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Zbl 1072.58008
Cattaneo, Alberto S.; Dherin, Benoit; Felder, Giovanni
Formal symplectic groupoid.
(English)
[J] Commun. Math. Phys. 253, No. 3, 645-674 (2005). ISSN 0010-3616; ISSN 1432-0916/e

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:\par Given a Poisson structure $\alpha$ on $\bbfR^d$ 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 $$S_h(p_1, p_2, x)= x(p_1+ p_2)+ \sum^\infty_{n=1} {h^n\over n!} \sum_{\Gamma\in T_{n,2}} W_\Gamma\widehat B_\Gamma(p_1, p_2,x),$$ where $T_{n,2}$ is the set of Kontsevich trees of type $(n, 2)$, $W_\Gamma$ is the Kontsevich weight of $\Gamma$ and $\widehat B_\Gamma$ is the symbol of the bidifferential operator $B_\Gamma$ associated to $\Gamma$.\par In case of a linear Poisson structure (i.e., the Kirillov-Kostant Poisson structure on the dual ${\Cal G}^*$ of a Lie algebra ${\Cal G}$), the generating function of the sympletic groupoid over ${\Cal G}^*$ reduces exactly to the familiar Campbell-Baker-Hausdorff formula $$S_h(p_1, p_2,x)= \biggl\langle {1\over h} CBH(hp_1, hp_2), x\biggr\rangle,$$ where $\langle,\rangle$ is the natural pairing between ${\Cal G}$ and ${\Cal G}^*$.\par Finally, the authors show that the star-product studied in [{\it 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.
[Hirokazu Nishimura (Tsukuba)]
MSC 2000:
*58H05 Pseudogroups on manifolds
53D05 Symplectic manifolds, general

Keywords: symplectic groupoid; Campbell-Baker-Hausdorff formula; quantization; Runge-Kutta method

Citations: Zbl 1058.53065

Cited in: Zbl 1101.58016

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