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

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.

39 pages, AMS-Latex; V2: Added comments and references concerning relation to Faddeev's modular double, minor corrections, version to be published in CMP

References in corpus (7)

• Discrete Heisenberg-Weyl Group and Modular Group
• Quantum Dilogarithm
• Liouville bootstrap via harmonic analysis on a noncompact quantum group
• Modular Double of Quantum Group
• Harmonic Analysis on the quantum Lorentz group
• Integral soluitons of q-difference equations of the hypergeometric type with |q|=1
• Fourier transforms on the quantum SU(1,1) group

Cited by in corpus (161)

• Liouville field theory on a pseudosphere
• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls
• The Annular Report on Non-Critical String Theory
• A lecture on the Liouville vertex operators
• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• Essays on the theory of elliptic hypergeometric functions
• The Schwarzian Theory - A Wilson Line Perspective
• Conformal Bootstrap, Universality and Gravitational Scattering
• Quantum Regge Trajectories and the Virasoro Analytic Bootstrap
• Liouville quantum gravity -- holography, JT and matrices
• Noncommutative Hypergeometry
• Universal Dynamics of Heavy Operators in CFT${}_2$
• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Crossing Symmetry in Alpha Space
• SUSY monopole potentials in 2+1 dimensions
• 5D partition functions, q-Virasoro systems and integrable spin-chains
• Nonperturbative Mellin Amplitudes: Existence, Properties, Applications
• R-operator, co-product and Haar-measure for the modular double of U_q(sl(2,R))
• Crossing Symmetry in the $H_3^{+}$ WZNW model
• Solving 3d Gravity with Virasoro TQFT
• Fine Structure of Jackiw-Teitelboim Quantum Gravity
• Lectures on Branes in Curved Backgrounds
• Higher-Level Appell Functions, Modular Transformations, and Characters
• On the relation between quantum Liouville theory and the quantized Teichm"uller spaces
• A CFT Perspective on Gravitational Dressing and Bulk Locality
• Reconstruction of Baxter Q-operator from Sklyanin SOV for cyclic representations of integrable quantum models
• S^3/Z_n partition function and dualities
• 3D Gravity, Point Particles and Liouville Theory
• Classical geometry from the quantum Liouville theory
• Painlevé VI connection problem and monodromy of c=1 conformal blocks
• Non-Gaussianities in the Statistical Distribution of Heavy OPE Coefficients and Wormholes
• OPE statistics from higher-point crossing
• Irregular conformal blocks and connection formulae for Painlevé V functions
• Modular bootstrap in Liouville field theory
• Completeness of Bethe Ansatz by Sklyanin SOV for Cyclic Representations of Integrable Quantum Models
• Dynamical actions and q-representation theory for double-scaled SYK
• Knot invariants from Virasoro related representation and pretzel knots
• On Boundary Perturbations in Liouville Theory and Brane Dynamics in Noncritical String Theories
• Unit circle elliptic beta integrals
• From Quantum Groups to Liouville and Dilaton Quantum Gravity
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Quantum geometry of 3-dimensional lattices
• From Conformal Blocks to Path Integrals in the Vaidya Geometry
• Decomposing Nekrasov Decomposition
• Gauge Theories Labelled by Three-Manifolds
• A slow review of the AGT correspondence
• On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models
• Intersecting Surface Defects and Two-Dimensional CFT
• Non-Rational 2D Quantum Gravity: I. World Sheet CFT
• The star-triangle relation, lens partition function, and hypergeometric sum/integrals
• Monodromy of solutions of the Knizhnik-Zamolodchikov equation: SL(2,C)/SU(2) WZNW model
• S-duality as a beta-deformed Fourier transform
• Non-Perturbative Topological Strings And Conformal Blocks
• Coupled minimal models revisited
• 3d gravity from Virasoro TQFT: Holography, wormholes and knots
• Twist accumulation in conformal field theory. A rigorous approach to the lightcone bootstrap
• Supergroup Structure of Jackiw-Teitelboim Supergravity
• Classical conformal blocks from TBA for the elliptic Calogero-Moser system
• Positive Representations of Split Real Simply-laced Quantum Groups
• The universal Racah-Wigner symbol for Uq(osp(1|2))
• Classical Liouville action on the sphere with three hyperbolic singularities
• S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
• $q$-Virasoro modular double and 3d partition functions
• Multiboundary wormholes and OPE statistics
• On the fusion matrix of the N=1 Neveu-Schwarz blocks
• On the crossing relation in the presence of defects
• Chern-Simons Theory and S-duality
• S-duality as Fourier transform for arbitrary ${\epsilon }_1,{\epsilon }_2$
• Braiding and fusion properties of the Neveu-Schwarz super-conformal blocks
• Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs
• Integrability of boundary Liouville conformal field theory
• Universal Dynamics of Heavy Operators in Boundary CFT${}_2$
• Elliptic beta integrals and solvable models of statistical mechanics
• On modular transformations of non-degenerate toric conformal blocks
• Liouville Quantum Gravity
• Continuous tensor categories from quantum groups I: algebraic aspects
• AdS/BCFT from Conformal Bootstrap: Construction of Gravity with Branes and Particles
• Braiding properties of the N=1 super-conformal blocks (Ramond sector)
