Hom-Lie-Rinehart Algebras

TL;DR

This paper introduces hom-Lie-Rinehart algebras as an algebraic framework extending hom-Lie algebroids, develops their cohomology theory, and explores their extensions and automorphisms with applications to Poisson algebras.

Contribution

It systematically defines hom-Lie-Rinehart algebras, constructs their cohomology complex, and characterizes their low-dimensional cohomology and extensions, including a canonical example related to Poisson algebras.

Findings

Defined hom-Lie-Rinehart algebras and their cohomology.

Characterized low-dimensional cohomology spaces.

Constructed a canonical example from Poisson algebra.

Abstract

We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterisation of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism.