弱Hopf代数上的扭积

在弱Hopf代数上,定义了交叉积概念,并且得到了它的两种特殊形式,冲积和扭积．特别地,给出了扭积为弱Hopf代数的一个充要条件,推广了Hopf代数的相应结论．     

Fundamental Theorems of Weak Doi–Hopf Modules and Semisimple Weak Smash Product Hopf Algebras

Abstract

This paper, mainly gives a Fundamental Theorem of weak Doi–Hopf modules, which is not only generalizes the Fundamental Theorem of weak Hopf modules but also generalizes the Fundamental Theorem of relative Hopf modules. Moreover, it gives a sufficient and necessary condition for weak smash product algebras to be weak bialgebras, and a sufficient condition for weak smash product algebras to be semisimple weak Hopf algebras.     

Hopf代数胚上的Smash积的Maschke型定理和Morita关系

本文研究了Hopf代数胚上的Smash积代数.利用Hopf代数胚的积分理论,获得了Hopf代数胚上的Smash的Maschke型定理并构造了一个Morita关系,推广了Cohen和Fishman在文献[1]中的相应结果.作为应用,获得了Hopf代数胚上的余模代数的Maschke型定理.     

Hom-弱Hopf代数上的Hom-smash积

本文研究了在Hom-Hopf代数上引入Hom-弱Hop代数的问题.通过建立弱左H-模Hom-代数的方法,构造Hom-smash积,证明Hom-smash积是Hom-代数,且给出使之成为Hom-弱Hopf代数的充分条件,推广了由Bohm等人定义的弱Hop代数.     

弱Hopf代数上的ω-交叉积

在弱Hopf代数上,定义了ω-交叉积概念,并给出了ω-交叉积是弱Hopf代数的充要条件,推广了弱Hopf代数的相应结论.     

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．     

Maschke-type theorem and Morita context over weak Hopf algebras

This paper gives a Maschke-type theorem over semisimple weak Hopf algebras, extends the well-known Maschke-type theorem given by Cohen and Fishman and constructs a Morita context over weak Hopf algebras.     

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.     

L-R smash products for multiplier Hopf algebras

The theory of L-R smash product is extended to multiplier Hopf algebras and a sufficient condition for L-R smash product to be regular multiplier Hopf algebras is given. In particular, the result of the paper implies Delvaux＇s main theorem in the case of smash products.     

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.     

$G$-交叉积在$G$-inner作用下的等价刻画

本文给出两个具有相同半格$Y$ 的半格分次弱Hopf代数的$G$-inner 作用的定义, 然后给出两个$G$-交叉积同构的充分必要条件.     

Semidirect products of weak multiplier Hopf algebras: Smash products and smash coproducts

In this paper, we will develop the smash product of weak multiplier Hopf algebras unifying the cases of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras. We will show that the smash product R#A has a regular weak multiplier Hopf algebra structure if R and A are regular weak multiplier Hopf algebras. We shall investigate integrals on R#A. We also consider the result in the ?-situation and new examples. Dually, we consider the smash coproduct of weak multiplier Hopf algebras under an appropriate form and integrals on the smash coproduct and we obtain results in the ?-situation.     

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．     

Crossed Products in Weak Contexts

We define the general notion of crossed products in a weak context, which generalizes the ones defined by Blattner, Cohen and Montgomery, Doi and Takeuchi in the context of Hopf algebras and the one given by Brzeziński. Also, the crossed products obtained by the authors, for weak Hopf algebras living in a symmetric monoidal category and weak C-cleft extensions associated to weak entwined structures, are particular instances of this theory.     

半拟三角弱Hopf代数

作为拟三角弱Hopf代数的推广,我们引入了半拟三角弱Hopf代数的概念.令(H,R,v)是一个半拟三角弱Hopf代数,其中,R是其半拟三角结构.我们指明R保持了拟三角弱Hopf代数中泛R-矩阵的许多基本性质.特别地,讨论了Drinfeld元的性质,证明其是可逆的并且是余作用v的余不变量.另外,证明了半拟三角弱Hopf代数的对极平方是对合的.     

Products of Linear Maps on Bialgebras,with Applications to Left Hopf Algebras and Hopf Algebroids

The Doi–Koppinen generalized smash product can be applied to linear maps on bialgebras, and we define a new associative product for linear maps on bialgebras. We comment on the applications of these products to left Hopf algebras and Hopf algebroids.     

相应于Borcherds-Cartan矩阵的双参数弱Hopf代数

In this paper a class of two-parameter weak Hopf algebras wrτ,s（g） corresponding to Borcherds-Cartan matrix is constructed. This class consists of noncommutative and noncocommutative weak Hopf algebras but not Hopf algebras. It can be viewed as a generalized class of one-parameter weak Hopf algebras wUq（g）.     

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.     

(ω,σ)-Smash积积和和(ν,α)-Smash余余积积

设B,H是两个Hopf代数,构造了(ω,σ)-Smash积Bω#σH和(ν,α)-Smash余积Bν■αH,并给出了Bω#σH是Hopf代数和Bν■αH是双代数的充要条件,证明了许多已知的积和余积是它们的特殊情况.     

A Class of Coquasitriangular Hopf Group Algebras

This article is devoted to constructing a class of coquasitriangular Hopf group algebras. First, we introduce the notion of group unified product. Then we prove the classification theorem for group unified products and discuss the sufficient and necessary conditions to make the group unified product to be a coquasitriangular Hopf group-algebra. Finally, some applications of the main results are considered.