Hom-groups, representations and homological algebra

Abstract

A Hom-group G is a nonassociative version of a group where associativity, invertibility, and unitality are twisted by a map α : G −→ 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.2010 Mathematics Subject Classification. 17D99, 06B15, 20J05.

Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

“…Throughout this paper, we consider regular Hom-groups that is the case when the structure map is invertible, and this notion can be traced back to Caenepeel and Goyvaerts's pioneering work [3]. The axioms in the following definition of Hom-group is different from the one in [7,8,13]. However, we show that if the structure map is invertible, then some axioms in the original definition are redundant and can be obtained from the Hom-associativity condition.…”

“…He also proved Lagrange's theorem for finite Hom-groups in [8]. The recent developments on Hom-groups (see [7,8,13]) make it natural to study Hom-Lie groups and to explore the relationship between Hom-Lie groups and Hom-Lie algebras.…”

“…Moreover, one can associate a Hom-group to any Hom-Lie algebra by considering group-like elements in its universal enveloping algebra. Recently, M. Hassanzadeh developed representations and a (co)homology theory for Hom-groups in [7]. He also proved Lagrange's theorem for finite Hom-groups in [8].…”

Free Hom-groups, Hom-rings and Semisimple modules

“…One can associate a Hom-group to any Hom-Lie algebra by considering group-like elements in its universal enveloping algebra. Recently, M. Hassanzadeh developed representations and a (co)homology theory for Hom-groups in [7]. He also proved Lagrange's theorem for finite Hom-groups in [8].…”

Matching Hom-Setting of Rota-Baxter Algebras, Dendriform Algebras, and Pre-Lie Algebras

“…It is shown that a Homassociative algebra gives rise to a Hom-Lie algebra using the commutator. Since then, various Hom-analogues of some classical algebraic structures have been introduced and studied intensively, such as Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras [24,25], Hom-groups [26,27], Hom-Hopf modules [28], Hom-Lie superalgebras [29,30], generalize Hom-Lie algebras [31], and Hom-Poisson algebras [32].…”