On 3-Hom-Lie-Rinehart algebras

TL;DR

This paper introduces 3-Hom-Lie-Rinehart algebras, develops their cohomology theory, explores extensions, and studies formal deformations, advancing the algebraic understanding of these structures.

Contribution

It systematically defines 3-Hom-Lie-Rinehart algebras, constructs their cohomology complex, and analyzes their extensions and deformations, providing a foundational framework.

Findings

Defined 3-Hom-Lie-Rinehart algebras and their cohomology.

Characterized extensions via first cohomology.

Studied formal deformations of these algebras.

Abstract

We introduce the notion of 3-Hom-Lie-Rinehart algebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we consider extensions of a 3-Hom-Lie-Rinehart algebra and characterize the first cohomology space in terms of the group of automorphisms of an $A$-split abelian extension and the equivalence classes of $A$-split abelian extensions. Finally, we study formal deformations of 3-Hom-Lie-Rinehart algebras.