Pairing and Quantum Double of Finite Hopf C^＊-Algebras

This paper defines a pairing of two finite Hopf C^＊-algebras A and B, and investigates the interactions between them. If the pairing is non-degenerate, then the quantum double construction is given. This construction yields a new finite Hopf C^＊-algebra D（A, B）. The canonical embedding maps of A and B into the double are both isometric.     

Galois Theory for Multiplier Hopf Algebras with Integrals

In this paper, we introduce a generalized Hopf Galois theory for regular multiplier Hopf algebras with integrals, which might be viewed as a generalization of the Hopf Galois theory of finite-dimensional Hopf algebras. We introduce the notion of a coaction of a multiplier Hopf algebra on an algebra. We show that there is a duality for actions and coactions of multiplier Hopf algebras with integrals. In order to study the Galois (co)action of a multiplier Hopf algebra with an integral, we construct a Morita context connecting the smash product and the coinvariants. A Galois (co)action can be characterized by certain surjectivity of a canonical map in the Morita context. Finally, we apply the Morita theory to obtain the duality theorems for actions and coactions of a co-Frobenius Hopf algebra.     

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.     

Traces on Multiplier Hopf Algebras

In this article we lay the algebraic foundations to establish the existence of trace functions on infinite-dimensional (multiplier) Hopf algebras. We solve the problem within the framework of multiplier Hopf algebra with integrals. By applying this theory to group-cograded multiplier Hopf algebras, we prove the existence of group-traces on group-cograded multiplier Hopf algebras with possibly infinite-dimensional components. We generalize the results as obtained by Virelizier in the case of finite-type Hopf group-coalgebras.     

Constructing Quasitriangular Multiplier Hopf Algebras By Twisted Tensor Coproducts

Let A and B be multiplier Hopf algebras, and let R ∈ M(B ? A) be an anti-copairing multiplier, i.e, the inverse of R is a skew-copairing multiplier in the sense of Delvaux [5 Delvaux , L. ( 2004 ). Twisted tensor coproduct of multiplier Hopf (*)-algebras . J. Algebra 274 : 751 – 771 . [Google Scholar]]. Then one can construct a twisted tensor coproduct multiplier Hopf algebra A ? _R  B. Using this, we establish the correspondence between the existence of quasitriangular structures in A ? _R  B and the existence of such structures in the factors A and B. We illustrate our theory with a profusion of examples which cannot be obtained by using classical Hopf algebras. Also, we study the class of minimal quasitriangular multiplier Hopf algebras and show that every minimal quasitriangular Hopf algebra is a quotient of a Drinfel’d double for some algebraic quantum group.     

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.     

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.     

Hopf Algebras of Dimension 12

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

Hopf代数的二重Ore扩张

本文的目的 是定义Hopf二重Ore扩张,讨论这种扩张的基本性质并研究Hopf代数的分次与Hopf二重Ore扩张之间的关系.作者还研究了连通分次Hopf代数的结构及其Hopf二重Ore扩张的同调性质.     

On the Twisting and Drinfel'd Double for Multiplier Hopf Algebras

We define the concept of “semiprime” for preradicals and for submodules, and we prove some properties that relate both of them. Related concepts are defined in article by Bican et al. [2 Bican , L. , Jambor , P. , Kepka , T. , Nemec , P. ( 1980 ). Prime and coprime modules . Fundamenta Mathematicae CVII , 33 – 45 . [Google Scholar]] and by Van den Berg and Wisbauer [9 Van den Berg , J. , Wisbauer , R. ( 2001 ). Duprime and dusemiprime modules . Journal of Pure and Applied Algebra 165 : 337 – 356 .[Crossref], [Web of Science ®] , [Google Scholar]]. For any ring, we compare the least semiprime preradical, the Jacobson radical and the join of all nilpotent preradicals, and we characterize V-rings in terms of these three preradicals. We study the least semiprime preradical above any preradical and we prove some of its properties. Using “Amitsur constructions” we define another related operators and prove some of their properties.     

Group-cograded Multiplier Hopf ${\left( { * {\text{ - }}} \right)}$algebras

Let G be a group and assume that (A _p ) _p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,q ∈ G, there is given a unital homomorphism Δ _p,q : A _pq → A _p ⊗ A _q satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps Δ _p,q can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras. Presented by Ken Goodearl.     

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.     

Multiple Noncommutative Tori and Hopf Algebras

Abstract

We derive the Kac-Paljutkin finite-dimensional Hopf algebras as finite fibrations of the quantum double torus and generalize the construction for quantum multiple tori.     

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.     

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.     

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.     

关于扭Hopf代数的一些注记

In this paper,we get some properties of the antipode of a twisted Hopf algebra.We proved that the graded global dimension of a twisted Hopf algebra coincides with the graded projective dimension of its trivial module k,which is also equal to the projective dimension of k.     

Locality Criteria for Cocommutative Hopf Algebras

We prove that a finite-dimensional cocommutative Hopf algebra H is local, if and only if the subalgebra generated by the first term of its coradical filtration H ₁ is local. In particular if H is connected, H is local if and only if all the primitive elements of H are nilpotent.     

On Semisimple Hopf Algebras of Dimension pq2, II

We obtain further classification results for semisimple Hopf algebras of dimension pq ² over an algebraically closed field k of characteristic zero. We complete the classification of semisimple Hopf algebras of dimension 28.     

Hopf Algebras on Schurian Quivers

Let H be a Hopf algebra with a finite-dimensional, nontrivial space of skew primitive elements, over an algebraically closed field of characteristic zero. We prove that H contains either the polynomial algebra as a Hopf subalgebra, or a certain Schurian simple-pointed Hopf subalgebra. As a consequence, a complete list of the locally finite, simple-pointed Hopf algebras is obtained. Also, the graded automorphism group of a Hopf algebra on a Schurian Hopf quiver is determined, and the relation between this group and the automorphism groups of the corresponding Hopf quiver, is clarified.