R-operator, co-product and Haar-measure for the modular double of U_q(sl(2,R))

Abstract

A certain class of unitary representations of U_q(sl(2,R)) has the property of being simultanenously a representation of U_{tilde{q}}(sl(2,R)) for a particular choice of tilde{q}(q). Faddeev has proposed to unify the quantum groups U_q(sl(2,R)) and U_{tilde{q}}(sl(2,R)) into some enlarged object for which he has coined the name ``modular double''. We study the R-operator, the co-product and the Haar-measure for the modular double of U_q(sl(2,R)) and establish their main properties. In particular it is shown that the Clebsch-Gordan maps constructed in [PT2] diagonalize this R-operator.

27 pages, LaTex (smfart.sty)

References in corpus (1)

• Noncommutative Hypergeometry

Cited by in corpus (40)

• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Liouville quantum gravity -- holography, JT and matrices
• Noncommutative Hypergeometry
• Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant
• Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices
• On the relation between quantum Liouville theory and the quantized Teichm"uller spaces
• Reconstruction of Baxter Q-operator from Sklyanin SOV for cyclic representations of integrable quantum models
• An analog of a modular functor from quantized Teichm"uller theory
• Completeness of Bethe Ansatz by Sklyanin SOV for Cyclic Representations of Integrable Quantum Models
• From Quantum Groups to Liouville and Dilaton Quantum Gravity
• On the form factors of local operators in the Bazhanov-Stroganov and chiral Potts models
• Yang-Baxter Maps, Discrete Integrable Equations and Quantum Groups
• Baxter Q-operators of the XXZ chain and R-matrix factorization
• Thermodynamics and conformal properties of XXZ chains with alternating spins
• Quantum Teichmüller space from quantum plane
• Quantum exponentials for the modular double and applications in gravity models
• Orthogonal Polynomials in Mathematical Physics
• From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation
• 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories
• Quantum Dilogarithm Identities at Root of Unity
• Representation of the Quantum Plane, its Quantum Double and Harmonic Analysis on $G{L}_q^{+}(2,R)$
• On the separation of variables for the modular XXZ magnet and the lattice Sinh-Gordon models
• SU(2)/SL(2) knot invariants and KS monodromies
• Quantum group origins of edge states in double-scaled SYK
• The R-operator for Modular Double
• On tensor products of positive representations of split real quantum Borel subalgebra $U_{q\tilde{q}}(b_R)$
• Gauss-Lusztig Decomposition for $G{L}_q^{+}(N,R)$ and Representation by q-Tori
• Finite-dimensional representations of the elliptic modular double
• A cluster realization of $U_q(\mathfrak{s}{\mathfrak{l}}_{\mathfrak{n}})$ from quantum character varieties
• On modular double of semisimple quantum groups
• On tensor product decomposition of positive representations of ${\mathcal{U}}_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$
• Positive representations, multiplier Hopf algebra, and continuous canonical basis
• Heisenberg double and Drinfeld double of the quantum superplane
• Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-$N$
• Baxterization of GL_q(2) and its application to the Liouville model and some other models on a lattice
• Positive Casimir and Central Characters of Split Real Quantum Groups
• Parabolic Positive Representations of ${\mathcal{U}}_q({\mathfrak{g}}_{\mathbb{R}})$
• On explicit realization of algebra of complex divided powers of $U_q(\mathfrak{sl}(2))$
• Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes
• On explicit realization of algebra of complex powers of generators of $U_q(\mathfrak{sl}(3))$