Mackey systems of monoidal categories and G-spectra

We construct an equivariant infinite loop space machine denned on certain class of monoidal O _G -categories which have built-in Mackey structure. Applications include the equivariant infinite delooping of the classifying space BF(G) for stable spherical G-fibrations and also the construction of an infinite loop G-space E(X, G) with ₀ ^HE (X, G) naturally isomorphic to the equivariant Whitehead groups Wh _H (X) of given G-space X.Dedicated to Professor Shôrô Araki on his sixtieth birthday     

Coherence for closed categories with biproducts

A coherence result for symmetric monoidal closed categories with biproducts is shown in this paper. It is also explained how to prove coherence for compact closed categories with biproducts and for dagger compact closed categories with dagger biproducts by using the same technique.     

The structure and construction of bi-Frobenius Hom-algebras

Bi-Frobenius Hom-algebras are introduced in this paper. They provide a common generalization of finite dimensional monoidal Hom-Hopf algebras and of bi-Frobenius algebras introduced by Doi and Takeuchi. The different conditions for finite dimensional monoidal Hom-algebras to be bi-Frobenius Hom-algebras are discussed. The substructures, quotient structures as well as morphisms of bi-Frobenius Hom-algebras are also studied. In addition, the Radford’s formula for S⁴ of a bi-Frobenius Hom-algebra is shown. The semisimplicity and separability for a special class of finite dimensional bi-Frobenius Hom-algebras are researched finally, which presents a version of Maschke’s theorem for this family.     

Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

We systematically study noncommutative and nonassociative algebras $A$ and their bimodules as algebras and bimodules internal to the representation category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of the monoidal category of $A$-bimodules by internal homomorphisms, and describe explicitly their evaluation and composition morphisms. For braided commutative algebras $A$ the full subcategory of symmetric $A$-bimodule objects is a braided closed monoidal category, from which we obtain an internal tensor product operation on internal homomorphisms. We describe how these structures deform under cochain twisting of the quasi-Hopf algebra, and apply the formalism to the example of deformation quantization of equivariant vector bundles over a smooth manifold. Our constructions set up the basic ingredients for the systematic development of differential geometry internal to the quasi-Hopf representation category, which will be tackled in the sequels to this paper, together with applications to models of noncommutative and nonassociative gravity such as those anticipated from non-geometric string theory.     

A Generalized Double Crossproduct and Drinfeld Double

In this paper we construct a new and more complicated algebra construction of two algebras B and H, a generalized double crossproduct B H. The left generalized smash product, the right generalized smash product, Majids double crossproduct, especially, the smash product, the Drinfeld Double D(H) and Doi-Takeuchi algebra B H are all special cases as our algebra structure. Next, we analyze conditions under which this new algebra B H is a Hopf algebra termed a generalized double crossproduct of Hopf algebra, and describe a coquasitriangular structure over the generalized double crossproduct Hopf algebra B H. Finally, what we do is to construct a new braided monoidal category ^J_JMod^Q_Q obtained from the structure of the generalized double crossproduct, and establish a kind of new quantum Yang-Baxter operators.AMS Subject Classification (1991): 16W20, 16D90, 16S40, 16W30     

Bialgebras Over Noncommutative Rings and a Structure Theorem for Hopf Bimodules

It is a key property of bialgebras that their modules have a natural tensor product. More precisely, a bialgebra over k can be characterized as an algebra H whose category of modules is a monoidal category in such a way that the underlying functor to the category of k-vector spaces is monoidal (i.e. preserves tensor products in a coherent way). In the present paper we study a class of algebras whose module categories are also monoidal categories; however, the underlying functor to the category of k-vector spaces fails to be monoidal. Instead, there is a suitable underlying functor to the category of B-bimodules over a k-algebra B which is monoidal with respect to the tensor product over B. In other words, we study algebras L such that for two L-modules V and W there is a natural tensor product, which is the tensor product V_BW over another k-algebra B, equipped with an L-module structure defined via some kind of comultiplication of L. We show that this property is characteristic for ×_B-bialgebras as studied by Sweedler (for commutative B) and Takeuchi. Our motivating example arises when H is a Hopf algebra and A an H-Galois extension of B. In this situation, one can construct an algebra L:=L(A,H), which was previously shown to be a Hopf algebra if B=k. We show that there is a structure theorem for relative Hopf bimodules in the form of a category equivalence . The category on the left hand side has a natural structure of monoidal category (with the tensor product over A) which induces the structure of a monoidal category on the right hand side. The ×_B-bialgebra structure of L that corresponds to this monoidal structure generalizes the Hopf algebra structure on L(A,H) known for B=k. We prove several other structure theorems involving L=L(A,H) in the form of category equivalences .     

Yetter—Drinfel'd H-Azumaya Monoids in Closed Categories

When C is a symmetric closed category with equalizers and coequalizers and H is a Hopf algebra in C, the category of Yetter—Drinfeld H-modules is a braided monoidal category.We develop a categorical version of the results in (10) constructing a Brauer group BQ(C,H) and studying its functorial properties.     

Quadratic algebra of square groups

Square groups are quadratic analogues of abelian groups. Many properties of abelian groups are shown to hold for square groups. In particular, there is a symmetric monoidal tensor product of square groups generalizing the classical tensor product.     

Symmetric centres of braided monoidal categories

This paper introduces the concept of ‘symmetric centres’ of braided monoidal categories. LetH be a Hopf algebra with bijective antipode over a fieldk. We address the symmetric centre of the Yetter-Drinfel’d module category: and show that a left Yetter-Drinfel’d moduleM belongs to the symmetric centre of and only ifM is trivial. We also study the symmetric centres of categories of representations of quasitriangular Hopf algebras and give a sufficient and necessary condition for the braid of, _Hℳ to induce the braid of , or equivalently, the braid of , whereA is a quantum commutativeH-module algebra     

K-Theory of Braided Monoidal Categories

We show that the commutativity constraint of a braided monoidal category gives rise to an algebraic structure on its K-theory known as a Gerstenhaber algebra. If, in addition, the braiding has a compatible balanced structure the Gerstenhaber bracket on the K-theory is generated by a Batalin–Vilkovisky differential. We use these algebraic structures to prove a generalization of the Anderson–Moore–Vafa theorem which says that the order of the twist, in a semi-simple balanced monoidal category with duals and finitely many simple objects, is finite.