Cohomology and crossed modules extension of Hom-Leibniz-Rinehart algebras

TL;DR

This paper develops the theory of crossed modules and cohomology for Hom-Leibniz-Rinehart algebras, establishing a correspondence between abelian extensions and second cohomology, and linking crossed modules to third cohomology.

Contribution

It introduces the concept of crossed modules for Hom-Leibniz-Rinehart algebras and explores their cohomological properties, providing new insights into their extension theory.

Findings

One-to-one correspondence between abelian extensions and second cohomology groups.

Natural map from crossed modules extension to third cohomology group.

Establishment of cohomology theory for Hom-Leibniz-Rinehart algebras.

Abstract

In this paper, we introduce the concept of crossed module for Hom-Leibniz-Rinehart algebras. We study the cohomology and extension theory of Hom-Leibniz-Rinehart algebras. It is proved that there is one-to-one correspondence between equivalence classes of abelian extensions of Hom-Leibniz-Rinehart algebras and the elements of second cohomology group. Furthermore, we prove that there is a natural map from $\alpha$-crossed modules extension of Hom-Leibniz-Rinehart algebras to the third cohomology group of Hom-Leibniz-Rinehart algebras.