Morita context,partial Hopf Galois extensions and partial entwining structure

We first introduce the notion of a right generalized partial smash product and explore some properties of such partial smash product. Based on these notions and properties, then we construct a Morita context for partial coactions of a co-Frobenius Hopf algebra. Finally, we prove that any Hopf partial Galois extension induces a unique partial entwining map compatible with the right partial coaction.     

Leavitt Path Algebras as Partial Skew Group Rings

We realize Leavitt path algebras as partial skew group rings and give new proofs, based on partial skew group ring theory of the Cuntz–Krieger uniqueness theorem and simplicity criteria for Leavitt path algebras.     

有足够幂等元的群分次环上的Morita对偶

首先定义了具有足够幂等元的群分次环的Ｓｍ ａｓｈ 积并讨论其性质,其次,借助Ｓｍ ａｓｈ 积给出了这类分环上的分次Ｍｏｒｉｔａ对偶的刻划．     

群分次环的本原性及分次本原性

设A是一个用有限群G分次的环,本文给出了Smash积A＃G~*为本原环的一个判别准则,并证明了群分次环的每一个本原理想必定包含一个分次本原理想。作为一个推论,得到已被Cohen和Montgomery证实的Bergman猜想的另一个证明。此外还得到了A_1为本原环或单环的判别。     

关于G－分次环与G－集的Smash积的几个结果

设Ｇ为任意群，本文借助于环的矩阵表示给出了Ｇ-分次环与任意可迁Ｇ-集的ｓｍａｓｈ积是素环或单环的刻画．     

关于G－分次环与G－集的Smash积的几个结果

设Ｇ为任意群，本文借助于环的矩阵表示给出了Ｇ－分次环与任意可迁Ｇ－集的Ｓｍａｓｈ积是素环或单环的刻画。     

Group Entwining Structures and Group Coalgebra Galois Extensions

Abstract

The notions of group coalgebra Galois extension and group entwining structure are defined. It is proved that any group coalgebra Galois extension induces a unique group-entwining map ψ = {ψ_{α, β}}_{α, β∈π} compatible with the right group coaction, generalizing the recent work of Brzeziński and Hajac [Brzeziński, T., Hajac, P. M. (1999). Coalgebra extensions and algebra coextensions of Galois type. Comm. Algebra 27:1347–1368].     

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.     

On Strongly Associative Group Algebras

Given a partial action α of a group G on the group algebra FH, where H is a finite group and F is a field whose characteristic p divides the order of H, we investigate the associativity question of the partial crossed product FH*_α G. If FH*_α G is associative for any G and any α, then FH is called “strongly associative.” Using a result of Dokuchaev and Exel (2005 Dokuchaev , M. , Exel , R. ( 2005 ). Associativity of crossed products by partial actions, enveloping actions and partial representations . Trans. Amer. Math. Soc. 357 : 1931 – 1952 .[Crossref], [Web of Science ®] , [Google Scholar]) we characterize the strongly associative modular group algebras FH for several classes of groups H.     

Strong associativity of a group algebra

The concept of strongly associative algebra was introduced in Dokuchaev and Exel [Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans Amer Math Soc, 2005] roughly 2 years ago. In Section 3 of Dokuchaev and Exel (to appear), the authors gave an example of a non-strongly associative algebra. Following that example, the strong associativity of the group algebra of the cyclic group of order four was claimed. In this paper, we will prove that this group algebra is not strongly associative.     

Skew category, Galois covering and smash product of a -category

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In the case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie's construction of a ring with local units. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery (1984), and we provide a unification with the results on coverings of quivers and relations by E. Green (1983). We obtain a confirmation in a quiver and relations-free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.

     

Integral Group Ring of the Mathieu Simple Group M 23

We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M ₂₃ using the Luthar–Passi method. This work is a continuation of the research that we carried out for Mathieu groups M ₁₁ and M ₁₂. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.     

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.     

关于有限群两个幂零子群积的问题

众所周知,有限群的两个幂零子群的积不一定是幂零的．本文研究了Ｅｎｇｅｌ条件对两个幂零子群的影响,得到两个幂零子群的积为幂零群的几个充分条件。     

一个与G－分次环和G－集的Smash积有关的Maschke－Type定理

对任意群Ｇ，［１］研究了有单位元１的Ｇ－分次环与有限可迁Ｇ－集的Ｓｍａｓｈ积．在本文中，我们对任意可迁Ｇ－集Ａ讨论了具有局部单位元的Ｇ－分次环与Ｇ－集Ａ的Ｓｍａｓｈ积，证明了有关的一个Ｍａｓｃｈｋｅ－ｔｙＰｅ定理．推广了［２］［３］中的一些重要结果．     

交错群A_5与二面体群D_6直积的整群环的挠单位

本文利用Luthar-Passi方法,研究了五次交错群A_5与六阶二面体群D_6直积的整群环的挠单位,得到了该群的Zassenhaus猜想成立.     

Constructions of Group Divisible Designs

In this paper we present constructions for group divisible designs from generalized partial difference matrices. We describe some classes of examples.     

Smash积为单环,素环,本原环的充要条件

本文对任意群Ｇ及任意的Ｇ－分环次Ａ（不必含有单位元），讨论了Ａ与Ｓｍａｓｈ积Ａ＃Ｇ＾的相关性质，给出了环Ａ＃Ｇ是单环，素环及本原环的刻划。