Harmonic Analysis on the quantum Lorentz group

Abstract

This work begins with a review of complexification and realification of Hopf algebras. We emphasize the notion of multiplier Hopf algebras for the description of different classes of functions (compact supported, bounded, unbounded) on complex quantum groups and the construction of the associated left and right Haar measure. Using a continuation of $6j$ symbols of $S{U}_q(2)$ with complex spins, we give a new description of the unitary representations of $SL_q (2,\CC)_{\RR}$ and find explicit expressions for the characters of $SL_q (2,\CC)_{\RR}$. The major theorem of this article is the Plancherel theorem for the Quantum Lorentz Group.

60 pages, tared gzipped Postscript file, major revision of the previous version, the Plancherel theorem is established in the more general sense and we delay the study of Fusion theory to the next part of this paper

Cited by in corpus (36)

• Spin Foam Models for Quantum Gravity
• A Lorentzian Signature Model for Quantum General Relativity
• Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))
• Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group
• 4-dimensional Spin-foam Model with Quantum Lorentz Group
• Quantum deformation of two four-dimensional spin foam models
• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• Cosmological Deformation of Lorentzian Spin Foam Models
• R-operator, co-product and Haar-measure for the modular double of U_q(sl(2,R))
• Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity
• A new twist on dS/CFT
• Tensor Products of Principal Unitary Representations of Quantum Lorentz Group and Askey-Wilson Polynomials
• A note on the geometrical interpretation of quantum groups and non-commutative spaces in gravity
• Typicality in spin network states of quantum geometry
• Hopf stars, twisted Hopf stars and scalar products on quantum spaces
• 6J Symbols Duality Relations
• Hidden classical symmetry in quantum spaces at roots of unity : From q-sphere to fuzzy sphere
• Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant
• Combinatorial quantisation of Euclidean gravity in three dimensions
• The new spin foam models and quantum gravity
• Cladistics of Double Yangians and Elliptic Algebras
• On the Asymptotics of Quantum Group Spinfoam Model
• Quantum group origins of edge states in double-scaled SYK
• On the Quasi-Hopf structure of deformed double Yangians
• Universal Solutions of Quantum Dynamical Yang-Baxter Equations
• An infinite product formula for $U_q(sl(2))$ dynamical coboundary element
• Chern-Simons Theory with Sources and Dynamical Quantum Groups I: Canonical Analysis and Algebraic Structures
• Quantum $SL(2,\mathbb{R})$ and its irreducible representations
• Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras
• Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-$N$
• Yangian and quantum universal solutions of Gervais--Neveu--Felder equations
• Invariant integrals on coideals and their Drinfeld doubles
• Statistical entropy of three-dimensional q-deformed Kerr-de Sitter space
• Knot Theory With The Lorentz Group
• q-Deformation of Lorentzian spin foam models
• On state sums, internalisation and unification