Formal symplectic groupoid (Q1770251)

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 \(\mathbb{R}^d\) 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 \[ S_h(p_1, p_2, x)= x(p_1+ p_2)+ \sum^\infty_{n=1} {h^n\over n!} \sum_{\Gamma\in T_{n,2}} W_\Gamma\widehat B_\Gamma(p_1, p_2,x), \] 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 \(\Gamma\). In case of a linear Poisson structure (i.e., the Kirillov-Kostant Poisson structure on the dual \({\mathcal G}^*\) of a Lie algebra \({\mathcal G}\)), the generating function of the sympletic groupoid over \({\mathcal G}^*\) 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 \({\mathcal G}\) and \({\mathcal G}^*\). Finally, the authors show that the star-product studied in [\textit{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.