Cohomology and deformation of compatible Hom-Leibniz algebras

TL;DR

This paper develops a cohomology theory and deformation analysis for compatible Hom-Leibniz algebras, revealing their algebraic structure and deformation properties through Maurer-Cartan elements and Nijenhuis operators.

Contribution

It introduces a graded Lie algebra framework for compatible Hom-Leibniz algebras and explores their cohomology, deformations, and representations.

Findings

Maurer-Cartan elements characterize compatible Hom-Leibniz structures

Cohomology relates to infinitesimal deformations and Nijenhuis operators

Cohomology with arbitrary coefficients is established

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.