QUANTIZATION OF THE ALGEBRA OF CHORD DIAGRAMS

JØRGEN ELLEGAARD ANDERSEN JOSEF MATTES NICOLAI RESHETIKHIN

Abstract. In this paper we define an algebra structure on the vector space L(Σ) generated by links in the manifold Σ × [0, 1] where Σ is an oriented surface. This algebra has a filtration and the associated graded algebra L_{Gr}(Σ) is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra of chord diagrams ch (Σ) on Σ to L_{Gr}(Σ).

We show that multiplication in L (Σ) provides a geometric way to define a deformation quantization of the algebra of chord diagrams, provided there is a universal Vassiliev invariant for links in Σ × [0, 1]. The quantization descends to a quantization of the moduli space of flat connections on Σ and it is universal with respect to group homomorphisms. If Σ is a compact with free fundamental group we construct a universal Vassiliev invariant.