Quantization of the algebra of chord diagrams

In this paper we study the algebra L(Sigma) generated by links in the manif old Sigma x[0, 1] where Sigma is an oriented surface. This algebra has a fi ltration and the associated graded algebra L-Gr (Sigma) is naturally a Pois son algebra. There is a Poisson algebra homomorphism from the algebra ch (S igma) of chord diagrams on Sigma to L-Gr(Sigma). We show that multiplication in L (Sigma) provides a geometric way to define a deformation quantization of the algebra of chord diagrams on Sigma, prov ided there is a universal Vassiliev invariant for links in Sigma x [0, 1]. If Sigma is compact with free I fundamental group me construct a universal Vassiliev invariant. The quantization descends to a quantization of the mod uli space of flat connections on Sigma and it is natural with respect to gr oup homomorphisms.