RECORD
Fourier transform and the Verlinde formula for the quantum double of a finite group
Quantum Algebra · Submitted 7 Apr 1999
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We use this to give a particularly short proof of the Verlinde formula for the fusion coefficients.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
A Fourier transform S is defined for the quantum double D(G) of a finite group G. Acting on characters of D(G), S and the central ribbon element of D(G) generate a unitary matrix representation of the group SL(2,Z). The characters form a ring over the integers under both the algebra multiplication and its dual, with the latter encoding the fusion rules of D(G). The Fourier transform relates the two ring structures. We use this to give a particularly short proof of the Verlinde formula for the fusion coefficients.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A Fourier transform S is defined for the quantum double D(G) of a finite group G. | contribution 0.94 | 0.84 | 0.07 | asserted 1.00 |
| s1 | Acting on characters of D(G), S and the central ribbon element of D(G) generate a unitary matrix representation of the group SL(2,Z). | result 0.94 | 0.78 | 0.24 | asserted 1.00 |
| s2 | The characters form a ring over the integers under both the algebra multiplication and its dual, with the latter encoding the fusion rules of D(G). | result 0.52 | 0.55 | 0.09 | asserted 1.00 |
| s3 | The Fourier transform relates the two ring structures. | result 0.73 | 0.68 | 0.12 | asserted 1.00 |
| s4 | We use this to give a particularly short proof of the Verlinde formula for the fusion coefficients. | contribution 0.78 | 0.95 | 0.56 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
A Fourier transform S is defined for the quantum double D(G) of a finite group G. Acting on characters of D(G), S and the central ribbon element of D(G) generate a unitary matrix representation of the group SL(2,Z). The characters form a ring over the integers under both the algebra multiplication and its dual, with the latter encoding the fusion rules of D(G). The Fourier transform relates the two ring structures. We use this to give a particularly short proof of the Verlinde formula for the fusion coefficients.
Human review
How the literature uses it
3 arXiv papers cite this in 5 sentences
Papers citing it · 2 sentences without a confident label
Cites it as background 2 papers
-
In this section, we review how this pair labels the irreducible representations of a well-known algebraic structure, namely the Drinfel'd double of the gauge group $G$ [ref] .
Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases · Tube algebra and Drinfel'd double · 0.95
-
They were presented in details in e.g. [ref] and numerous identities were also proven in [ref] .
On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity · Drinfel'd double · 1.00
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.