COMBINATORIAL QUANTIZATION OF THE HAMILTONIAN CHERN-SIMONS THEORY .2.

This paper further develops the combinatorial approach to quantization of the Hamiltonian Chern Simons theory advertised in [1]. Using the t heory of quantum Wilson lines, we show how the Verlinde algebra appear s within the context of quantum group gauge theory. This allows to dis cuss flatness of quantum connections so that we can give a mathematica lly rigorous definition of the algebra of observables A(CS) of the Che rn Simons model. It is a -algebra of ''functions on the quantum modul i space of flat connections'' and comes equipped with a positive funct ional omega (''integration''). We prove that this data does not depend on the particular choices which have been made in the construction. F ollowing ideas of Fock and Rosly [2], the algebra A(CS) provides a def ormation quantization of the algebra of functions on the moduli space along the natural Poisson bracket induced by the Chern Simons action. We evaluate a volume of the quantized moduli space and prove that it c oincides with the Verlinde number. This answer is also interpreted as a partition partition function of the lattice Yang-Mills theory corres ponding to a quantum gauge group.