Cohomology and deformation of compatible Hom-Leibniz algebras

Abstract: In this paper, we consider compatible Hom-Leibniz algebra where the Hom map twists the operations in the compatible system. We consider a suitably graded Lie algebra whose Maurer-Cartan elements characterize the structure of compatible Hom-Leibniz algebras. Using this, we study cohomology, infinitesimal deformations, the Nijenhuis operator, and their relation for compatible Hom-Leibniz algebras. Finally we see the cohomology of compatible Hom-Leibniz algebra with coefficients in an arbitrary representation.