Non-abelian extensions of Hom-Jacobi-Jordan algebras

TL;DR

This paper introduces a cohomology framework for Hom-Jacobi-Jordan algebras, enabling classification of non-abelian extensions via second cohomology groups, generalizing classical algebraic theories.

Contribution

It develops a cohomology theory for Hom-Jacobi-Jordan algebras and classifies non-abelian extensions using second cohomology, extending classical algebraic results.

Findings

Establishes a bijection between split extensions and second cohomology classes.

Provides explicit characterization of extensions through 2-cocycles.

Classifies low-dimensional non-abelian extensions.

Abstract

This paper develops a cohomology theory for Hom-Jacobi-Jordan algebras using and applies it to classify non-abelian extensions. The main result establishes that equivalence classes of split extensions of a Hom-Jacobi-Jordan algebra $J$ by $V$ are in bijection with the second cohomology group $H^2(J,V)$, generalizing classical results from Lie and Leibniz algebra theory. We characterize extensions explicitly through 2-cocycles $(\rho, \theta)$ satisfying compatibility conditions, and provide complete classifications of low-dimensional cases.