Quantization of the algebra of chord diagrams
Jørgen Ellegaard Andersen, Josef Mattes, and Nicolai Reshetikhin
In this paper we define an algebra structure on the vector space $L(\Sigma )$ generated by links in the manifold $\Sigma \times [0,1]$ where $\Sigma $ is an oriented surface. This algebra has a filtration and the associated graded algebra $L_{Gr}(\Sigma )$ is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra of chord diagrams $ch\left( \Sigma \right) $ on $\Sigma $ to $L_{Gr}(\Sigma )$.
We show that multiplication in $L\left( \Sigma \right) $ 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 $% \Sigma \times [0,1]$. The quantization descends to a quantization of the moduli space of flat connections on $\Sigma $ and it is universal with respect to group homomorphisms. If $\Sigma $ is a compact with free fundamental group we construct a universal Vassiliev invariant.