The Crossed Burnside Ring, the Drinfel’d Double, and the Dade Group of a p-Group

2009 ◽  
Vol 13 (2) ◽  
pp. 231-242
Author(s):  
Fumihito Oda
Keyword(s):  
Burnside Ring ◽  
Drinfel’D Double ◽  
Dade Group

scholarly journals 3d gravity and quantum deformations: a Drinfel'd double approach

2012 ◽  
Vol 360 ◽  
pp. 012010
Author(s):  
A Ballesteros ◽  
F J Herranz ◽  
C Meusburger
Keyword(s):  
Drinfel’D Double ◽  
Quantum Deformations ◽  
Double Approach ◽  
3D Gravity

Stable cohomotopy and cobordism of abelian groups

1981 ◽  
Vol 90 (2) ◽  
pp. 273-278 ◽  
Author(s):  
C. T. Stretch
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Formal Group ◽  
Characteristic Class ◽  
Burnside Ring ◽  
Classifying Space ◽  
Formal Group Law ◽  
Stable Cohomotopy ◽  
Group Law

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

scholarly journals A collection of subgroups for the generalized Burnside ring

2009 ◽  
Vol 222 (1) ◽  
pp. 307-317 ◽  
Author(s):  
Fumihito Oda ◽  
Masato Sawabe
Keyword(s):  
Burnside Ring

scholarly journals Non-Abelian U -duality for membranes

2020 ◽  
Vol 2020 (7) ◽  
Author(s):  
Yuho Sakatani ◽  
Shozo Uehara
Keyword(s):  
String Theory ◽  
Isometry Group ◽  
Standard Treatment ◽  
Abelian Extension ◽  
Target Space ◽  
Membrane Theory ◽  
Drinfel’D Double ◽  
M Theory

Abstract The $T$-duality of string theory can be extended to the Poisson–Lie $T$-duality when the target space has a generalized isometry group given by a Drinfel’d double. In M-theory, $T$-duality is understood as a subgroup of $U$-duality, but the non-Abelian extension of $U$-duality is still a mystery. In this paper we study membrane theory on a curved background with a generalized isometry group given by the $\mathcal E_n$ algebra. This provides a natural setup to study non-Abelian $U$-duality because the $\mathcal E_n$ algebra has been proposed as a $U$-duality extension of the Drinfel’d double. We show that the standard treatment of Abelian $U$-duality can be extended to the non-Abelian setup. However, a famous issue in Abelian $U$-duality still exists in the non-Abelian extension.

scholarly journals On the lifting of the Dade group

2019 ◽  
Vol 22 (3) ◽  
pp. 441-451
Author(s):  
Caroline Lassueur ◽  
Jacques Thévenaz
Keyword(s):  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Group Homomorphism ◽  
Complete Discrete Valuation Ring ◽  
Dade Group

Abstract For the group of endo-permutation modules of a finite p-group, there is a surjective reduction homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characteristic p. We prove that this reduction map always has a section which is a group homomorphism.

Irreducible components and isomorphisms of the Burnside ring

2008 ◽  
Vol 11 (6) ◽  
Author(s):  
Wolfgang Kimmerle ◽  
Florian Luca ◽  
Alberto Gerardo Raggi-Cárdenas
Keyword(s):  
Burnside Ring ◽  
Irreducible Components
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