scholarly journals THE DRINFEL'D DOUBLE AND TWISTING IN STRINGY ORBIFOLD THEORY

2009 ◽  
Vol 20 (05) ◽  
pp. 623-657 ◽  
Author(s):  
RALPH M. KAUFMANN ◽  
DAVID PHAM
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Geometric Realization ◽  
Braided Category ◽  
Drinfel’D Double ◽  
Frobenius Algebras ◽  
K Theory ◽  
Orbifold Cohomology ◽  
The One ◽  
A Finite Group

This paper exposes the fundamental role that the Drinfel'd double D(k[G]) of the group ring of a finite group G and its twists Dβ(k[G]), β ∈ Z3(G,k*) as defined by Dijkgraaf–Pasquier–Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that G-Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of D(k[G])-modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold K-theory of global quotient given by the inertia variety of a point with a G action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full K-theory of the stack [pt/G]. Finally, we show how one can use the co-cycles β above to twist the global orbifold K-theory of the inertia of a global quotient and more importantly, the stacky K-theory of a global quotient [X/G]. This corresponds to twistings with a special type of two-gerbe.

On the Morava K-theory of some finite 2-groups

1997 ◽  
Vol 121 (1) ◽  
pp. 7-13 ◽  
Author(s):  
BJÖRN SCHUSTER
Keyword(s):  
Finite Group ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Wreath Products ◽  
Classifying Space ◽  
Generalized Cohomology ◽  
K Theory ◽  
A Finite Group ◽  
Generalized Quaternion

For any fixed prime p and any non-negative integer n there is a 2(pn − 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

The Inverse Multiplier for Abelian Group Difference Sets

1964 ◽  
Vol 16 ◽  
pp. 787-796 ◽  
Author(s):  
E. C. Johnsen
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Collineation Group ◽  
Difference Sets ◽  
Difference Set ◽  
The One ◽  
A Finite Group

In(1)Bruck introduced the notion of a difference set in a finite group. LetGbe a finite group ofvelements and let D = {di},i= 1, . . . ,kbe ak-subset ofGsuch that in the set of differences {di-1dj} each element ≠ 1 inGappears exactly λ times, where 0 < λ <k<v— 1. When this occurs we say that (G,D) is av,k,λ group difference set. Bruck showed that this situation is equivalent to the one where the differences {didj-1} are considered instead, and that av,k, λ group difference set is equivalent to a transitivev,k,λconfiguration, i.e., av,k,λconfiguration which has a collineation group which is transitive and regular on the elements (points) and on the blocks (lines) of the configuration. Among the parametersv,kandλ, then, we have the relation shown by Ryser(5)

scholarly journals Integral Group Rings with Nilpotent Unit Groups

1976 ◽  
Vol 28 (5) ◽  
pp. 954-960 ◽  
Author(s):  
César Polcino Milies
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Unit Element ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
The Subject ◽  
A Finite Group

Let R be a ring with unit element and G a finite group. We denote by RG the group ring of the group G over R and by U(RG) the group of units of this group ring.The study of the nilpotency of U(RG) has been the subject of several papers.

Natural maps of extension functors and a theorem of R. G. Swan

1961 ◽  
Vol 57 (3) ◽  
pp. 489-502 ◽  
Author(s):  
P. J. Hilton ◽  
D. Rees
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Group Ring ◽  
Projective Module ◽  
Additive Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Natural Maps ◽  
Image Position ◽  
A Finite Group

The present paper has been inspired by a theorem of Swan(5). The theorem can be described as follows. Let G be a finite group and let Γ be its integral group ring. We shall denote by Z an infinite cyclic additive group considered as a left Γ-module by defining gm = m for all g in G and m in Z. By a Tate resolution of Z is meant an exact sequencewhere Xn is a projective module for − ∞ < n < + ∞, and.

scholarly journals On Galois projective group rings

1991 ◽  
Vol 14 (1) ◽  
pp. 149-153
Author(s):  
George Szeto ◽  
Linjun Ma
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Projective Group ◽  
Group Rings ◽  
Inner Automorphism ◽  
A Finite Group

LetAbe a ring with1,Cthe center ofAandG′an inner automorphism group ofAinduced by {Uαin​A/αin a finite groupGwhose order is invertible}. LetAG′be the fixed subring ofAunder the action ofG′.IfAis a Galcis extension ofAG′with Galois groupG′andCis the center of the subring∑αAG′UαthenA=∑αAG′Uαand the center ofAG′is alsoC. Moreover, if∑αAG′Uαis Azumaya overC, thenAis a projective group ring.

scholarly journals On the group ring of a finite abelian group

1969 ◽  
Vol 1 (2) ◽  
pp. 245-261 ◽  
Author(s):  
Raymond G. Ayoub ◽  
Christine Ayoub
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Free Abelian Group ◽  
Finite Abelian Group ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Group Of Units ◽  
Direct Sum ◽  
A Finite Group

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G. It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

scholarly journals Units of group rings of groups of order 16

1993 ◽  
Vol 35 (3) ◽  
pp. 367-379 ◽  
Author(s):  
E. Jespers ◽  
M. M. Parmenter
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Abelian Groups ◽  
Unit Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Group Of Units ◽  
Abelian Case ◽  
A Finite Group

LetGbe a finite group,(ZG) the group of units of the integral group ring ZGand1(ZG) the subgroup of units of augmentation 1. In this paper, we are primarily concerned with the problem of describing constructively(ZG) for particular groupsG.This has been done for a small number of groups (see [11] for an excellent survey), and most recently Jespers and Leal [3] described(ZG) for several 2-groups. While the situation is clear for all groups of order less than 16, not all groups of order 16 were discussed in their paper. Our main aim is to complete the description of(ZG) for all groups of order 16. Since the structure of the unit group of abelian groups is very well known (see for example [10]), we are only interested in the non-abelian case.

scholarly journals On automorphism groups of low complexity subshifts

2015 ◽  
Vol 36 (1) ◽  
pp. 64-95 ◽  
Author(s):  
SEBASTIÁN DONOSO ◽  
FABIEN DURAND ◽  
ALEJANDRO MAASS ◽  
SAMUEL PETITE
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Polynomial Complexity ◽  
Automorphism Groups ◽  
Low Complexity ◽  
Topological Dynamics ◽  
Main Technique ◽  
Shift Map ◽  
The One ◽  
A Finite Group

In this article, we study the automorphism group$\text{Aut}(X,{\it\sigma})$of subshifts$(X,{\it\sigma})$of low word complexity. In particular, we prove that$\text{Aut}(X,{\it\sigma})$is virtually$\mathbb{Z}$for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a$d$-step nilsystem is nilpotent of order$d$and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually$\mathbb{Z}$.

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