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On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds. (English) Zbl 1059.53064
The author shows how the integration of Lie algebroids can be seen as a particular case of the integration of Lie algebras. It was shown by the author and Felder that Poisson manifolds can be integrated to symplectic groupoids via symplectic reduction. Independently, it was proved by Crainic-Fernandes [M. Crainic and R. L. Fernandes, Ann. Math. (2) 157, No. 2, 575–620 (2003; Zbl 1037.22003)] and by P. Severa, that this integration can be generalized to any Lie algebroid.
The present paper shows that the construction of Crainic-Fernandes and Severa can also be viewed as a particular case of the integration of Poisson manifolds. In other words, the two constructions are in fact equivalent. Moreover, the above correspondence is extended to a more general correspondence between coisotropic submanifolds of a Poisson manifold and Lagrangian Lie subgroupoids of its symplectic groupoids. The case of twisted Poisson manifolds is also discussed at the end of the paper.

##### MSC:
 53D17 Poisson manifolds; Poisson groupoids and algebroids 53D20 Momentum maps; symplectic reduction 22A22 Topological groupoids (including differentiable and Lie groupoids) 58H05 Pseudogroups and differentiable groupoids
##### Keywords:
Poisson manifolds; Lie algebroids
Zbl 1037.22003
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