On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds.

*(English)*Zbl 1059.53064The 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.

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.

Reviewer: Angela Gammella (Nogent sur Oise)

##### 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 |