In this paper we study the algebra L([sum ]) generated
by links in the manifold
[sum ]×[0, 1] where [sum ] is an oriented surface. This
algebra has a filtration and the
associated graded algebra LGr([sum ]) is naturally
a Poisson algebra. There is a Poisson
algebra homomorphism from the algebra ch ([sum ]) of chord diagrams on [sum ]
to LGr([sum ]).
We show that multiplication in L([sum ]) provides a geometric way to
define a
deformation quantization of the algebra of chord diagrams on [sum ], provided
there is a
universal Vassiliev invariant for links in [sum ]×[0, 1].
If [sum ] is compact with free
fundamental group we construct a universal Vassiliev invariant. The quantization
descends to a quantization of the moduli space of flat connections on [sum ]
and it is
natural with respect to group homomorphisms.