On Lie-holomorphs of 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 $2$.

Final version, accepted for publication

References in corpus (8)

• Algebras with representable representations
• Two-step nilpotent Leibniz algebras
• Derivations of two-step nilpotent algebras
• On the representability of actions of Leibniz algebras and Poisson algebras
• Biderivations of low-dimensional Leibniz algebras
• Weak representability of actions of non-associative algebras
• Action accessible and weakly action representable varieties of algebras
• Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra