On Semisimple Hopf Algebras of Dimension 2 m

In this paper we classify all nontrivial semisimple Hopf algebras of dimension 2 ^{n +1} with the group of grouplikes isomorphic to _{2 n–1}×₂. Moreover, we extend some results on irreducible representations from groups to semisimple Hopf algebras and prove that certain semisimple Hopf algebras, including the ones classified in this paper, satisfy the generalized power map property.     

一类半单Hopf代数的结构

设k是特征为零的代数闭域,H是k上的pq~2维Frobenius型半单Hopf代数,其中p,q为不同的素数.本文证明了,如果p>q且H~*也是Frobenius型Hopf代数,则H是q~2维群代数A与A上p维Yetter-Drinfeld Hopf代数R的双积,即H≌R#A.作为例子,本文还证明了任意63维或68维的半单Hopf代数均为Frobenius型Hopf代数.     

Structure Theorems for Semisimple Hopf Algebras of Dimension pq 3

Let p, q be prime numbers with p > q ³, and k an algebraically closed field of characteristic 0. In this article, we obtain the structure theorems for semisimple Hopf algebras of dimension pq ³.     

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.     

低维的Frobenius 型半单Hopf 代数

Let k be an algebraically closed field of characteristic zero.This paper proves that semisimple Hopf algebras over k of dimension 66,70 and 78 are of Frobenius type.     

基于弱Hopf代数的半单范畴的构造

本文研究并刻画了交换环上弱Hopf代数、Yetter-Drinfeld模范畴的一些性质,给出了其能够做成半单范畴的充分条件.     

高维的Frobenius型半单Hopf代数

设H是特征为零的代数闭域k上的半单Hopf代数.本文证明了如果dimкH 是小于351的奇数,则H是Frobenius型Hopf代数.     

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.     

On Hopf Algebras and Their Generalizations

We survey Hopf algebras and their generalizations. In particular, we compare and contrast three well-studied generalizations (quasi-Hopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (Hopf monads and hopfish algebras). Each of these notions was originally introduced for a specific purpose within a particular context; our discussion favors applicability to the theory of dynamical quantum groups. Throughout the note, we provide several definitions and examples in order to make this exposition accessible to readers with differing backgrounds.     

On Limits of Mishchenko--Fomenko Subalgebras in Poisson Algebras of Semisimple Lie Algebras

In this paper, the commutative (with respect to the Poisson bracket) subalgebras in the Poisson algebras of the semisimple Lie algebras are considered on condition that these subalgebras are limits of Mishchenko--Fomenko subalgebras. We study the case of the degeneration within a fixed Cartan subalgebra. The structure of the limit subalgebras is described (i.e., it is proved that these subalgebras are free, and their generators are found). The classification of the limit subalgebras of the above type is also established.     

Quasitriangular Hopf Algebras of Dimension pq

Let p and q be odd prime numbers. It is shown that all quasitriangularHopf algebras of dimension pq over an algebraically closed fieldk of characteristic zero are semisimple and therefore isomorphicto a group algebra.     

On Galois Extension of Hopf Algebras

Let H be a cosemisimple Hopf algebra over a field k, and π ： A→ H be a surjective cocentral bialgebra homomorphism of bialgebras. The authors prove that if A is Galois over its coinvariants B=LH Ker π and B is a sub-Hopf algebra of A, then A is itself a Hopf algebra. This generalizes a result of Cegarra [3] on group-graded algebras.     

On Hopf Galois Extension of Separable Algebras

In this paper, the classical Galois theory to the H*-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra,and A/AHa right H*-Galois extension. The authors prove that, if AHis a separable kalgebra, then for any right coideal subalgebra B of H, the B-invariants A~B= {a ∈ A |b · a = ε(b)a, b∈ B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing AHas in the classical case. The results are applied to the case H =(kG)*for a finite group G to get a Galois 1-1 correspondence.     

Hopf Algebras of Dimension 12

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

Coset Decomposition for Semisimple Hopf Algebras

The notion of double coset for semisimple finite dimensional Hopf algebras is introduced. This is done by considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. As an application formulae for the restriction of the irreducible characters to normal Hopf subalgebras are given.     

Structure Theorems for Bicomodule Algebras Over Quasi-Hopf Algebras,Weak Hopf Algebras,and Braided Hopf Algebras

Let H be a quasi-Hopf algebra, a weak Hopf algebra, or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v: H → B. Then we can define an object B^co(H), which is a left-left Yetter–Drinfeld module over H, having extra properties that allow to make a smash product B^co(H)#H, which is an H-bicomodule algebra, isomorphic to B.     

半单弱Hopf代数的作用

令H是半单弱Hopf代数, A是左H-模代数.我们证明了正则A-模的内射维数, A#H-模A的内射维数和正则A#H-模的内射维数三者是相等的. 而且,利用H在A上的不动点代数我们给出了A是Gorenstein代数的充要条件.     

Multiplier Hopf Algebras in Categories and the Biproduct Construction

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct. Presented by K. Goodearl.     

On Hopf Algebras with Nonzero Integral

Let S be the model of a semigroup with an associate subgroup whose identity is a medial idempotent constructed by Blyth and Martins considered as a unary semigroup. For another such semigroup T, we construct all unary homomorphisms of S into T in terms of their parameters. On S we construct all unary congruences again directly from its parameters. This construction leads to a characterization of congruences in terms of kernels and traces. We describe the K, T, L, U and V relations on the lattice of all unary congruences on S, again in terms of parameters of S.