scholarly journals A Generalization for n-Cocycles

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Beishang Ren ◽  
Shixun Lin
Keyword(s):  
Tensor Product ◽  
Type Ii ◽  
Module Algebra ◽  
Product Algebra

We will give generalized definitions called type II n-cocycles and weak quasi-bialgebra and also show properties of type II n-cocycles and some results about weak quasi-bialgebras, for instance, construct a new structure of tensor product algebra over a module algebra on weak quasi-bialgebras.

scholarly journals A basis for the GLn tensor product algebra

2005 ◽  
Vol 196 (2) ◽  
pp. 531-564 ◽  
Author(s):  
Roger E. Howe ◽  
Eng-Chye Tan ◽  
Jeb F. Willenbring
Keyword(s):  
Tensor Product ◽  
Product Algebra

scholarly journals Weak amenability for subfactors

2015 ◽  
Vol 26 (07) ◽  
pp. 1550048 ◽  
Author(s):  
Arnaud Brothier
Keyword(s):  
Tensor Product ◽  
Free Product ◽  
Finite Index ◽  
Finite Family ◽  
Type Ii ◽  
Weak Amenability

We define the notions of weak amenability and the Cowling–Haagerup constant for extremal finite index subfactors s of type II1. We prove that the Cowling–Haagerup constant only depends on the standard invariant of the subfactor. Hence, we define the Cowling–Haagerup constant for standard invariants. We explicitly compute the constant for Bisch–Haagerup subfactors and prove that it is equal to the constant of the group involved in the construction. Given a finite family of amenable standard invariants, we prove that their free product in the sense of Bisch–Jones is weakly amenable with constant 1. We show that the Cowling–Haagerup of the tensor product of a finite family of standard invariants is equal to the product of their Cowling–Haagerup constants.

SMASH PRODUCTS, SEPARABLE EXTENSIONS AND A MORITA CONTEXT OVER HOPF ALGEBROIDS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350124
Author(s):  
YONG WANG ◽  
GUANGQUAN GUO
Keyword(s):  
Type Theorem ◽  
Smash Product ◽  
Module Algebra ◽  
Morita Context ◽  
Hopf Algebroid ◽  
Hopf Algebroids ◽  
Maschke Type Theorem ◽  
Module Algebras ◽  
Product Algebra ◽  
Separable Extensions

Let [Formula: see text] be a Hopf algebroid, and A a left [Formula: see text]-module algebra. This paper is concerned with the smash product algebra A#H over Hopf algebroids. In this paper, we investigate separable extensions for module algebras over Hopf algebroids. As an application, we obtain a Maschke-type theorem for A#H-modules over Hopf algebroids, which generalizes the corresponding result given by Cohen and Fischman in [Hopf algebra actions, J. Algebra100 (1986) 363–379]. Furthermore, based on the work of Kadison and Szlachányi in [Bialgebroid actions on depth two extensions and duality, Adv. Math.179 (2003) 75–121], we construct a Morita context connecting A#H and [Formula: see text] the invariant subalgebra of [Formula: see text] on A.

An extended form of Majid’s double biproduct

2017 ◽  
Vol 16 (04) ◽  
pp. 1750061 ◽  
Author(s):  
Tianshui Ma ◽  
Huihui Zheng
Keyword(s):  
Hopf Algebra ◽  
Sufficient Conditions ◽  
Smash Product ◽  
Algebra Structure ◽  
Module Algebra ◽  
Extended Form ◽  
Linear Map ◽  
Product Algebra ◽  
Necessary And Sufficient ◽  
Crossed Coproducts

Let [Formula: see text] be a bialgebra. Let [Formula: see text] be a linear map, where [Formula: see text] is a left [Formula: see text]-module algebra, and a coalgebra with a left [Formula: see text]-weak coaction. Let [Formula: see text] be a linear map, where [Formula: see text] is a right [Formula: see text]-module algebra, and a coalgebra with a right [Formula: see text]-weak coaction. In this paper, we extend the construction of two-sided smash coproduct to two-sided crossed coproduct [Formula: see text]. Then we derive the necessary and sufficient conditions for two-sided smash product algebra [Formula: see text] and [Formula: see text] to be a bialgebra, which generalizes the Majid’s double biproduct in [Double-bosonization of braided groups and the construction of [Formula: see text], Math. Proc. Camb. Philos. Soc. 125(1) (1999) 151–192] and the Wang–Wang–Yao’s crossed coproduct in [Hopf algebra structure over crossed coproducts, Southeast Asian Bull. Math. 24(1) (2000) 105–113].

scholarly journals On the structures of hive algebras and tensor product algebras for general linear groups of low rank

2019 ◽  
Vol 29 (07) ◽  
pp. 1193-1218
Author(s):  
Donggyun Kim ◽  
Sangjib Kim ◽  
Euisung Park
Keyword(s):  
Tensor Product ◽  
Irreducible Polynomial ◽  
Tensor Products ◽  
Low Rank ◽  
General Linear ◽  
Generators And Relations ◽  
General Linear Groups ◽  
Highest Weight ◽  
Polynomial Representations ◽  
Product Algebra

