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Formal symplectic groupoid. (English) Zbl 1072.58008

The principal objective in this paper is to give a positive answer to the deformation problem for symplectic groupoids. Its exact formulation goes as follows:

Given a Poisson structure $\alpha$ on ${ℝ}^{d}$ there exists a unique natural deformation of the trivial generating function such that the first-order is precisely $\alpha$. Moreover, we have an explicit formula for this deformation

${S}_{h}\left({p}_{1},{p}_{2},x\right)=x\left({p}_{1}+{p}_{2}\right)+\sum _{n=1}^{\infty }\frac{{h}^{n}}{n!}\sum _{{\Gamma }\in {T}_{n,2}}{W}_{{\Gamma }}{\stackrel{^}{B}}_{{\Gamma }}\left({p}_{1},{p}_{2},x\right),$

where ${T}_{n,2}$ is the set of Kontsevich trees of type $\left(n,2\right)$, ${W}_{{\Gamma }}$ is the Kontsevich weight of ${\Gamma }$ and ${\stackrel{^}{B}}_{{\Gamma }}$ is the symbol of the bidifferential operator ${B}_{{\Gamma }}$ associated to ${\Gamma }$.

In case of a linear Poisson structure (i.e., the Kirillov-Kostant Poisson structure on the dual ${𝒢}^{*}$ of a Lie algebra $𝒢$), the generating function of the sympletic groupoid over ${𝒢}^{*}$ reduces exactly to the familiar Campbell-Baker-Hausdorff formula

${S}_{h}\left({p}_{1},{p}_{2},x\right)=〈\frac{1}{h}CBH\left(h{p}_{1},h{p}_{2}\right),x〉,$

where $〈,〉$ is the natural pairing between $𝒢$ and ${𝒢}^{*}$.

Finally, the authors show that the star-product studied in [M. Kontsevich, Lett. Math. Phys. 66, No. 3, 167–216 (2003; Zbl 1058.53065)] can be considered as a suitable exponentiation of a deformation of the Poisson structure.

##### MSC:
 58H05 Pseudogroups and differentiable groupoids on manifolds 53D05 Symplectic manifolds, general
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