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

## TL;DR

This paper introduces and studies Lie-derivations and Lie-central derivations in Leibniz algebras, characterizing Lie-stem Leibniz algebras and analyzing properties of derivations in Lie-nilpotent cases.

## Contribution

It generalizes derivations to non-Lie Leibniz algebras, introduces ${
m ID}_*^{Lie}$-derivations, and explores their properties and classifications.

## Key findings

- Characterization of Lie-stem Leibniz algebras via Lie-central derivations
- Properties of Lie-central derivations in Lie-nilpotent Leibniz algebras of class 2
- Upper bounds for the dimension of ${
m ID}_*^{Lie}$-derivation Lie algebras

## Abstract

In this paper, we introduce the notion 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, 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 ${\sf ID}_*-Lie$-derivations. A ${\sf ID}_*-Lie$-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 $ID_*^{Lie}(G)$ are isomorphic for any two Lie-isoclinic Leibniz algebras G and Q.