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**Extensions of symplectic groupoids and quantization.**
*(English)*
Zbl 0722.58021

As part of a program to construct noncommutative algebras from Poisson manifolds, the authors study the prequantization of the symplectic groupoids of Poisson manifolds. The main results in the paper may be summarized as follows. A suitable prequantization of the symplectic groupoid \(\Gamma\) of a Poisson manifold is an extension of that groupoid by the circle group. Such an extension corresponds, by a slight variation of the construction by J. Renault in “A groupoid approach to \(C^*\)-algebras (1980; Zbl 0433.46049), to a groupoid 2-cocycle. This cocycle is represented by the symplectic form on \(\Gamma\), and the corresponding Lie algebroid 2-cocycle is precisely the one attached to the Poisson structure itself by J. Huebschmann in J. Reine Angew. Math. 408, 57-113 (1990; Zbl 0699.53037).

The first section of the paper is a general discussion of the relation between Lie groupoid cohomology and Lie algebroid cohomology and may be of independent interest. It turns out that, in order to use Lie groupoid cohomology to describe smooth groupoid extensions which are topologically nontrivial circle bundles, one must consider cocycles which are smooth only in the neighborhood of the “identity diagonal” of the groupoid. This “identity smooth” cohomology theory, inspired by that of Tuynman and Wiegerinck [G. M. Tuynman and W. A. J. J. Wiegerinck, J. Geom. Phys. 4, No.2, 207-258 (1987; Zbl 0649.58014)] in the case of groups, seems to be new.

The first section of the paper is a general discussion of the relation between Lie groupoid cohomology and Lie algebroid cohomology and may be of independent interest. It turns out that, in order to use Lie groupoid cohomology to describe smooth groupoid extensions which are topologically nontrivial circle bundles, one must consider cocycles which are smooth only in the neighborhood of the “identity diagonal” of the groupoid. This “identity smooth” cohomology theory, inspired by that of Tuynman and Wiegerinck [G. M. Tuynman and W. A. J. J. Wiegerinck, J. Geom. Phys. 4, No.2, 207-258 (1987; Zbl 0649.58014)] in the case of groups, seems to be new.

Reviewer: A.Weinstein

### MSC:

53D50 | Geometric quantization |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

17B56 | Cohomology of Lie (super)algebras |

55R25 | Sphere bundles and vector bundles in algebraic topology |

58H99 | Pseudogroups, differentiable groupoids and general structures on manifolds |

20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |