The Structure Theorems of Weak Hopf Algebras

In this article, we mainly give the structure theorem of endomorphism algebras of weak Hopf algebras, and give another structure theorem as well as some applications for weak Doi–Hopf modules.     

The Structure of Hopf Algebras with a Weak Projection

We analyze the structure of a Hopf algebra H that has a Hopf subalgebra H " and a left H "-module coalgebra projection onto H ". In this situation H H " Q for Q = H / H "⁺ H, and the Hopf algebra structure on H can be recovered from suitable structures on Q, among others an in general nonassociative multiplication. The construction of H from H " and Q generalizes Radford biproduct, double crossproducts, and certain bicrossproducts. Further examples are Hopf algebras with a triangular decomposition, like all quantized enveloping algebras. In an appendix, we improve a standard criterion for a bicrossproduct A B of two Hopf algebras to be a Hopf algebra, and we show that in this case the antipode of the bicrossproduct is bijective if the antipodes of the factors are.     

The Duality Theorem for Weak Hopf Algebra (Co) Actions

In this article, we mainly study a new notion of a generalized smash product for weak Hopf comodule algebras and provide a new version of the duality theorem for weak smash products as an application.     

双边弱smash积的Maschke型定理

本文主要给出半单弱Hopf代数上双边弱Smash积的Maschke型定理.     

半单弱Hopf代数的作用

令H是半单弱Hopf代数, A是左H-模代数.我们证明了正则A-模的内射维数, A#H-模A的内射维数和正则A#H-模的内射维数三者是相等的. 而且,利用H在A上的不动点代数我们给出了A是Gorenstein代数的充要条件.     

模余代数的结构定理(英文)

This paper,mainly gives the structure theorem for module coalgebras by a kind of new method,and deletes the condition that the antipode S of the Hopf algebra H is bijective.     

A Generalized Drinfeld Quantum Double Construction Based on Weak Hopf Algebras

Let B and H be weak Hopf algebras with bijective antipodes S _B and S _H , respectively. Based on a compatible weak Hopf dual pairing (B, H, σ), we construct a generalized Drinfeld quantum double 𝔻(B, H) which is a weak T-coalgebra over a twisted semi-direct square of groups. In particular, when B and H are finite dimensional and the above pairing map σ is nondegenerate, 𝔻(B, H) admits a nontrivial quasitriangular structure. Some explicit examples are given as an application of our theory.     

Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras

Abstract

In this paper we construct a cylindrical module A ? ? for an ?-comodule algebra A, where the antipode of the Hopf algebra ? is bijective. We show that the cyclic module associated to the diagonal of A ? ? is isomorphic with the cyclic module of the crossed product algebra A ? ?. This enables us to derive a spectral sequence for the cyclic homology of the crossed product algebra. We also construct a cocylindrical module for Hopf module coalgebras and establish a similar spectral sequence to compute the cyclic cohomology of crossed product coalgebras.     

Twisted Comodule Algebras and the Generalized Quantum Double

In this article we show that the notion of twisted comodule algebras is a generalization of Drinfeld's cocycle twisting construction of bialgebras or Hopf algebras, and that several other twisting constructions, such as Ferrer's twisting product and Lu's one-sided twisting, are also special examples of twisted comodule algebras. Some general properties on twisted comodule algebras are investigated, and applications to the generalized quantum double are given.     

带弱内射的Hopf代数结构

利用已知Hopf代数构造新的Hopf代数是Hopf代数理论中最基本的问题之一．该文给出了Smash积A#H为Hopf代数，H是A#H的商Hopf代数， 且具有弱内射H→A#H的充分必要条件．易证，此种构造推广了Radford和Majid等人所构造的双积和双交叉积等结构．     

弱Hopf代数作用与冲积

本文研究了弱Hopf代数上的冲积并讨论了它约性质．设H是弱Hopf代数,A是左H-摸代数．我们给出了冲积A#H是弱双代数的一个充分条件以及A#H是A可分扩张的一个判定条件．另外,利用积分理论研究了Hopf模代数的有限性条件．     

Crossed Products over Weak Hopf Algebras Related to Cleft Extensions and Cohomology

The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H.For a right H-comodule algebra,they obtain a bijective correspondence between the isomorphisms classes of H-cleft extensions AH → A,where AH is the subalgebra of coinvariants,and the equivalence classes of crossed systems for H over AH.Finally,they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group HφZ(AH)2(H,Z(AH)),where Z(AH)is the center of AH.     

Duality theorem for weak L-R smash products

This paper gives a duality theorem for weak L-R smash products, which extends the duality theorem for weak smash products given by Nikshych.     

Structure Theorems for Bicomodule Algebras Over Quasi-Hopf Algebras,Weak Hopf Algebras,and Braided Hopf Algebras

Let H be a quasi-Hopf algebra, a weak Hopf algebra, or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v: H → B. Then we can define an object B^co(H), which is a left-left Yetter–Drinfeld module over H, having extra properties that allow to make a smash product B^co(H)#H, which is an H-bicomodule algebra, isomorphic to B.     

弱Hopf代数上的弱Alternative Doi-Hopf模

该文首先引入了弱Hopf代数上的弱Alternative Doi-Hopf模,然后构造了从弱Alternative Doi-Hopf模范畴到模范畴(余模范畴)忘却函子的伴随函子.     

The Structure of Weak Hopf Algebras Corresponding to U q (sl 2)

The structure of weak Hopf algebras corresponding to U _q (sl ₂) are classified by their algebra structure and coalgebra structure. The algebra structure of weak Hopf algebras corresponding to U _q (sl ₂) can be written as the direct sum of U _q (sl ₂) and an algebra of polynomials. The coalgebra structure of weak Hopf algebras corresponding to U _q (sl ₂) are classified by their Ext quiver. There are four types of such structures.     

Crossed Products for Weak Hopf Algebras

Weak crossed products by a weak bialgebra are defined. The resulting structure is not that of a unital algebra but an associative algebra with preunit. A general formula of such a product is given in terms of a weak 2-cocycle and a weak measuring. The relation with weak cleft extensions is studied. Equivalences of weak crossed products are defined and are related to gauge transformations. A relation between cleaving maps is described in terms of gauge transformations.     

重模代数和量子Yang-Baxter模代数的量子化

In this paper, we mainly construct quantization of dimodule algebras and quantum Yang-Baxter H-module algebras, and give some results of smash products and braided products.     

弱HOPF代数的中心构建与基本定理

范畴D(H)^M与范畴H^M的中心是同构的，并将Hopf模的结构定理推广到弱Hopf代数上。     

Hopf代数上的Smash积代数的K0群结构

本文主要给出了Smash积代数的K_0群结构,以及余交换且点化Hopf代数的K_0群结构及其正合性质;并利用一种新的有限对偶函子H()~0证明了K_0(A＃H)≌κ_0(_HA~0×H~0).