Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 088, 12 pages      arXiv:1604.08450

A Duflo Star Product for Poisson Groups

Adrien Brochier
MPIM Bonn, Germany

Received May 18, 2016, in final form September 05, 2016; Published online September 08, 2016

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.

Key words: quantum groups; knot theory; Duflo isomorphism.

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