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Symmetric Hom–Leibniz algebras

Part of: General nonassociative rings

Published online by Cambridge University Press:  29 September 2025

Samiha Hidri  and
Fahmi Mhamdi
Samiha Hidri
Affiliation:
University of Gabes, Faculty of Sciences, Riadh Zerig, 6029, Gabes, Tunisia
Fahmi Mhamdi*
Affiliation:
University of Gafsa, Faculty of Sciences, Department of Mathematics, Zarroug, Gafsa, Tunisia
*
Corresponding author: Fahmi Mhamdi; Email: fahmi.mhamdi@gmail.com
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Abstract

This paper focuses on quadratic Hom–Leibniz algebras, defined as (left or right) Hom–Leibniz algebras equipped with symmetric, non-degenerate, and invariant bilinear forms. In particular, we demonstrate that every quadratic regular Hom–Leibniz algebra is symmetric, meaning that it is simultaneously a left and a right Hom–Leibniz algebra. We provide characterizations of symmetric (resp. quadratic) Hom–Leibniz algebras. We also investigate the $\mathrm{T}^*$-extensions of Hom–Leibniz algebras, establishing their compatibility with solvability and nilpotency. We study the equivalence of such extensions and provide the necessary and sufficient conditions for a nilpotent quadratic Hom–Leibniz algebra to be isometric to a $\mathrm{T}^*$-extension. Furthermore, through the procedure of double extension, which is a central extension followed by a generalized semi-direct product, we get an inductive description of all quadratic regular Hom–Leibniz algebras, allowing us to reduce their study to that of quadratic regular Hom–Lie algebras. Finally, we construct several non-trivial examples of symmetric (resp. quadratic) Hom–Leibniz algebras.

Keywords

Hom–Leibniz algebrasrepresentations of Hom–Leibniz algebrassymmetric Hom–Leibniz algebrasquadratic Hom–Leibniz algebrasT*-extensiondouble extension

MSC classification

Primary: 17A30: Algebras satisfying other identities 17A32: Leibniz algebras
Secondary: 17A45: Quadratic algebras (but not quadratic Jordan algebras) 17A60: Structure theory

Information

Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 24
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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