The Partition Algebra as a Centralizer Algebra of the Alternating Group

ABSTRACT

In this article, we focus on the result of V.F.R. Jones which says that the partition algebra is the algebra of all transformations commuting with the action of the symmetric group on tensor products of its permutation representation. In particular, we restrict the action of the symmetric group to the action of the alternating group. In this context, we compute a basis for the centralizer algebra and show when the centralizer is isomorphic to the partition algebra.     

Finite Dimensional Representations of Symplectic Reflection Algebras Associated to Wreath Products II

In this note we refine the methods of Etingof and Montarani (Represent. Theory, 9: 457–467, 2005) in order to extend the main result of that article to a wider class of finite dimensional representations of wreath product symplectic reflection algebras. Presented by Claus M. Ringel.     

圈积的词长度公式

利用Fordham的技巧证明了圈积群HιZ的一个词长度公式,其中H是一个有限生成群,Z是整数群.     

On Some Finite Dimensional Representations of Symplectic Reflection Algebras Associated to Wreath Products

Let Γ _N : = S _N  ? Γ ^N be the wreath product of Γ, a finite subgroup of SL(2,C), by the symmetric group of degree N. In this article we classify all the irreducible representations of S _N  ? Γ ^N that can be extended to a representation of the associated symplectic reflection algebra H _1,k,c (Γ _N ) (where k is a complex number and c a class function on the nontrivial elements of Γ) for nonzero values of k.     

Specht Modules and Simple Modules for Centralizer Algebras of the Symmetric Group

Let Σ_n be the symmetric group on n letters. For l ≤ n identify Σ_l with a subgroup of Σ_n in the natural way. Let k be an algebraically closed field of characteristic p. This article begins to develop a theory for modules over the centralizer algebras kΣ_n^Σ_l that is analogous to James's theory of permutation modules, Specht modules, and simple modules over kΣ_n. We make a conjecture about how to construct all simple kΣ_n^Σ_l-modules, we develop tools to test the conjecture, and we prove that it is correct for all n when l < p.     

Probabilistic Generation of Wreath Products of Non-abelian Finite Simple Groups

Abstract

We show that the probability of generating an iterated standard wreath product of non-abelian finite simple groups is positive and tends to 1 as the order of the first simple group tends to infinity. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated. Information depending on the Classification of Finite Simple Groups is used throughout.     

Centralizer algebras of the group associated to Z4-codes

The purpose of this paper is to investigate the finite group which appears in the study of the Type II ${\mathbb{Z}}_4$-codes. To be precise, it is characterized in terms of generators and relations, and we determine the structure of the centralizer algebras of the tensor representations of this group.     

Constructions of Stratified Algebras

In this article, a construction to build recursively all basic finite dimensional standardly stratified algebras is given. In comparison to the construction described by Dlab and Ringel for the quasi-hereditary case ([15 Dlab , V. , Ringel , C. M. ( 1989 ). A construction for quasi-hereditary algebras . Compositio Math. 70 : 155 – 175 . [Google Scholar]]) some new features appear here.     

Smash Products of H-Simple Module Algebras

Let H be a finite-dimensional Hopf algebra and A a finite-dimensional H-simple left H-module algebra. We show that the smash product A#H is isomorphic to End _A′(V ? H*), where V ≠ 0 is a finite-dimensional left A-module and (A′, V′) the stabilizer of (A, V). As an application it is proved that A#H is isomorphic to a full matrix algebra over A′ when H is semisimple and dim V|dim A.     

弱Hopf代数上的扭积

在弱Hopf代数上,定义了交叉积概念,并且得到了它的两种特殊形式,冲积和扭积．特别地,给出了扭积为弱Hopf代数的一个充要条件,推广了Hopf代数的相应结论．     

Quantum Yang-Baxter H-Module Algebras and Their Braided Products

Abstract

We give a necessary and sufficient condition of the braided product to be a bialgebra or a Hopf algebra and two interesting examples to show that the conditions in Theorem 2.4: “(H1) and (H2)” weaken the commutativity and cocommutativity of H in Caenepeel et al. (Caenepeel, S., Oystaeyen, F. Van, Zhang, Yin-huo (1994). Quantum Yang-Baxter module algebra. K-Theorem 8:231–255.). Dually, we introduce the concept of a braided coproduct and give the distinguished conditions.     

