On Lie-holomorphs of Leibniz algebras

TL;DR

This paper explores the concept of Lie-holomorphs in Leibniz algebras, linking it to biderivations and classifying low-dimensional cases, thus extending the understanding of algebraic structures beyond Lie algebras.

Contribution

It establishes a connection between Lie-holomorphs and biderivations, and classifies low-dimensional non-Lie Leibniz algebras' Lie-holomorphs.

Findings

Lie-derivations are equivalent to derivations and anti-derivations.

Connected Lie-holomorphs with biderivations.

Classification of low-dimensional non-Lie Leibniz algebras.

Abstract

We study the notion of the Lie-holomorph of a Leibniz algebra, recently introduced by N. P. Souris as a generalisation of the classical holomorph construction for Lie algebras. We establish a connection between the Lie-holomorph construction and the Leibniz algebra of biderivations defined by J.-L. Loday, and we prove that a linear endomorphism is a Lie-derivation if and only if it is simultaneously a derivation and an anti-derivation. As an application, we classify the Lie-holomorph algebras of all low-dimensional non-Lie Leibniz algebras over a field of characteristic different from $2$.