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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 α on d there exists a unique natural deformation of the trivial generating function such that the first-order is precisely α. Moreover, we have an explicit formula for this deformation

S h (p 1 ,p 2 ,x)=x(p 1 +p 2 )+ n=1 h n n! ΓT n,2 W Γ B ^ Γ (p 1 ,p 2 ,x),

where T n,2 is the set of Kontsevich trees of type (n,2), W Γ is the Kontsevich weight of Γ and B ^ Γ is the symbol of the bidifferential operator B Γ associated to Γ.

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 (p 1 ,p 2 ,x)=1 h C B H (hp 1 ,hp 2 ) , 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:
58H05Pseudogroups and differentiable groupoids on manifolds
53D05Symplectic manifolds, general
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