Hom-groups, representations and homological algebra
Volume 158 / 2019
Abstract
A Hom-group $G$ is a nonassociative generalization of a group where associativity, invertibility, and unitality are twisted by a map $\alpha : G\rightarrow G$. Introducing the Hom-group algebra $\mathbb {K}G$, we observe that Hom-groups provide examples of Hom-algebras, Hom-Lie algebras and Hom-Hopf algebras. We introduce two types of modules over a Hom-group $G$. To find out more about those modules, we introduce Hom-group (co)homology with coefficients in those modules. Our (co)homology theories generalize group (co)homology for groups. In contrast to the associative case, the coefficients of Hom-group homology are 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 is shown that our (co)homology theories for Hom-groups with coefficients could be reduced to Hochschild (co)homology of Hom-group algebras. For certain coefficients the functoriality of Hom-group (co)homology is shown.
