Intégration Symplectique des Variétés de Poisson Régulières |
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Authors: | F Alcalde Cuesta G Hector |
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Institution: | (1) Departamento de Xeometria e Topoloxia, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain;(2) Laboratoire de Géométrie et Analyse — U.R.A. 746, Université Claude Bernard-Lyon 1, 69622 Villeurbanne, France |
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Abstract: | Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ0 with the induced Poisson structure Λ0 is isomorphic to (M, Λ). This notion was introduced by A. Weinstein in 28] in order to quantize Poisson manifolds by quantizing their symplectic
integration. Any Poisson manifold can be integrated by alocal symplectic groupoid (4], 13]) but already for regular Poisson manifolds there are obstructions to global integrability
(2], 6], 11], 17], 28]).
The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability
of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will
provide necessary and sufficient if we require Γ to be Hausdorff, which is a suitable condition to proceed to Weinstein’s
program of quantization. These integrability results may be interpreted as a generalization of the Cartan-Smith proof of Lie’s
third theorem in the infinite dimensional case.
Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90) |
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