| Preface | p. xi |
| Generalities on Poisson Structures | |
| Poisson brackets | p. 1 |
| Poisson tensors | p. 5 |
| Poisson morphisms | p. 9 |
| Local canonical coordinates | p. 13 |
| Singular symplectic foliations | p. 16 |
| Transverse Poisson structures | p. 21 |
| Group actions and reduction | p. 23 |
| The Schouten bracket | p. 27 |
| Schouten bracket of multi-vector fields | p. 27 |
| Schouten bracket on Lie algebras | p. 31 |
| Compatible Poisson structures | p. 33 |
| Symplectic realizations | p. 34 |
| Poisson Cohomology | |
| Poisson cohomology | p. 39 |
| Definition of Poisson cohomology | p. 39 |
| Interpretation of Poisson cohomology | p. 40 |
| Poisson cohomology versus de Rham cohomology | p. 41 |
| Other versions of Poisson cohomology | p. 42 |
| Computation of Poisson cohomology | p. 43 |
| Normal forms of Poisson structures | p. 44 |
| Cohomology of Lie algebras | p. 49 |
| Chevalley-Eilenberg complexes | p. 49 |
| Cohomology of linear Poisson structures | p. 51 |
| Rigid Lie algebras | p. 53 |
| Spectral sequences | p. 54 |
| Spectral sequence of a filtered complex | p. 54 |
| Leray spectral sequence | p. 56 |
| Hochschild-Serre spectral sequence | p. 57 |
| Spectral sequence for Poisson cohomology | p. 59 |
| Poisson cohomology in dimension 2 | p. 60 |
| Simple singularities | p. 61 |
| Cohomology of Poisson germs | p. 63 |
| Some examples and remarks | p. 68 |
| The curl operator | p. 69 |
| Definition of the curl operator | p. 69 |
| Schouten bracket via curl operator | p. 71 |
| The modular class | p. 72 |
| The curl operator of an affine connection | p. 73 |
| Poisson homology | p. 74 |
| Levi Decomposition | |
| Formal Levi decomposition | p. 78 |
| Levi decomposition of Poisson structures | p. 81 |
| Construction of Levi decomposition | p. 84 |
| Normed vanishing of cohomology | p. 88 |
| Proof of analytic Levi decomposition theorem | p. 92 |
| The smooth case | p. 98 |
| Linearization of Poisson Structures | |
| Nondegenerate Lie algebras | p. 105 |
| Linearization of low-dimensional Poisson structures | p. 107 |
| Two-dimensional case | p. 107 |
| Three-dimensional case | p. 108 |
| Four-dimensional case | p. 110 |
| Poisson geometry of real semisimple Lie algebras | p. 112 |
| Nondegeneracy of aff(n) | p. 117 |
| Some other linearization results | p. 122 |
| Equivariant linearization | p. 122 |
| Linearization of Poisson-Lie tensors | p. 122 |
| Poisson structures with a hyperbolic <$>op {R}^k<$>-action | p. 124 |
| Transverse Poisson structures to coadjoint orbits | p. 125 |
| Finite determinacy of Poisson structures | p. 126 |
| Multiplicative and Quadratic Poisson Structures | |
| Multiplicative tensors | p. 129 |
| Poisson-Lie groups and r-matrices | p. 132 |
| The dual and the double of a Poisson-Lie group | p. 136 |
| Actions of Poisson-Lie groups | p. 139 |
| Poisson actions of Poisson-Lie groups | p. 139 |
| Dressing transformations | p. 142 |
| Momentum maps | p. 144 |
| r-matrices and quadratic Poisson structures | p. 145 |
| Linear curl vector fields | p. 147 |
| Quadratization of Poisson structures | p. 150 |
| Nonhomogeneous quadratic Poisson structures | p. 156 |
| Nambu Structures and Singular Foliations | |
| Nambu brackets and Nambu tensors | p. 159 |
| Integrable differential forms | p. 165 |
| Frobenius with singularities | p. 168 |
| Linear Nambu structures | p. 171 |
| Kupka's phenomenon | p. 178 |
| Linearization of Nambu structures | p. 182 |
| Decomposability of | p. 184 |
| Formal linearization of the associated foliation | p. 185 |
| The analytic case | p. 188 |
| Formal linearization of | p. 188 |
| The smooth elliptic case | p. 190 |
| Integrable 1-forms with a non-zero linear part | p. 192 |
| Quadratic integrable 1-forms | p. 197 |
| Poisson structures in dimension 3 | p. 199 |
| Lie Groupoids | |
| Some basic notions on groupoids | p. 203 |
| Definitions and first examples | p. 203 |
| Lie groupoids | p. 206 |
| Germs and slices of Lie groupoids | p. 208 |
| Actions of groupoids | p. 208 |
| Haar systems | p. 209 |
| Morita equivalence | p. 210 |
| Proper Lie groupoids | p. 213 |
| Definition and elementary properties | p. 213 |
| Source-local triviality | p. 215 |
| Orbifold groupoids | p. 216 |
| Linearization of Lie groupoids | p. 217 |
| Linearization of Lie group actions | p. 217 |
| Local linearization of Lie groupoids | p. 218 |
| Slice theorem for Lie groupoids | p. 222 |
| Symplectic groupoids | p. 223 |
| Definition and basic properties | p. 223 |
| Proper symplectic groupoids | p. 227 |
| Hamiltonian actions of symplectic groupoids | p. 232 |
| Some generalizations | p. 233 |
| Lie Algebroids | |
| Some basic definitions and properties | p. 235 |
| Definition and some examples | p. 235 |
| The Lie algebroid of a Lie groupoid | p. 237 |
| Isotropy algebras | p. 238 |
| Characteristic foliation of a Lie algebroid | p. 239 |
| Lie pseudoalgebras | p. 239 |
| Fiber-wise linear Poisson structures | p. 240 |
| Lie algebroid morphisms | p. 242 |
| Lie algebroid actions and connections | p. 243 |
| Splitting theorem and transverse structures | p. 246 |
| Cohomology of Lie algebroids | p. 249 |
| Linearization of Lie algebroids | p. 252 |
| Integrability of Lie brackets | p. 257 |
| Reconstruction of groupoids from their algebroids | p. 257 |
| Integrability criteria | p. 259 |
| Integrability of Poisson manifolds | p. 262 |
| Appendix | |
| Moser's path method | p. 263 |
| Division theorems | p. 269 |
| Reeb stability | p. 271 |
| Action-angle variables | p. 273 |
| Normal forms of vector fields | p. 276 |
| Poincaré-Dulac normal forms | p. 276 |
| Birkhoff normal forms | p. 278 |
| Toric characterization of normal forms | p. 280 |
| Smooth normal forms | p. 282 |
| Normal forms along a singular curve | p. 283 |
| The neighborhood of a symplectic leaf | p. 286 |
| Geometric data and coupling tensors | p. 286 |
| Linear models | p. 290 |
| Dirac structures | p. 292 |
| Deformation quantization | p. 294 |
| Bibliography | p. 299 |
| Index | p. 317 |
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