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Self-dual Continuous Series of Representations for U_q(sl(2)) and U_q(osp(1|2))
• A Relativistic Conical Function and its Whittaker Limits
• Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries
• Thermodynamics and conformal properties of XXZ chains with alternating spins
• Elliptic recursion for 4-point superconformal blocks and bootstrap in N=1 SLFT
• Quantum Liouville theory versus quantized Teichmüller spaces
• Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation
• Quantum Teichmüller space from quantum plane
• Orthogonal Polynomials in Mathematical Physics
• Quantum exponentials for the modular double and applications in gravity models
• Matching branches of non-perturbative conformal block at its singularity divisor
• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• Hartle-Hawking state and its factorization in 3d gravity
• Semiclassical Gravity from Averaged Boundaries in two-dimensional BCFTs
• From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation
• 3d Gravity as a random ensemble
• 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories
• On the unitary representations of the affine $ax+b$-group, $\widehat{sl}(2,\mathbb{R})$ and their relatives
• Hidden classical symmetry in quantum spaces at roots of unity : From q-sphere to fuzzy sphere
• Check-Operators and Quantum Spectral Curves
• Rigid Fuchsian systems in 2-dimensional conformal field theories
• Zamolodchikov relations and Liouville hierarchy in SL(2,R)_k WZNW model
• Semiclassical limit of the FZZT Liouville theory
• The complex Liouville string: worldsheet boundaries and non-perturbative effects
• A parafermionic hypergeometric function and supersymmetric 6j-symbols
• Baxter equations and Deformation of Abelian Differentials
• S-duality wall of SQCD from Toda braiding
• Renyi Entropies, the Analytic Bootstrap, and 3D Quantum Gravity at Higher Genus
• Representation of the Quantum Plane, its Quantum Double and Harmonic Analysis on $G{L}_q^{+}(2,R)$
• Recent progress on Liouville Field Theory
• The hyperbolic modular double and the Yang-Baxter equation
• On fusion kernel in Liouville theory
• Colored HOMFLY-PT for hybrid weaving knot ${\hat{W}}_3(m,n)$
• Aspects of the S transformation Bootstrap
• Elliptic hypergeometric function and $6j$-symbols for the SL(2,$\mathbb{C}$) group
• On the 6j-symbols for SL(2,C) group
• Virasoro blocks at large exchange dimension
• The R-operator for Modular Double
• Quantum group origins of edge states in double-scaled SYK
• Ryu-Takayanagi Formula for Multi-Boundary Black Holes from 2D Large-$c$ CFT Ensemble
• On tensor products of positive representations of split real quantum Borel subalgebra $U_{q\tilde{q}}(b_R)$
• On the spectral theory of one functional-difference operator from conformal field theory
• On the Virasoro fusion and modular kernels at any irrational central charge
• Universal Dynamics in Non-Orientable CFT${}_2$
• Anyon Condensation in Virasoro TQFT: Wormhole Factorization
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$
• A cluster realization of $U_q(\mathfrak{s}{\mathfrak{l}}_{\mathfrak{n}})$ from quantum character varieties
• The Virasoro fusion kernel and Ruijsenaars' hypergeometric function
• Confluent Conformal Blocks of the Second Kind
• On the Virasoro fusion kernel at $c=25$
• Probabilistic conformal blocks for Liouville CFT on the torus
• Supersymmetry and the Modular Double
• On the continuous series for affine sl(2,R)
• Joining-splitting interaction of non-critical string
• Finite-dimensional representations of the elliptic modular double
• Heisenberg double and Drinfeld double of the quantum superplane
• Positive representations, multiplier Hopf algebra, and continuous canonical basis
• Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-$N$
• On tensor product decomposition of positive representations of ${\mathcal{U}}_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$
• Three-dimensional quantum gravity from the quantum pseudo-Kähler plane
• Open-closed 3d gravity as a random ensemble
• Complex rational Ruijsenaars model. The two-particle case
• On modular double of semisimple quantum groups
• Discrete series of representations for the modular double of U_q(sl(2,R))
• Non-critical string pentagon equations and their solutions
• Modular matrices of 2d Carrollian and warped CFTs
• Tug-the-hook symmetry for quantum 6j-symbols
• Baxterization of GL_q(2) and its application to the Liouville model and some other models on a lattice
• Parabolic Positive Representations of ${\mathcal{U}}_q({\mathfrak{g}}_{\mathbb{R}})$
• Four-point correlation modular bootstrap for OPE densities
• Holographic Black Hole Formation and Scrambling in Time-Ordered Correlators
• On explicit realization of algebra of complex divided powers of $U_q(\mathfrak{sl}(2))$
• Braiding Operator via Quantum Cluster Algebra
• Positive Casimir and Central Characters of Split Real Quantum Groups
• Open strings in Lie groups and associative products
• Regge symmetry of 6j-symbols of the Lorentz group
• A universal sum over topologies in 3d gravity
• Non-critical string, Liouville theory and geometric bootstrap hypothesis
• From hyperbolic to complex Euler integrals
• Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes
• On explicit realization of algebra of complex powers of generators of $U_q(\mathfrak{sl}(3))$