Non-abelian cohomology and extensions of Hom-algebras via the \boldsymbolβ-Nijenhuis--Richardson bracket

2020/12/02 by Nejib Saadaoui, Saadaoui, Nejib
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2012.01561

openalex publication_date 2020/12/02 · openalex created_date 2022/07/25 · openalex updated_date 2026/08/01

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 H²(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.

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