A Duflo Star Product for Poisson Groups

TL;DR

This paper generalizes Duflo's theorem to Poisson groups by showing the center of their quantization aligns with the Poisson center, using Etingof-Kazhdan functors and graphical calculus.

Contribution

It extends Duflo's theorem to Poisson algebraic groups via a novel application of Etingof-Kazhdan quantization and graphical calculus methods.

Findings

Center of quantized Poisson group is isomorphic to Poisson center

Generalizes Duflo's theorem to broader classes of groups

Connects to Kashiwara-Vergne conjecture and Kontsevich integral

Abstract

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 $\mathfrak{a}$ and the subalgebra of ad-invariant in the symmetric algebra of $\mathfrak{a}$. 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.