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Tensor product representations of the quantum double of a compact group

T. H. Koornwinder, F. A. Bais, N. M. Muller

TL;DR

This work develops a detailed representation-theoretic framework for the quantum double $\mathcal{D}(G)$ of a compact group $G$, using the explicit comultiplication and universal $R$-matrix to define tensor products, braiding, and fusion of irreducible $*$-representations. The authors formulate a constructive decomposition of tensor products into direct integrals of irreducibles via double coset analysis, establishing implicit fusion rules and multiplicities that depend on the centralizers and associated measures. They provide a rigorous SU(2) case with complete Clebsch–Gordan data: the generic tensor products decompose into integrals over conjugacy-parameterised sectors, with explicit intertwiners yielding Clebsch–Gordan coefficients built from standard SU(2) data. The results offer a concrete toolkit for modeling topological interactions in 2+1-dimensional theories and related quantum gravity contexts, where defects and anyonic excitations can be described by representations of $\mathcal{D}(G)$.

Abstract

We consider the quantum double D(G) of a compact group G, following an earlier paper. We use the explicit comultiplication on D(G) in order to build tensor products of irreducible *-representations. Then we study their behaviour under the action of the R-matrix, and their decomposition into irreducible *-representations. The example of D(SU(2)) is treated in detail, with explicit formulas for direct integral decomposition (`Clebsch-Gordan series') and Clebsch-Gordan coefficients. We point out possible physical applications.

Tensor product representations of the quantum double of a compact group

TL;DR

This work develops a detailed representation-theoretic framework for the quantum double of a compact group , using the explicit comultiplication and universal -matrix to define tensor products, braiding, and fusion of irreducible -representations. The authors formulate a constructive decomposition of tensor products into direct integrals of irreducibles via double coset analysis, establishing implicit fusion rules and multiplicities that depend on the centralizers and associated measures. They provide a rigorous SU(2) case with complete Clebsch–Gordan data: the generic tensor products decompose into integrals over conjugacy-parameterised sectors, with explicit intertwiners yielding Clebsch–Gordan coefficients built from standard SU(2) data. The results offer a concrete toolkit for modeling topological interactions in 2+1-dimensional theories and related quantum gravity contexts, where defects and anyonic excitations can be described by representations of .

Abstract

We consider the quantum double D(G) of a compact group G, following an earlier paper. We use the explicit comultiplication on D(G) in order to build tensor products of irreducible *-representations. Then we study their behaviour under the action of the R-matrix, and their decomposition into irreducible *-representations. The example of D(SU(2)) is treated in detail, with explicit formulas for direct integral decomposition (`Clebsch-Gordan series') and Clebsch-Gordan coefficients. We point out possible physical applications.

Paper Structure

This paper contains 13 sections, 13 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. The Hopf algebra structure of ${\cal D}(G)$
  3. Irreducible representations
  4. Tensor products
  5. Braiding of two representations
  6. Tensor product decomposition
  7. Explicit results for $G = SU(2)$
  8. Parametrisation and notation
  9. Irreducible representations
  10. Clebsch--Gordan series
  11. Clebsch--Gordan coefficients
  12. Conclusion
  13. Some measure theoretical results

Key Result

Theorem 3.2

For $A\in {\rm Conj}(G)$ and $\alpha\in\widehat{N_A}$ we have mutually inequivalent irreducible $*$-representations $\Pi^A_{\alpha}$ of ${\cal D}(G) = C(G\times G)$ on $L^2_{\alpha} (G,V_{\alpha})$ given by These representations are moreover $\|.\|_1$-bounded (see for this notion formula (33) in KM). All irreducible $\|.\|_1$-bounded $*$-representations of ${\cal D}(G)$ are equivalent to some $\P

Theorems & Definitions (14)

  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 6.3
  • Proposition 6.4
  • Lemma 6.5
  • Lemma 6.6
  • Corollary 6.8
  • Theorem 6.10
  • Theorem A.1
  • ...and 4 more