Lie-stem Leibniz algebras and their relationship with the properties of the Lie algebra of Lie-derivations.…”