Nichols algebras over weak Hopf algebras

In this paper, we study a Yetter-Drinfeld module V over a weak Hopf algebra H.Although the category of all left H-modules is not a braided tensor category, we can define a Yetter-Drinfeld module. Using this Yetter-Drinfeld modules V, we construct Nichols algebra B(V) over the weak Hopf algebra H, and a series of weak Hopf algebras. Some results of [8] are generalized.     

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 .     

弱辫子T-范畴（英文）

Recently Turaev generalized the notion of a tensor category to that of a crossed group category.In[5]the authors constructed the representation category Rep(H) of a T-coalgebra H.In[2]the authors introduced the notions of a weak tensor category to characterize a weak bialgebra and a weak Hopf algebra.This paper is based on these ideas to naturally introduce the notions of a weak T-category and a weak braided T-category which are not under the usual way and prove that the categories of representations of a weak T-coalgebra and a weak braided T-coalgebra are a weak T-category and a weak braided T-category respectively.Furthermore we also discuss some properties of weak T-category.     

关于(f,τ) -相容Hopf代数对(B,H)

该文定义了(f,τ) -相容Hopf代数对(B,H),利用这样的对(B,H),给出了左H -余模范畴^HM的一个辫子张量子范畴,从而得到一个量子Yang-Baxter算子,并且通过扭曲Hopf代数$B$的乘法,构造出Yetter-Drinfeld范畴中^H_HYD的Hopf代数.     

关于(f,T)-相容对(B,H)和_H~HYD中的Hopf代数

本文定义了(f,T)-相容对(B,H),利用这样的相容对可以给出一个辫子张量 范畴和一个量子Yang-Baxter方程的解,并且通过扭曲Hopf代数B的乘法,构造 Yetter-Drinfeld范畴中HHyD的Hopf代数.     

(f,ω)-Compatible Pair (B,H) for ω-Smash Coproduct Hopf Algebras

In this paper we introduce the notion of (f,ω)-compatible pair (B,H), by which we construct a Hopf algebra in the category HHYD of Yetter-Drinfeld H-modules by twisting the comultiplication of B. We also study the property of ω-smash coproduct Hopf algebras Bω H.     

Invariants of piecewise-linear 3-manifolds

This paper presents an algebraic framework for constructing invariants of closed oriented 3-manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A significant feature is that the tensor category is not required to be braided. The main examples are constructed from the categories of representations of involutive Hopf algebras and of quantised enveloping algebras at a root of unity.

     

Quantum doubles in monoidal categories

Let H be a Hopf algebra in a rigid symmetric monoidal category C then the evaluation map τis a convolution-invertible skew pairing. In the previous paper, we constructed a Hopf algebra D(H)=H ? _r H ^?cop in C. In this paper, we first show that D(H) is a quasitriangular Hopf algebra in C. Next, let H be an ordinary triangular finite-dimensional Hopf algebra. Then one can form quasitriangular Hopf algebras B(H,H) and B(H,D(H)) (in a rigid braided monoidal category) by Majid’s method associated to the ordinary Hopf algebra maps H →H and i_H H → D(H), where D(H) is the Drin-fePd quantum double. We show that D (B(H,H)) and B(H,D(H)) are isomorphic Hopf algebras in the braided monoidal category.     

关于$\omega$-Smash余积Hopf代数的$(f,\omega)$相容对$(B,,H)$

In this paper we introduce the notion of （f,ω）-compatible pair （B,H）, by which we construct a Hopf algebra in the category HHYD of Yetter-Drinfeld H-modules by twisting the comultiplication of B. We also study the property of ω-smash coproduct Hopf algebras Bω H.     

H-SEPARABLE RINGS AND THEIR HOPF-GALOIS EXTENSIONS

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弱Hopf代数的扭曲理论

本文研究了弱Hopf代数的扭曲理论的对偶问题.利用了弱Hopf代数上的弱Hopf双模的(辫子)张量范畴与扭曲弱Hopf代数上的弱Hopf双模的(辫子)张量范畴等价方法,得到Long模范畴是Yetter-Drinfel'd模范畴的辫子张量子范畴.推广了Oeckl(2000)的结果.     

Hopf monads on monoidal categories

We define Hopf monads on an arbitrary monoidal category, extending the definition given in Bruguières and Virelizier (2007) [5] for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode.Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid.Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C. The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford–Majid bosonization of Hopf algebras).We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedler?s Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).     

Weak Hopf Algebra in Yetter-Drinfeld Categories and Weak Biproducts

The Yetter-Drinfeld category of the Hopf algebra over a field is a pre braided category. In this paper we prove this result for the weak Hopf algebra. We study the smash product and smash coproduct, weak biproducts in the weak Hopf algebra over a field k. For a weak Hopf algebra A in left Yetter-Drinfeld category HHYD. we prove that the weak biproducts of A and H is a weak Hopf algebra.     

三角Hopf余模范畴上的张量余代数

在三角Ｈｏｐｆ代数余模范畴上研究张量余代数．主要给出三角Ｈｏｐｆ代数余模范畴上的张量余代数的结构．     

Braided bosonization and inhomogeneous quantum groups

We consider quasitriangular Hopf algebras in braided tensor categories introduced by Majid. It is known that a quasitriangular Hopf algebra H in a braided monoidal category C induces a braiding in a full monoidal subcategory of the category of H-modules in C. Within this subcategory, a braided version of the bosonization theorem with respect to the category C will be proved. An example of braided monoidal categories with quasitriangular structure deviating from the ordinary case of symmetric tensor categories of vector spaces is provided by certain braided supersymmetric tensor categories. Braided inhomogeneous quantum groups like the dilaton free q-Poincaré group are explicit applications.Supported in part by the Deutsche Forschungsgemeinschaft (DFG) through a research fellowship.     

The formal theory of hopf algebras Part I: Hopf monoids in a monoidal category

Abstract

The category Hopf ? of Hopf monoids in a symmetric monoidal category ?, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ? preserve directed colimits one has the following results: (1) If, in ?, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ? and comonadic over the category of monoids in C. (2) If, in ?, extremal monomorphisms are stable under tensor squaring, then Hopf ? is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ?.     

Making non-trivially associated tensor categories from left coset representatives

The paper begins by giving an algebraic structure on a set of coset representatives for the left action of a subgroup on a group. From this we construct a non-trivially associated tensor category. Also a double construction is given, and this allows the construction of a non-trivially associated braided tensor category. In this category we explicitly reconstruct a braided Hopf algebra, whose representations comprise the category itself.     

弱Hopf群T-余代数上的弱Doi-Hopf群模

在弱Hopf群T-余代数情形下,弱量子Yetter-Drinfeld群模的概念被引入,并证明了弱量子Yetter-Drinfeld群模是特殊的弱Doi-Hopf群模.接着建立了弱量子Yetter Drinfeld群模范畴与弱Hopf群双余模代数的余不动点子代数B上模范畴之间的伴随对.最后考虑了弱量子Yetter-Drinfeld群模的积分.