Non-abelian cohomology and extensions of Hom-algebras via the $\boldsymbol{\beta}$-Nijenhuis--Richardson bracket

TL;DR

This paper introduces a cohomology framework for Hom-Leibniz algebras using the $eta$-Nijenhuis--Richardson bracket, enabling classification of non-abelian extensions and generalizing classical algebraic results.

Contribution

It develops a new cohomology theory for Hom-Leibniz algebras with graded Lie algebra structures, extending classical extension classification methods.

Findings

Establishes a bijection between split extensions and second cohomology classes.

Defines explicit 2-cocycles for characterizing extensions.

Provides classifications for low-dimensional cases.

Abstract

This paper develops a cohomology theory for Hom-Leibniz algebras using the $\beta$-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 $\beta$-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 $(\lambda_l, \lambda_r, \theta)$ and provide complete classifications of low-dimensional cases.