On Lie-holomorphs of Leibniz algebras

2025/12/22 by Gianmarco La Rosa, La Rosa, Gianmarco, Manuel Mancini +1
Mathematics · Physics and Astronomy · #15B30 #16W25 #17A32 #17A36 #17B40 #Advanced Differential Geometry Research #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2512.19276

openalex publication_date 2025/12/22 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28

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.

Citations

• A characterization of perfect Leibniz algebras
• Action accessible and weakly action representable varieties of algebras
• On the representability of actions of Leibniz algebras and Poisson algebras
• Central extensions of associative algebras and weakly action representable categories
• Complete Leibniz Algebras
• On Classification of Four Dimensional Nilpotent Leibniz Algebras
• On Lie-central extensions of Leibniz algebras
• The classification of 4-dimensional Leibniz algebras
• Classification of 4-dimensional nilpotent complex Leibniz algebras

Related

• Complete Leibniz Algebras
• Lie-central derivations, Lie-centroids and Lie-stem Leibniz algebras
• A characterization of nilpotent Leibniz algebras
• Some characterizations of Hom-Leibniz algebras
• On some structures of Leibniz algebras