On Smash Products of Transitive Module Algebras

Let H be a semisimple Hopf algebra over a field of characteristic 0, and A a finite-dimensional transitive H-module algebra with a l-dimensional ideal. It is proved that the smash product A#H is isomorphic to a full matrix algebra over some right coideal subalgebra N of H. The correspondence between A and such N, and the special case A = k（X） of function algebra on a finite set X are considered.     

Construction of CPs-Stratified Algebras

The results of [7 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras. In: Tachikawa, H., Brenner, S. eds. Representations of Algebras and Related Topics, London Math. Society Lecture Note Series 168:200–224 . [Google Scholar]] and [2 Ágoston , I. , Dlab , V. , Lukács , E. ( 2011 ). Constructions of stratified algebras . Comm. Algebra 39 : 2545 – 2553 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] gave a recursive construction for all quasi-hereditary and standardly stratified algebras starting with local algebras and suitable bimodules. Using the notion of stratifying pairs of subcategories, introduced in [3 Ágoston , I. , Lukács , E. Stratifying pairs of subcategories for CPS-stratified algebras . To appear in Journal of Algebra and Its Applications , p. 11 . [Google Scholar]], we generalize these earlier results to construct recursively all CPS-stratified algebras.     

Writing Commutators of Commutators as Products of Cubes in Groups

We study the cube length of certain elements of the derived subgroup of a group G. By the cube length Cu(γ) of an element γ of a group G, we mean the least natural number k such that γ is a products of k cubes. We find an upper bound for the cube length of a commutator of commutators. If W = F?C _∞ is the wreath product of a free group F by the infinite cyclic group, we show that every element of W″ is a product of at most three cubes in W.     

一类特征单的n-李代数

利用结合代数与n-李代数的张量积构造了一类无限维特征单的n-李代数,且证明了除n=3的情形以外,这类特征单n-李代数的内导子代数是特征单李代数.     

Commutator Calculus for Wreath Product Groups

We introduce a class of function spaces consisting of integer-valued functions of several integers which coordinatize the elements of certain subgroups of some finite regular wreath product groups. On each function space, we define operators which correspond to forming certain commutators relevent to computing the upper central series. We define an automorphism of the function space which enables us to define a class of subgroups that is useful for describing the upper central series of certain finite regular wreath product p-groups. Our results describe fundamental, interesting, and useful relationships between the automorphism and the operators. We describe some applications.     

On Quasi-Bicrossed Products of Weak Hopf Algebras

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.     

Hopf代数的双交叉积

本文定义并详细讨论了交叉余积，考虑交叉积与交叉余积合起未成为双代数的问题，讨论了由内作用，内余作用构造的双交叉积．我们还用正合序列刻划了双交叉积．     

TENSOR PRODUCTS OF JACOBSON RADICALS IN NEST ALGEBRAS

This paper studies the tensor product RN RM of Jacobson radicals in nest algebras, and obtains that RN RM = {T∈B(H1 H2) : T(N M)(?)N_ M_, N∈N,M∈M}; and based on the characterization of rank-one operators in RN RM,it is proved that if N, M are non-trivial then RN RM=R if and only if N, M are continuous.     

Crossed Products for Weak Hopf Algebras

Weak crossed products by a weak bialgebra are defined. The resulting structure is not that of a unital algebra but an associative algebra with preunit. A general formula of such a product is given in terms of a weak 2-cocycle and a weak measuring. The relation with weak cleft extensions is studied. Equivalences of weak crossed products are defined and are related to gauge transformations. A relation between cleaving maps is described in terms of gauge transformations.