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.     

Coquasitriangular Hopf group coalgebras and braided monoidal categories

Let ?? be a group, and let H be a Hopf ??-coalgebra. We first show that the category M ^H of right ??-comodules over H is a monoidal category and there is a monoidal endofunctor (F _?? , id, id) of M ^H for any ?? ?? ??. Then we give the definition of coquasitriangular Hopf ??-coalgebras. Finally, we show that H is a coquasitriangular Hopf ??-coalgebra if and only if M ^H is a braided monoidal category and (F _?? , id, id) is a braided monoidal endofunctor of M ^H for any ?? ?? ??.     

扭曲Smash余积的辫Monoidal范畴

主要讨论扭曲Smash余积余模范畴c×Hll,得到c×Hll是辫monoidal范畴的一个充要条件.     

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     

A monoidal approach to splitting morphisms of bialgebras

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra such that its Jacobson radical is a nilpotent Hopf ideal and is a semisimple algebra. We prove that the canonical projection of on has a section which is an -colinear algebra map. Furthermore, if is cosemisimple too, then we can choose this section to be an -bicolinear algebra morphism. This fact allows us to describe as a `generalized bosonization' of a certain algebra in the category of Yetter-Drinfeld modules over . As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. Let be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of into which is an -linear coalgebra morphism. Furthermore, if is semisimple too, then we can choose this retraction to be an -bilinear coalgebra morphism. Then, also in this case, we can describe as a `generalized bosonization' of a certain coalgebra in the category of Yetter-Drinfeld modules over .

     

Quantum commutative algebras and their duals

If an algebraA is quantum commutative with respect to the action of a quasitriangular Hopf algebraH, then the monoidal structure on the category_H ^ℳ of modules overH induces a rnonoidal structure on the category_A#H ^ℳ of modules over the associated smash productA # H. The condition under which the braiding structure of_H ^ℳ induces a braiding structure on_A#H ^ℳ is further investigated. Dually, the notion of quantum cocommutativity is introduced, and similar result in this dual situation is obtained.     

Loop spaces, and coherence for monoidal and braided monoidal bicategories

We prove a coherence theorem for braided monoidal bicategories and relate it to the coherence theorem for monoidal bicategories. We show how coherence for these structures can be interpreted topologically using up-to-homotopy operad actions and the algebraic classification of surface braids.     

Centre of an algebra

Motivated by algebraic structures appearing in Rational Conformal Field Theory we study a construction associating to an algebra in a monoidal category a commutative algebra (full centre) in the monoidal centre of the monoidal category. We establish Morita invariance of this construction by extending it to module categories.As an example we treat the case of group-theoretical categories.     

Hochschild cohomology of ring objects in monoidal categories

We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.     

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.     

Calculus and Quantizations Over Hopf Algebras

In this paper we outline an approach to calculus over quasitriangular Hopf algebras. We construct braided differential operators and introduce a general notion of quantizations in monoidal categories. We discuss some applications to quantizations of differential operators.     

Binary operations suffice to test collapsing of monoidal intervals

It will be shown that the stabilizer clone of a transformation monoid is either trivial, i.e., it is generated by the monoid itself, or it contains an essentially binary function. Received September 16, 1996; accepted in final form May 21, 1997.     

On the Structure of Modular Categories

For a braided tensor category C and a subcategory K there isa notion of a centralizer C_C K, which is a full tensor subcategoryof C. A pre-modular tensor category is known to be modular inthe sense of Turaev if and only if the center Z₂C C_CC (not tobe confused with the center Z₁ of a tensor category, relatedto the quantum double) is trivial, that is, consists only ofmultiples of the tensor unit, and dimC 0. Here , the X_i being the simple objects. We prove several structural properties of modular categories.Our main technical tool is the following double centralizertheorem. Let C be a modular category and K a full tensor subcategoryclosed with respect to direct sums, subobjects and duals. ThenC_CC_CK = K and dim K·dim C_CK = dim C. We give several applications. (1) If C is modular and K is a full modular subcategory,then L=C_CK is also modular and C is equivalent as a ribbon categoryto the direct product: . Thus every modular category factorizes (non-uniquely, in general)into prime modular categories. We study the prime factorizationsof the categories D(G)-Mod, where G is a finite abelian group. (2) If C is a modular *-category and K is a full tensorsubcategory then dim C dim K · dim Z₂K. We give exampleswhere the bound is attained and conjecture that every pre-modularK can be embedded fully into a modular category C with dim C=dimK·dim Z₂K. (3) For every finite group G there is a braided tensor*-category C such that Z₂CRep,G and the modular closure/modularization is non-trivial. 2000 MathematicsSubject Classification 18D10.     

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.     

Topological Hopf Algebras and Braided Monoidal Categories

The relation between a monoidal category which has an exact faithful monoidal functor to a category of finite rank projective modules over a Dedekind domain, and the category of continuous modules over a topological bialgebra is discussed. If the monoidal category is braided, the bialgebra is topologically quasitriangular. If the monoidal category is rigid monoidal, the bialgebra is a Hopf algebra.     

辫子范畴中的bi-Frobenius代数

在辫子范畴中考察Doi的关于bi-Probenius代数的结果．证明了辫子bi-Frobenius代数的同态基本定理．     

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     

余拟三角弱Hopf群代数与辫子Monoidal范畴

In this paper, we first give the definitions of a crossed left π-H-comodules over a crossed weak Hopf π-algebra H, and show that the category of crossed left π-H-comodules is a monoidal category. Finally, we show that a family σ = {σα,β: Hα Hβ→ k}α,β∈πof k-linear maps is a coquasitriangular structure of a crossed weak Hopf π-algebra H if and only if the category of crossed left π-H-comodules over H is a braided monoidal category with braiding defined by σ.     

On categories associated to a Quasi-Hopf algebra

A quasi-Hopf algebra H can be seen as a commutative algebra A in the center 𝒵(H-Mod) of H-Mod. We show that the category of A-modules in 𝒵(H-Mod) is equivalent (as a monoidal category) to H-Mod. This can be regarded as a generalization of the structure theorem of Hopf bimodules of a Hopf algebra to the quasi-Hopf setting.     

The Giry monad is not strong for the canonical symmetric monoidal closed structure on Meas

We show that the Giry monad is not strong with respect to the canonical symmetric monoidal closed structure on the category Meas of all measurable spaces and measurable functions.