乘子Hopf代数的 L-R Smash余积

设 $A$ 是正则乘子Hopf代数, $R$ 是 $A$-\!\!双余模代数, 首先定义了 $A$-\!\!双余模双代数, 并利用它构造了 L-R Smash 积的对偶形式, 即 $R\otimes A$ 上一种非平凡的乘子Hopf代数结构, 称之为 L-R Smash余积. 然后给出了 L-R Smash余积上的积分和$*$-\!\!结构.     

H－弱余模余代数和交叉余积

引进了交叉积的对偶交叉余积，证明了：余Ｃｌｅｆｔ模余代数的结构定理（作为余代数）；如果为Ｈｏｐｆ代数余可裂正合序列，那么作为Ｈｏｐｆ代数，由此有强增广余代数Ｃ的结构定理（作为双代数）；如果为Ｈｏｐｆ代数可裂正合序列，那么作为Ｈｏｐｆ代数并简单地讨论了Ｃ×αＨ的余半单性．     

Hopf代数的扭曲余积和H^R型Hopf代数

本文引进了Ｈｏｐｆ代数的扭曲余积，推广了广义偶交叉积，使得一般的右Ｓｍａｓｈ余积也是这里的特殊情况，讨论了Ｈ＾Ｒ型Ｈｏｐｆ代数与扭曲余积的关系。     

拟三角Hopf代数的Ore -扩张

该文主要考虑了拟三角Hopf代数的某种Ore -扩张问题. 对拟三角Hopf代数的Ore -扩张何时保持相同的拟三角结构给出了充分必要条件. 最后作为应用, 文章讨论了Sweedler Hopf代数和Lusztig小量子群的Ore -扩张结构.     

Twist积和Twist余积Hopf代数

设H,A是两个Hopf代数,构造了twist积A#σH和twist余积A#τH,证明了文[1]中的double twist和S.Majid构造的Bicrossproduct结构以及通常的smash积都是A#σH的一种特殊情况;文[2,3]中的twist Hopf代数以及通常的smash余积是A#τH的特殊情况,最后讨论了A#τH上的拟三角结构.     

Hopf代数交叉余积的对偶定理

证明了交叉余积的对偶定理,该定理具有与交叉积对偶定理(概括了群分次环的相应结果)相类似的意义。     

三角Hopf余模范畴上的线量余代数

在三角Hopf代数余模范畴上研究张量余代数,主要给出三角Hopf代数作模范畴上的张量余代数的结构。     

用quiver构造拟三角Hopf代数

利用quiver方法确定了一个广义Taft代数具有拟三角Hopf结构当且仅当它是Sweedler 4维Hopf代数.用不同于文[15]的方法,对任意的正整数n,构造出一类拟三角Hopf代数H(n).     

关于(f,т)-相容Hopf代数对(B,H)

该文定义了(f,τ)-相容Hopf代数对(B,H),利用这样的对(B,H),给出了左H-余模范畴HM的一个辫子张量子范畴,从而得到一个量子Yang-Baxter算子,并且通过扭曲Hopf代数B的乘法,构造出Yetter-Drinfeld范畴中H HYD的Hopf代数．     

关于(f,τ) -相容Hopf代数对(B,H)

该文定义了(f,τ) -相容Hopf代数对(B,H),利用这样的对(B,H),给出了左H -余模范畴^HM的一个辫子张量子范畴,从而得到一个量子Yang-Baxter算子,并且通过扭曲Hopf代数$B$的乘法,构造出Yetter-Drinfeld范畴中^H_HYD的Hopf代数.     

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.     

Homological Dimension of Coalgebras and Crossed Coproducts

Let H be a Hopf k-algebra. We study the global homological dimension of the underlying coalgebra structure of H. We show that gl.dim(H) is equal to the injective dimension of the trivial right H-comodule k. We also prove that if D = C H is a crossed coproduct with invertible , then gl.dim(D) gl.dim(C) + gl.dim(H). Some applications of this result are obtained. Moreover, if C is a cocommutative coalgebra such that C ^* is noetherian, then the global dimension of the coalgebra C coincides with the global dimension of the algebra C ^*.     

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.     

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.     

Hopf Algebra Structure Over Crossed Coproducts

The smash coproduct coalgebra has been generalized to crossed coproduct coalgebra in [3]. It is natural to replace the smash coproduct by the crossed coproduct and consider the conditions under which the smash product algebra structure and the crossed coproduct coalgebra structure will inherit a bialgebra structure or a Hopf algebra structure. We derive necessary and sufficient conditions for this problem. This generalizes the corresponding results in [7]. Finally, we characterize this new structure by introducing a concept of (H, )-comodule and prove that Heisenberg double [4] and smash coproduct do not make a bialgebra.AMS Subject Classification (1991): 16S40 16W30The first and second authors were partially supported by NNSF of China. The first author was also supported by the NSF of Henan Province and the second author by the YSF of Shandong Province (No. Q98A05113).     

准三角Hopf代数与双积

本文首先给出准三角Ｈｏｐｆ代数与Ｙｅｔｔｅｒ－Ｄｒｉｎｆｅｌｄ模的关系，推广了Ｒａｄｆｏｒｄ关于双积的构造，讨论二重Ｓｍａｓｈ积与双积的关系．     

Hopf Algebras of Dimension 12

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

Quasitriangular Pointed Hopf Algebras Constructed by Ore Extensions

We study the quasitriangular structures for a family of pointed Hopf algebras which is big enough to include Taft's Hopf algebras H _n ², Radford's Hopf algebras H _N,n,q, and E(n). We give necessary and sufficient conditions for the Hopf algebras in our family to be quasitriangular. For the case when they are, we determine completely all the quasitriangular structures. Also, we determine the ribbon elements of the quasitriangular Hopf algebras and the quasi-ribbon elements of their Drinfel'd double.     

Algebras Versus Coalgebras

Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and computer scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neighbouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s. Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between the base category and the category of the associated bimodules. For a set G, the endofunctor G× – on the category of sets shares these properties if and only if G admits a group structure. Finally, we report about the possibility of subsuming algebras and coalgebras in the notion of ( F , G )-dimodules associated to two functors between different categories. This observation, due to Tatsuya Hagino, was an outcome from the theory of categorical data types and may also be of use in classical algebra.