The tensor product algebra [Formula: see text] for the complex general linear group [Formula: see text], introduced by Howe et al., describes the decomposition of tensor products of irreducible polynomial representations of [Formula: see text]. Using the hive model for the Littlewood–Richardson (LR) coefficients, we provide a finite presentation of the algebra [Formula: see text] for [Formula: see text] in terms of generators and relations, thereby giving a description of highest weight vectors of irreducible representations in the tensor products. We also compute the generating function of certain sums of LR coefficients.

scholarly journals TWISTING ALL THE WAY: FROM ALGEBRAS TO MORPHISMS AND CONNECTIONS

2012 ◽  
Vol 13 ◽  
pp. 1-19 ◽  
Author(s):  
PAOLO ASCHIERI
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Commutative Algebra ◽  
Tensor Products ◽  
Module Algebra ◽  
Linear Maps ◽  
Definition Of ◽  
Quasitriangular Hopf Algebra ◽  
Just Right ◽  
The Way

Given a Hopf algebra H and an algebra A that is an H-module algebra we consider the category of left H-modules and A-bimodules [Formula: see text], where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist [Formula: see text] of H we then quantize (deform) H to [Formula: see text], A to A⋆ and correspondingly the category [Formula: see text] to [Formula: see text]. If we consider a quasitriangular Hopf algebra H, a quasi-commutative algebra A and quasi-commutative A-bimodules, we can further construct and study tensor products over A of modules and of morphisms, and their twist quantization. This study leads to the definition of arbitrary (i.e., not necessarily H-equivariant) connections on quasi-commutative A-bimodules, to extend these connections to tensor product modules and to quantize them to A⋆-bimodule connections. Their curvatures and those on tensor product modules are also determined.

Relations of A*H and AH

2010 ◽  
Vol 143-144 ◽  
pp. 828-831
Author(s):  
Yan Yan ◽  
Cui Lan Mi ◽  
Xin Chun Wang
Keyword(s):  
Hopf Algebra ◽  
Weak Hopf Algebra ◽  
Smash Product ◽  
Trace Function ◽  
Module Algebra ◽  
Finiteness Conditions ◽  
Module Algebras ◽  
The Right ◽  
Product Algebra

In this paper, we study the concept of the right twisted smash product algebra A*H over weak Hopf algebra. Let H be a weak Hopf algebra and A an H-module algebra, using the properties of the trace function we describe the finiteness conditions for H-module algebras.

scholarly journals TOPOLOGICAL STABLE RANK OF INCLUSIONS OF UNITAL C*-ALGEBRAS

2006 ◽  
Vol 17 (01) ◽  
pp. 19-34 ◽  
Author(s):  
HIROYUKI OSAKA ◽  
TAMOTSU TERUYA
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Finite Type ◽  
Stable Rank ◽  
C Algebras ◽  
Rank One ◽  
Topological Stable Rank ◽  
Crossed Product Algebra ◽  
A Finite Group ◽  
Product Algebra

Let 1 ∈ A ⊂ B be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute the topological stable rank of B (= tsr (B)) when A has topological stable rank one. We show that tsr (B) ≤ 2 when A is a tsr boundedly divisible algebra, in particular, A is a C*-minimal tensor product UHF ⊗ D with tsr (D) = 1. When G is a finite group and α is an action of G on UHF, we know that a crossed product algebra UHF ⋊α G has topological stable rank less than or equal to two. These results are affirmative data to a generalization of a question by Blackadar in 1988.

Spectra for Infinite Tensor Product Type Actions of Compact Groups

1996 ◽  
Vol 48 (2) ◽  
pp. 330-342
Author(s):  
Elliot C. Gootman ◽  
Aldo J. Lazar
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Abelian Groups ◽  
Product Type ◽  
Crossed Product ◽  
Compact Groups ◽  
Ideal Structure ◽  
Crossed Product Algebra ◽  
Product Algebra ◽  
The Ideal

AbstractWe present explicit calculations of the Arveson spectrum, the strong Arveson spectrum, the Connes spectrum, and the strong Connes spectrum, for an infinite tensor product type action of a compact group. Using these calculations and earlier results (of the authors and C. Peligrad) relating the various spectra to the ideal structure of the crossed product algebra, we prove that the topology of G influences the ideal structure of the crossed product algebra, in the following sense: if G contains a nontrivial connected group as a direct summand, then the crossed product algebra may be prime, but it is never simple; while if G is discrete, the crossed product algebra is simple if and only if it is prime. These results extend to compact groups analogous results of Bratteli for abelian groups. In addition, we exhibit a class of examples illustrating that for compact groups, unlike the case for abelian groups, the Connes spectrum and strong Connes spectrum need not be stable.

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