The Structure Theorem of Weak Comodule Algebras

This article mainly gives the structure theorem of weak comodule algebras, that is, assume that H is a weak Hopf algebra, and B a weak right H-comodule algebra, if there exists a morphism φ: H → B of a weak right H-comodule algebras, then there exists an algebra isomorphism: B ? B ^coH #H, where B ^coH denotes the coinvariant subalgebra of B, and B ^coH #H denotes the weak smash product.     

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.     

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．     

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.     

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代数的余枢纽元

本文研究并刻画了弱Hopf代数的余枢纽元,给出了其余表示伴有枢纽结构的等价条件,并刻画了余枢纽元和余带状元之间的关系,给出了相对应的Deligne型定理.     

Hopf Algebras of Dimension 12

We conclude the classification of Hopf algebras of dimension 12 over an algebraically closed field of characteristic zero.     

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).     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

A Characterization of Quasitriangular Hopf Algebras

In this paper,we show that if H is a finite dimensional Hopf algebra then H is quasitri-angular if and only if H is coquasi-triangular. As a consequentility ,we obtain a generalized result of Sauchenburg.     

基于弱Hopf代数的半单范畴的构造

本文研究并刻画了交换环上弱Hopf代数、Yetter-Drinfeld模范畴的一些性质,给出了其能够做成半单范畴的充分条件.     

On a Class of Finite-dimensional Semisimple Hopf Algebras

Let H be a finite-dimensional and semisimple Hopf algebra over an algebraically closed field of characteristic 0 such that H has exactly one isomorphism class of simple modules that have not dimension 1. These Hopf algebras were the object of study in, for instance, [1 Artamonov , V. A. ( 2007 ). Semisimple finite-dimensional Hopf algebras . Sbornik: Mathematics 198 ( 9 ): 1221 – 1245 .[Crossref], [Web of Science ®] , [Google Scholar]] and [9 Mukhatov , R. B. ( 2009 ). On semisimple finite-dimensional Hopf algebras . Fundamentalnaya i Prikladnaya Matematika 15 ( 2 ): 133 – 143 . [Google Scholar]]. In this paper we study this property in the context of certain abelian extensions of group algebras and give a group theoretical criterion for such Hopf algebras to be of the above type. We also give a classification result in a special case thereof.     

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。     

On Quasi-Bicrossed Products of Weak Hopf Algebras

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.     

A Class of Multiplier Hopf Algebras

We compute the Drinfel’d double for the bicrossproduct multiplier Hopf algebra A = k[G] ⋊ K(H) associated with the factorization of an infinite group M into two subgroups G and H. We also show that there is a basis-preserving self-duality structure for the multiplier Hopf algebra A = k[G] ⋊ K(H) if there is a factor-reversing group isomorphism. Presented by A. Verschoren.     

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.     

Cyclic Homology of Hopf Algebras

A cyclic cohomology theory adapted to Hopf algebras has been introduced recently by Connes and Moscovici. In this paper, we consider this object in the homological framework, in the spirit of Loday and Quillen and Karoubi's work on the cyclic homology of associative algebras. In the case of group algebras, we interpret the decomposition of the classical cyclic homology of a group algebra in terms of this homology. We also compute both cyclic homologies for truncated quiver algebras.     

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.     

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.