Self-dual weak Hopf algebras

In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually simple forms in the sense of E. L. Green and E. N. Morcos. We also give the definition of self-dual weak Hopf quiver and apply these types of quivers to classify the finite- dimensional self-dual semilattice graded weak Hopf algebras. Finally, we prove partially the conjecture given by N. Andruskiewitsch and H.-J. Schneider in the case of finite-dimensional pointed semilattice graded weak Hopf algebra H when grH is self-dual.     

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.     

A maschke type theorem for weak Hopf algebras

In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.     

扭曲余积和广义相关Hopf模

本文在学习［1,2］之后,首先给出了扭曲余积关于量子余交换的一个重要性质;在学习［3］之后,又证明了广义相关Hopf模的对偶模仍是广义相关Hopf模,从而推广了［3］中的相关结论.     

Projections of weak braided Hopf algebras

In this paper we study the projections of weak braided Hopf algebras using the notion of Yetter-Drinfeld module associated with a weak braided Hopf algebra. As a consequence, we complete the study ofthe structure of weak Hopf algebras with a projection in a braiding setting obtaining a categorical equivalencebetween the category of weak Hopf algebra projections associated with a weak Hopf algebra H living in abraided monoidal category and the category of Hopf algebras in the non-strict braided monoidal cate...     

Construct weak Hopf algebras by using Borcherds matrix

We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra by adding a new generator J satisfying jm = j for some integer m. We denote this algebra by wUqT（A）. This algebra is a weak Hopf algebra if and only if m = 2,3. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usual quantum envelope algebra Uq （A） of a generalized Kac-Moody algebra A.     

Transmutation theory of a coquasitriangular weak Hopf algebra

Let H be a coquasitriangular quantum groupoid. In this paper, using a suitable idempotent element e in H, we prove that eH is a braided group (or a braided Hopf algebra in the category of right H-comodules), which generalizes Majid’s transmutation theory from a coquasitriangular Hopf algebra to a coquasitriangular weak Hopf algebra.     

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.     

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.     

Morita contexts and Galois theory for weak Hopf comodulelike algebras

We investigate the Morita context and graded cases for weak group corings and derive some equivalent conditions for μ to be surjective. Furthermore, we develop Galois theory for weak group corings. As an application, we give Galois theory for comodulelike algebras over a weak Hopf group coalgebra.     

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.     

二面体群上Hopf路余代数的结构分类

从Hopf quiver出发,借助于右kZ_u(c)-模的直积范畴■ Mkz_(u(C))与kG-Hopf双模范畴kG/kG M kG/kG之间的同构,当G是二面体群D_3时,给出了Hopf路余代数kQ~c的同构分类及其子Hopf代数kG[kQ_1]结构.     

The weak braided Hopf algebra structure of some Cayley–Dickson algebras

It has been shown by Albuquerque and Majid that a class of unital k-algebras, not necessarily associative, obtained through the Cayley–Dickson process can be viewed as commutative associative algebras in some suitable symmetric monoidal categories. In this note we will prove that they are, moreover, commutative and cocommutative weak braided Hopf algebras within these categories. To this end we first define a Cayley–Dickson process for coalgebras. We then see that the k-vector space of complex numbers, of quaternions, of octonions, of sedenions, etc. fit to our theory, hence they are all monoidal coalgebras as well, and therefore weak braided Hopf algebras.     

Baxter algebras and Hopf algebras

By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.

     

弱量子代数wUq((g))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wUq((g)),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.     

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 connecting the smash product A#H and the invariant subalgebra A~H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.     

Multiplier Hopf algebroids: Basic theory and examples

Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable base algebras. The main structure maps of a multiplier Hopf algebroid are a left and a right comultiplication. We show that bijectivity of two associated canonical maps is equivalent to the existence of an antipode, discusses invertibility of the antipode, and presents some examples and special cases.     

Construction of Rota-Baxter algebras via Hopf module algebras

We propose the notion of Hopf module algebra and show that the projection onto the subspace of coinvariants is an idempotent Rota-Baxter operator of weight-1. We also provide a construction of Hopf module algebras by using Yetter-Drinfeld module algebras. As an application,we prove that the positive part of a quantum group admits idempotent Rota-Baxter algebra structures.