Fundamentals of quasigroup Hopf group-coalgebras

A large class of algebras(possibly nonassociative)with group-coalgebraic structures,called quasigroup Hopf group-coalgebras,is introduced and studied.Quasigroup Hopf group-coalgebras provide a unifying framework for the classical Hopf algebras and Hopf group-coalgebras as well as Hopf quasigroups.Then,basic results similar to those in Hopf algebras H are proved,such as anti-(co)multiplicativity of the antipode S:H→H,and S^(2)=id if H is commutative or cocommutative.

由弱H-Hopf模代数构造Rota-Baxter代数

给出弱H-Hopf模代数的定义及其构造定理,进而由弱H-Hopf模代数出发构造Rota-Baxter代数并提供具体的例子。

Structure theorem for Hopf group-coalgebra

Let π be a group with a unit 1; H is a Hopf π- coalgebra and A is a right π-H-comodule algebra. First, the notion of a two-sided relative （A, H）-Hopf π-comodule is introduced; then it is obtained that Hom A H （M, N） H and HOMA（M, N） are isomorphic as right Hopf π-H-comodules, where Hom A H（M, N） denotes the space of right A-module fight H-comodule morphisms and HOMa （M, N） denotes the rational space of a space Hom A（M, N） of right A-module morphisms. Secondly, the structure theorem of endomorphism algebras of two-sided relative （A, H）-Hopf π--comodules is established; that is, End A H （M）#H and END A（M, N） are isomorphic as fight Hopf π-H-comodules and algebras.

Morita context of weak Hopf group coalgebras

The concept of weak Hopf group coalgebras is a natural generalization of the notions of both weak Hopf algebras（quantum groupoids） and Hopf group coalgebras.Let π be a group.The Morita context is considered in the sense of weak Hopf π-coalgebras.Let H be a finite type weak Hopf π-coalgebra,and A a weak right π-H-comodule algebra.It is constructed that a Morita context connects A#H＊ which is a weak smash product and the ring of coinvariants AcoH.This result is the generalization of that of Wang＇s in the paper ＂Morita contexts,π-Galois extensions for Hopf π-coalgebras＂ in 2006.Furthermore,the result is important for constructing weak π-Galois extensions.

TWISTED PRODUCTS AND SMASH PRODUCTS OVER WEAK HOPF ALGEBRAS

This paper gives a sufficient and necessary condition for twisted products to be weak Hopf algebras, moreover, gives a description for smash products to be weak Hopf algebras. It respectively generalizes R.K.Molnar's major result and I.Boca's result.

Twisting theory for weak Hopf algebras

The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the （braided） monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra to generalize the result from Oeckl （2000）.

Weak Hopf Algebra in Yetter-Drinfeld Categories and Weak Biproducts

The Yetter-Drinfeld category of the Hopf algebra over a field is a prebraided 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.

MORITA CONTEXT OF WEAK HOPF ALGEBRAS

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context 〈A^H,A#H,A,A,τ,μ〉 connecting the smash product A#H and the invariant subalgebra A^H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.

SYMMETRIES IN YETTER-DRINFELD CATEGORIES OVER WEAK HOPF ALGEBRAS

This article is devoted to the study of the symmetry in the Yetter-Drinfeld category of a finite-dimensional weak Hopf algebra.It generalizes the corresponding results in Hopf algebras.

CONSTRUCTING POINTED WEAK HOPF ALGEBRAS BY ORE-EXTENSION

The main work of this article is to give a nontrivial method to construct pointed semilattice graded weak Hopf algebra from a Clifford monoid S =[Y; Gα. φα,β]by Ore-extensions, and to obtain a co-Frobenius semilattice graded weak Hopf algebra H（S, n, c, x, a, b） through factoring At by a semilattice graded weak Hopf ideal.

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.

A Maschke Type Theorem for Weak Hopf π-Comodules

In this paper, we mainly generalize a Maschke type theorem to the setting of a weak Hopf group coalgebra. First we introduce the notion of a weak Hopf group coalgebra as a generalization of Hopf group coalgebra introduced in [7] and a weak Hopf algebra introduced in [2]. And we study some basic properties of weak Hopf group coalgebras. Next we aim at finding some sucient conditions under which an epimorphism of weak （H, A） Hopf π-comodule splits if it splits as an A-module morphism and give an application of our results.

Gorenstein dimensions for weak Hopf-Galois extensions

The representation of weak Hopf algebras is studied by investigating the Gorenstein dimensions of weak Hopf algebras and weak Hopf-Galois extensions.Let H be a weak Hopf algebra with a bijective antipode,A a weak right H-comodule algebra and B the H-coinvariant subalgebra of A.First,some properties of Gorenstein projective H-modules in the representation category are studied,and the fact that Gorenstein global dimension of H is the same as the Gorenstein projective dimension of its left unital subalgebra is demonstrated.Secondly,by applying the integral theory of weak Hopf algebras,on the one hand,a sufficient and necessary condition that a projective A-module is a projective B-module is given;on the other hand,the separability of the functor AB-and that of the restriction of scalar function B(-)are described,respectively.Finally,as a mean result,the Gorenstein global dimension of a weak Hopf-Galois extension is investigated under the condition that H is both semisimple and cosemisimple.

弱Hopf群余模及其结构基本定理

构造了弱Hopf群余代数,证明了弱Hopf群余模结构基本定理.

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．

交叉余积上的弱Hopf代数结构

利用左H-模、弱双代数、交叉积等工具给出了交叉余积成为弱双代数与Hopf代数的充要条件.

弱Hopf Galois扩张的Cotorsion维数

考虑在Ext群上构造Grothendieck谱序列揭示弱Hopf Galois扩张的cotorsion维数.设H为有限维弱Hopf代数,A/B为弱H-Galois扩张,给出A,B的左cotorsion维数与H的右整体维数之间的关系,并讨论当B为可换的或H*为半单时,A,B的左cotorsion维数的性质.

弱Hopf群余代数Kegel定理(英文)

本文研究了余三角弱Hopfπ-余代数H的左弱π-H-余模代数.通过构造左弱π-H-余模代数的导出π-σ-李代数,得到了弱Hopfπ-余代数Kegel定理,推广了文献[4]的结果.

弱Doi-Hopf群模的Maschke型定理

设π为一个群,讨论了弱Doi-Hopfπ-模的半单性或可约性.设(H,A,C)是一个弱Doi-Hopfπ-数据,利用忘却函子将Doi-Hopfπ-模范畴π-CUA中的对象映为右A-模范畴MA中对象,通过对MA中可分单同态进行变形,建立了Doi-Hopfπ-数据积分概念.借助此积分,证明了弱Doi-Hopfπ-模的Mas-chke型定理,推广了一些文献的结果.

弱量子代数wU_p(■)的弱Hopf代数结构

设G为有限维半单李代数,参数q不是单位根.定义了一个具有弱Hopf代数结构的弱量子代数wU_q(■),构造了它的类群元素集,并给出了两个不同参数的弱量子代数同构的条件.