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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, right, and symmetric versions of the bracket, each defining a graded Lie algebra structure on the space of $\beta$-cochains, and prove that the coboundary operator $D_k(f) = [d,f]$ satisfies $D_{k+1} \circ D_k = 0$. 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 group $H^2(L,V)$, generalizing classical results from Lie and Leibniz algebra theory. We characterize extensions explicitly through 2-cocycles $(\lambda_l, \lambda_r, \theta)$ satisfying compatibility conditions, and provide complete classifications of low-dimensional cases, including all two-dimensional regular Hom-Leibniz algebras.