A Duflo Star Product for Poisson Groups

Symmetry Integrability and Geometry-Methods and Applications

article

Adrien Brochier¹ 

[1] MPIM

关键词: quantum groups;    knot theory;    Duflo isomorphism;   

DOI  :  10.3842/SIGMA.2016.088

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown