In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category C endowed with a symmetric 2-trace, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the cyclic (co)homology of the (co)algebra "with coefficients in F ". We observe that if M is a C-bimodule category equipped with a stable central pair then C acquires a symmetric 2-trace. The dual notions of symmetric 2-contratraces and stable central contrapairs are derived as well. As an application we can recover all Hopf cyclic type (co)homology theories, obtain a conceptual understanding of anti-Yetter-Drinfeld modules, and give a formula-free definition of cyclic cohomology. The machinery can also be applied in settings more general than Hopf algebra modules and comodules.2010 Mathematics Subject Classification. monoidal category (18D10), abelian and additive category (18E05), cyclic homology (19D55), Hopf algebras (16T05).

Hom-groups are nonassociative generalizations of groups where the unitality and associativity are twisted by a map. We show that a Hom-group (G, α) is a pointed idempotent quasigroup (pique). We use Cayley table of quasigroups to introduce some examples of Hom-groups. Introducing the notions of Hom-subgroups and cosets we prove Lagrange's theorem for finite Hom-groups. This states that the order of any Hom-subgroup H of a finite Hom-group G divides the order of G. We linearize Homgroups to obtain a class of nonassociative Hopf algebras called Hom-Hopf algebras. As an application of our results, we show that the dimension of a Hom-sub-Hopf algebra of the finite dimensional Hom-group Hopf algebra KG divides the order of G.The new tools introduced in this paper could potentially have applications in theories of quasigroups, nonassociative Hopf algebras, Hom-type objects, combinatorics, and cryptography.

In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Homassociative setting.

We define the notion of equivariant ×-Hopf Galois extension and apply it as a functor between the categories of SAYD modules of the ×-Hopf algebras involving in the extension. This generalizes the result of Jara-Stefan and Böhm-Stefan on associating a SAYD modules to any ordinary Hopf Galois extension.

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.