## Lectures on the Geometry of Poisson ManifoldsThis book is addressed to graduate students and researchers in the fields of mathematics and physics who are interested in mathematical and theoretical physics, differential geometry, mechanics, quantization theories and quantum physics, quantum groups etc., and who are familiar with differentiable and symplectic manifolds. The aim of the book is to provide the reader with a monograph that enables him to study systematically basic and advanced material on the recently developed theory of Poisson manifolds, and that also offers ready access to bibliographical references for the continuation of his study. Until now, most of this material was dispersed in research papers published in many journals and languages. The main subjects treated are the Schouten-Nijenhuis bracket; the generalized Frobenius theorem; the basics of Poisson manifolds; Poisson calculus and cohomology; quantization; Poisson morphisms and reduction; realizations of Poisson manifolds by symplectic manifolds and by symplectic groupoids and Poisson-Lie groups. The book unifies terminology and notation. It also reports on some original developments stemming from the author's work, including new results on Poisson cohomology and geometric quantization, cofoliations and biinvariant Poisson structures on Lie groups. |

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### Contents

1 | |

The symplectic foliation of a Poisson manifold | 19 |

Examples of Poisson manifolds | 31 |

Poisson calculus | 41 |

Poisson cohomology | 63 |

An introduction to quantization | 83 |

Poisson morphisms coinduced structures reduction | 97 |

Symplectic realizations of Poisson manifolds | 115 |

Realizations of Poisson manifolds | 134 |

PoissonLie groups | 161 |

189 | |

202 | |

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### Common terms and phrases

1-forms action of G Ann(TJF bialgebra bundle coad cofoliation coinduced coisotropic computation condition connected const constant contravariant derivative coordinates corresponding defined Definition denote differentiable manifold distribution dual endowed equation equivalent exists fibers foliation follows formula functions Furthermore Hamiltonian vector fields Hence homomorphism implies induced isomorphism isotropic realization Jacobi identity left invariant Lie algebra Lie algebroid Lie group Lie group G Lie groupoid Lie-Poisson structure linear LP-cohomology Math momentum map morphism multiplication operation orbits Poisson algebra Poisson bivector Poisson bracket Poisson manifold Poisson mapping Poisson structure Poisson-Lie group prequantization Proof Proposition quantization regular Poisson manifold Remark result Riemannian right invariant satisfies Schouten-Nijenhuis bracket submersion symplectic foliation symplectic form symplectic geometry symplectic groupoid symplectic leaves symplectic manifold symplectic orthogonal symplectic realizations symplectic structure Theorem theory transversal unit manifold vanishes Weinstein yields