Hom-Groups, Representations and Homological Algebra

TL;DR

This paper introduces Hom-groups, a nonassociative generalization of groups twisted by a map, and develops their (co)homology theories, connecting them to Hom-algebras and Hochschild (co)homologies.

Contribution

It defines Hom-groups and their modules, develops associated (co)homology theories, and relates these to existing algebraic structures like Hochschild (co)homology.

Findings

Hom-groups generalize classical groups with twisted associativity.

Hom-group (co)homology differs from classical group (co)homology in coefficients.

Hom-group (co)homology relates to Hochschild (co)homology of Hom-group algebras.

Abstract

A Hom-group G is a nonassociative version of a group where associativity, invertibility, and unitality are twisted by a map \alpha: G\longrightarrow G. Introducing the Hom-group algebra KG, we observe that Hom-groups are providing examples of Homalgebras, Hom-Lie algebras and Hom-Hopf algebras. We introduce two types of modules over a Hom-group G. To find out more about these modules, we introduce Hom-group (co)homology with coefficients in these modules. Our (co)homology theories generalizes group (co)homologies for groups. Despite the associative case we observe that the coefficients of Hom-group homology is different from the ones for Hom-group cohomology. We show that the inverse elements provide a relation between Hom-group (co)homology with coefficients in right and left G-modules. It will be shown that our (co)homology theories for Hom-groups with coefficients could be reduced to the Hochschild (co)homologies of Hom-group algebras. For certain coefficients the functoriality of Hom-group (co)homology will be shown.