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Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))

B. Ponsot, J. Teschner

TL;DR

The paper develops a rigorous harmonic-analysis framework for a continuous family of unitary representations of the noncompact quantum group $U_q(sl(2,R))$, with $q=e^{ ext{pi} i b^2}$. It constructs explicit integral kernels that generalize Clebsch-Gordan coefficients and Racah-Wigner coefficients, formulated via the self-dual special function $S_b$, and demonstrates that the subfamily of representations closes under tensor products through a modular-double perspective. The authors derive the spectral measure and intertwiners for binary tensor products, prove unitarity of the Clebsch-Gordan maps, and compute Racah-Wigner coefficients, including their orthogonality, completeness, and pentagon relations. Their results offer a concrete technical foundation for a potential $C^*$-algebraic quantum group structure tied to $SL_q(2,R)$ and illuminate connections to Liouville theory and conformal blocks via integral-operator realizations.

Abstract

The decomposition of tensor products of representations into irreducibles is studied for a continuous family of integrable operator representations of $U_q(sl(2,R)$. It is described by an explicit integral transformation involving a distributional kernel that can be seen as an analogue of the Clebsch-Gordan coefficients. Moreover, we also study the relation between two canonical decompositions of triple tensor products into irreducibles. It can be represented by an integral transformation with a kernel that generalizes the Racah-Wigner coefficients. This kernel is explicitly calculated.

Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))

TL;DR

The paper develops a rigorous harmonic-analysis framework for a continuous family of unitary representations of the noncompact quantum group , with . It constructs explicit integral kernels that generalize Clebsch-Gordan coefficients and Racah-Wigner coefficients, formulated via the self-dual special function , and demonstrates that the subfamily of representations closes under tensor products through a modular-double perspective. The authors derive the spectral measure and intertwiners for binary tensor products, prove unitarity of the Clebsch-Gordan maps, and compute Racah-Wigner coefficients, including their orthogonality, completeness, and pentagon relations. Their results offer a concrete technical foundation for a potential -algebraic quantum group structure tied to and illuminate connections to Liouville theory and conformal blocks via integral-operator realizations.

Abstract

The decomposition of tensor products of representations into irreducibles is studied for a continuous family of integrable operator representations of . It is described by an explicit integral transformation involving a distributional kernel that can be seen as an analogue of the Clebsch-Gordan coefficients. Moreover, we also study the relation between two canonical decompositions of triple tensor products into irreducibles. It can be represented by an integral transformation with a kernel that generalizes the Racah-Wigner coefficients. This kernel is explicitly calculated.

Paper Structure

This paper contains 42 sections, 39 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite difference operators
  4. Fourier-transformation
  5. Distributions
  6. A useful Lemma from complex analysis
  7. A class of representations of $U_q(sl(2,{\mathbb R}))$
  8. Definintion
  9. Intertwining operators
  10. The Clebsch-Gordan decomposition of tensor products
  11. Spectral measure
  12. Intertwining property
  13. Racah-Wigner coefficients for ${\mathcal{U}}_q({\mathfrak s}{\mathfrak l}(2,{\mathbb R}))$
  14. Canonical decompositions for triple tensor products
  15. Relation between ${\mathcal{C}}_{3(21)}$ and ${\mathcal{C}}_{(32)1}$
  16. ...and 27 more sections

Key Result

Theorem 1

(Paley-Wiener) Let $f$ be in $L^2({\mathbb R})$. Then $(e^{2\pi xa_+}+e^{-2\pi xa_-})f\in L^2({\mathbb R})$, $a_{\pm}>0$ if and only if $\tilde{f}$ has an analytic continuation to the strip $\{\omega\in{\mathbb C};\text{Im}(\omega)\in(-a_-,a_+)\}$ such that for any $\omega_2\in(-a_-,a_+)$, $\tilde{f

Theorems & Definitions (50)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Remark 1
  • Lemma 3
  • Remark 2
  • Lemma 4
  • Proposition 1
  • Remark 3
  • Lemma 5
  • ...and 40 more