On Hom-Gerstenhaber algebras and Hom-Lie algebroids

TL;DR

This paper introduces new algebraic structures called hom-Gerstenhaber and hom-Lie algebroids, explores their representations and cohomology, and connects them to geometric structures and homology theories in differential geometry.

Contribution

It defines hom-Batalin-Vilkovisky and strong differential hom-Gerstenhaber algebras, studies their representations and cohomology, and links these structures to geometric and homological concepts.

Findings

Defined hom-Batalin-Vilkovisky algebras and strong differential hom-Gerstenhaber algebras.

Established a cohomology theory for regular hom-Lie algebroids.

Connected hom-Gerstenhaber algebras to geometric structures and introduced homology theories.

Abstract

We define the notion of hom-Batalin-Vilkovisky algebras and strong differential hom-Gerstenhaber algebras as a special class of hom-Gerstenhaber algebras and provide canonical examples associated to some well-known hom-structures. Representations of a hom-Lie algebroid on a hom-bundle are defined and a cohomology of a regular hom-Lie algebroid with coefficients in a representation is studied. We discuss about relationship between these classes of hom-Gerstenhaber algebras and geometric structures on a vector bundle. As an application, we associate a homology to a regular hom-Lie algebroid and then define a hom-Poisson homology associated to a hom-Poisson manifold.