Combinatorial quantization of Euclidean gravity in three dimensions

In the Chern-Simons formulation of Einstein gravity in 2+1 dimensions, the phase space is the moduli space of flat G-connections on a two-dimensional surface, where G is a typically non-compact Lie group which depends on the signature of space-time and the cosmological constant. For Euclidean signature and vanishing cosmological constant, G is the three-dimensional Euclidean group. For this case the Poisson structure of the moduli space is given explicitly in terms of a classical r-matrix. It is shown that the quantum R-matrix of the quantum double D(SU(2)) provides a quantization of that Poisson structure.