A Duflo Star Product for Poisson Groups

Let G be a finite-dimensional Poisson algebraic, Lie or formal group. We show that the center of the quantization of G provided by an Etingof–Kazhdan functor is isomorphic as an algebra to the Poisson center of the algebra of functions on G. This recovers and generalizes Duflo’s theorem which gives an isomorphism between the center of the enveloping algebra of a finite-dimensional Lie algebra a and the subalgebra of ad-invariant in the symmetric algebra of a. As our proof relies on Etingof–Kazhdan construction it ultimately depends on the existence of Drinfeld associators, but otherwise it is a fairly simple application of graphical calculus. This shed some lights on Alekseev–Torossian proof of the Kashiwara–Vergne conjecture, and on the relation observed by Bar-Natan–Le–Thurston between the Duflo isomorphism and the Kontsevich integral of the unknot.