Quantum groups and quantum moduli spaces of flat connections

Abstract

Abstract:We give an introduction to the theory of dynamical quantum groups and precise its relation to the harmonic analysis on non-compact quantum groups through the example of the quantum Lorentz-Group.Keywords: Dynamical quantum groups, Non-compact quantum groups, Racah coefficients.Representation of non compact quantum groups is of central importance in different theoretical physics systems. We can at least give three examples of this fact:1. Chern-Simons theory with non compact group G. This is particularly important in view of its applications to quantum gravity in 2+1 dimensions where the group G is equal to SO(3, 1) (Λ > 0), ISO(2, 1) (Λ = 0) or SL(2, R) × SL(2, R), (Λ < 0) depending on the sign of the cosmological constant Λ.2. Discretization of Lorentzian gravity in the spirit of Ponzano-Regge.3. Liouville theory in the weak and probably strong coupling regime.In these three cases, there is an associated "non compact" quantum group, which is a star Hopf algebra, with a category of unitary representations. The computation of physical quantities in these three systems amounts to compute explicitely, or to have a good understanding of the