Hom-Lie-Rinehart algebras

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 characterization 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.

KEYWORDS:

• Cohomology of Lie- Rinehart algebras
• hom-algebras
• Lie algebroids
• Lie- Rinehart algebras

2010 AMS MATHEMATICS SUBJECT CLASSIFICATION:

• 17A32
• 17B66
• 53D17

Acknowledgements

We would like to thank the anonymous referee for various comments, which improved the presentation of the paper. In particular, we thank the referee for remark on Definition 5.2 and definition of the center for hom-Lie-Rinehart algebras.