Link invariants and combinatorial quantization of Hamiltonian Chern Simons theory

We define and study the properties of observables associated to any link in Sigma xR (where Sigma is a compact surface) using the combinatorial quantization of hamiltonian Chem-Simons theory. These observables are traces of holonomies in a non-commutative Yang-Mills theory where the gauge symmetry is ensured by a quantum group. We show that these observables are link invariants taking values in a non-commutative algebra, the so-called Moduli Algebra. When Sigma=S-2 these link invariants are pure numbers and are equal to Reshetikhin-Turaev link invariants.