• Corpus ID: 119152339

# Integrability of Jacobi structures

```@article{Crainic2004IntegrabilityOJ,
title={Integrability of Jacobi structures},
author={Marius Crainic and C. Zhu},
journal={arXiv: Differential Geometry},
year={2004}
}```
• Published 16 March 2004
• Mathematics
• arXiv: Differential Geometry
We discuss the integrability of Jacobi manifolds by contact groupoids, and then look at what the Jacobi point of view brings new into Poisson geometry. In particular, using contact groupoids, we prove a Kostant-type theorem on the prequantization of symplectic groupoids, which answers a question posed by Weinstein and Xu \cite{prequan}. The methods used are those of Crainic-Fernandes on \$A\$-paths and monodromy group(oid)s of algebroids. In particular, most of the results we obtain are valid…

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## References

SHOWING 1-10 OF 20 REFERENCES

### Sur l'intégration des algèbres de Lie locales et la préquantification

This self-containt paper is devoted to the study of Lie groups and algebras in infinite dimension by mean of a method built on the theories of Lie groupoids, Lie algebroids and diffeologies: we are

### On integrability of infinitesimal actions

• Mathematics
• 2000
We use foliations and connections on principal Lie groupoid bundles to prove various integrability results for Lie algebroids. In particular, we show, under quite general assumptions, that the

### Contact reduction and groupoid actions

• Mathematics
• 2004
We introduce a new method to perform reduction of contact manifolds that extends Willett's and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map J: M → Γ 0

### Integration of twisted Dirac brackets

• Mathematics
• 2003
Given a Lie groupoid G over a manifold M, we show that multiplicative 2-forms on G relatively closed with respect to a closed 3-form phi on M correspond to maps from the Lie algebroid of G into T* M

### Extensions of symplectic groupoids and quantization.

• Mathematics
• 1991
An important role of Poisson manifolds is äs intermediate objects between ordinary manifolds, with their commutative algebras of functions, and the "noncommutative spaces" of quantum mechanics. Up to

### Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes

AbstractIn the first section we discuss Morita invariance of differentiable/algebroid cohomology.In the second section we extend the Van Est isomorphism to groupoids. As a first application we

### Poisson sigma models and symplectic groupoids

• Mathematics
• 2000
We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the

### Geometric Models for Noncommutative Algebras

• Mathematics
• 1999
UNIVERSAL ENVELOPING ALGEBRAS Algebraic constructions The Poincare-Birkhoff-Witt theorem POISSON GEOMETRY Poisson structures Normal forms Local Poisson geometry POISSON CATEGORY Poisson maps