On Deformation Quantization of Poisson-Lie Groups and Moduli Spaces of Flat Connections

Abstract

We give simple explicit formulas for deformation quantization of Poisson-Lie groups and of similar Poisson manifolds which can be represented as moduli spaces of flat connections on surfaces. The star products depend on a choice of Drinfe\v{l}d associator and are obtained by applying certain monoidal functors (fusion and reduction) to commutative algebras in Drinfe\v{l}d categories. From a geometric point of view this construction can be understood as a quantization of the quasi-Poisson structures on moduli spaces of flat connections.