Non-abelian cohomology and extensions of Hom-algebras via the $\boldsymbolβ$-Nijenhuis--Richardson bracket
Nejib Saadaoui
TL;DR
The paper develops a cohomology framework for non-abelian extensions of Hom-Leibniz algebras via the $β$-Nijenhuis--Richardson bracket, unifying left, right, and symmetric cases. It shows that split extensions of a Hom-Leibniz algebra $L$ by a module $V$ are classified by the second cohomology group $H^2_β(L,V)$, extending classical Lie/Leibniz results to the Hom setting. The authors construct explicit $2$-cocycle data $(\lambda_l,\lambda_r,\theta)$ encoding extensions, derive coboundary operators, and establish a bijection between extension classes and $H^2_β(L,V)$, including deformation-theoretic perspectives. An application to two-dimensional regular Hom-Leibniz algebras demonstrates the method's concreteness and yields a usable algorithm for non-simple algebra classifications, highlighting the practical impact of this cohomological approach.
Abstract
This paper develops a cohomology theory for Hom-Leibniz algebras using the $β$-Nijenhuis--Richardson bracket and applies it to classify non-abelian extensions. We introduce left, and right versions of the bracket, each defining a graded Lie algebra structure on the space of $β$-cochains. The main result establishes that equivalence classes of split extensions of a Hom-Leibniz algebra $L$ by $V$ are in bijection with the second cohomology space $H^2(L,V)$, generalizing classical results from Lie and Leibniz algebra theory. We characterize extensions explicitly through 2-cocycles $(λ_l, λ_r, θ)$ and provide complete classifications of low-dimensional cases.