UNIVERSAL ENVELOPING ALGEBRAS Algebraic constructions The Poincare-Birkhoff-Witt theorem POISSON GEOMETRY Poisson structures Normal forms Local Poisson geometry POISSON CATEGORY Poisson maps Hamiltonian actions DUAL PAIRS Operator algebras Dual pairs in Poisson geometry Examples of symplectic realizations GENERALIZED FUNCTIONS Group algebras Densities GROUPOIDS Groupoids Groupoid algebras Extended groupoid algebras ALGEBROIDS Lie algebroids Examples of Lie algebroids Differential geometry for Lie algebroids DEFORMATIONS OF ALGEBRAS OF FUNCTIONS Algebraic deformation theory Weyl algebras Deformation quantization.

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Geometric Models for Noncommutative Algebras

In this paper we present the solution to a longstanding problem of differential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications.

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Integrability of Lie brackets

In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications.

The modular automorphism group of a Poisson manifold

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.240.940&rep=rep1&type=pdf

Lie Algebroids, Holonomy and Characteristic Classes

Feuilletages à singularités

Poisson structures are contravariant structures. Nevertheless there is a good description of regular Poisson manifolds by means of foliated symplectic forms. This point of view makes it easy to lift these structures to the holonomy or homotopy groupoid of their symplectic (regular) foliation; defining the Poisson realization of the Poisson structure.

Intégration Symplectique des Variétés de Poisson Totalement Asphériques

Resume En geometrie globale les groupoides de Lie sont incontournables : ainsi l'utilisation du groupoide d'holonomie d'un feuilletage regulier conduit-elle a une solution satisfaisante de nombreux problemes d'existence globale. Trois exemples sont donnes : le theoreme de Palais sur les actions locales de groupes, la methode des caracteristiques et l'etude des liens entre morphismes de groupoides et d'algebroides de Lie. Des demonstrations detaillees seront donnees par ailleurs.

Groupoïde d'holonomie et géométrie globale

The purpose of this note is to give an example of a singular Poisson structure on R2 which admits a symplectic realization by a Lie groupoid.

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Pierre Dazord

Papers

Feuilletages à singularités

Intégration Symplectique des Variétés de Poisson Totalement Asphériques

Groupoïde d'holonomie et géométrie globale

Sur l'intégration symplectique de la structure de poisson singuliére Λ = (x2 + y2) ∂/∂x Λ ∂/∂y de R2