Poisson structures and their normal forms

Poisson manifolds play a fundamental role in Hamiltonian dynamics, where they serve as phase spaces. They also arise naturally in other mathematical problems, and form a bridge from the "commutative world" to the "noncommutative world". The aim of this book is twofold: On the one hand, it gives a quick, self-contained introduction to Poisson geometry and related subjects, including singular foliations, Lie groupoids and Lie algebroids. On the other hand, it presents a comprehensive treatment of the normal form problem in Poisson geometry. Even when it comes to classical results, the book gives new insights. It contains results obtained over the past 10 years which are not available in other books.

Preface	p. xi
1 Generalities on Poisson Structures
1.1 Poisson brackets	p. 1
1.2 Poisson tensors	p. 5
1.3 Poisson morphisms	p. 9
1.4 Local canonical coordinates	p. 13
1.5 Singular symplectic foliations	p. 16
1.6 Transverse Poisson structures	p. 21
1.7 Group actions and reduction	p. 23
1.8 The Schouten bracket	p. 27
1.8.1 Schouten bracket of multi-vector fields	p. 27
1.8.2 Schouten bracket on Lie algebras	p. 31
1.8.3 Compatible Poisson structures	p. 33
1.9 Symplectic realizations	p. 34
2 Poisson Cohomology
2.1 Poisson cohomology	p. 39
2.1.1 Definition of Poisson cohomology	p. 39
2.1.2 Interpretation of Poisson cohomology	p. 40
2.1.3 Poisson cohomology versus de Rham cohomology	p. 41
2.1.4 Other versions of Poisson cohomology	p. 42
2.1.5 Computation of Poisson cohomology	p. 43
2.2 Normal forms of Poisson structures	p. 44
2.3 Cohomology of Lie algebras	p. 49
2.3.1 Chevalley-Eilenberg complexes	p. 49
2.3.2 Cohomology of linear Poisson structures	p. 51
2.3.3 Rigid Lie algebras	p. 53
2.4 Spectral sequences	p. 54
2.4.1 Spectral sequence of a filtered complex	p. 54
2.4.2 Leray spectral sequence	p. 56
2.4.3 Hochschild-Serre spectral sequence	p. 57
2.4.4 Spectral sequence for Poisson cohomology	p. 59
2.5 Poisson cohomology in dimension 2	p. 60
2.5.1 Simple singularities	p. 61
2.5.2 Cohomology of Poisson germs	p. 63
2.5.3 Some examples and remarks	p. 68
2.6 The curl operator	p. 69
2.6.1 Definition of the curl operator	p. 69
2.6.2 Schouten bracket via curl operator	p. 71
2.6.3 The modular class	p. 72
2.6.4 The curl operator of an affine connection	p. 73
2.7 Poisson homology	p. 74
3 Levi Decomposition
3.1 Formal Levi decomposition	p. 78
3.2 Levi decomposition of Poisson structures	p. 81
3.3 Construction of Levi decomposition	p. 84
3.4 Normed vanishing of cohomology	p. 88
3.5 Proof of analytic Levi decomposition theorem	p. 92
3.6 The smooth case	p. 98
4 Linearization of Poisson Structures
4.1 Nondegenerate Lie algebras	p. 105
4.2 Linearization of low-dimensional Poisson structures	p. 107
4.2.1 Two-dimensional case	p. 107
4.2.2 Three-dimensional case	p. 108
4.2.3 Four-dimensional case	p. 110
4.3 Poisson geometry of real semisimple Lie algebras	p. 112
4.4 Nondegeneracy of aff(n)	p. 117
4.5 Some other linearization results	p. 122
4.5.1 Equivariant linearization	p. 122
4.5.2 Linearization of Poisson-Lie tensors	p. 122
4.5.3 Poisson structures with a hyperbolic \op {{R}}^k -action	p. 124
4.5.4 Transverse Poisson structures to coadjoint orbits	p. 125
4.5.5 Finite determinacy of Poisson structures	p. 126
5 Multiplicative and Quadratic Poisson Structures
5.1 Multiplicative tensors	p. 129
5.2 Poisson-Lie groups and r-matrices	p. 132
5.3 The dual and the double of a Poisson-Lie group	p. 136
5.4 Actions of Poisson-Lie groups	p. 139
5.4.1 Poisson actions of Poisson-Lie groups	p. 139
5.4.2 Dressing transformations	p. 142
5.4.3 Momentum maps	p. 144
5.5 r-matrices and quadratic Poisson structures	p. 145
5.6 Linear curl vector fields	p. 147
5.7 Quadratization of Poisson structures	p. 150
5.8 Nonhomogeneous quadratic Poisson structures	p. 156
6 Nambu Structures and Singular Foliations
6.1 Nambu brackets and Nambu tensors	p. 159
6.2 Integrable differential forms	p. 165
6.3 Frobenius with singularities	p. 168
6.4 Linear Nambu structures	p. 171
6.5 Kupka's phenomenon	p. 178
6.6 Linearization of Nambu structures	p. 182
6.6.1 Decomposability of ¿	p. 184
6.6.2 Formal linearization of the associated foliation	p. 185
6.6.3 The analytic case	p. 188
6.6.4 Formal linearization of ¿	p. 188
6.6.5 The smooth elliptic case	p. 190
6.7 Integrable 1-forms with a non-zero linear part	p. 192
6.8 Quadratic integrable 1-forms	p. 197
6.9 Poisson structures in dimension 3	p. 199
7 Lie Groupoids
7.1 Some basic notions on groupoids	p. 203
7.1.1 Definitions and first examples	p. 203
7.1.2 Lie groupoids	p. 206
7.1.3 Germs and slices of Lie groupoids	p. 208
7.1.4 Actions of groupoids	p. 208
7.1.5 Haar systems	p. 209
7.2 Morita equivalence	p. 210
7.3 Proper Lie groupoids	p. 213
7.3.1 Definition and elementary properties	p. 213
7.3.2 Source-local triviality	p. 215
7.3.3 Orbifold groupoids	p. 216
7.4 Linearization of Lie groupoids	p. 217
7.4.1 Linearization of Lie group actions	p. 217
7.4.2 Local linearization of Lie groupoids	p. 218
7.4.3 Slice theorem for Lie groupoids	p. 222
7.5 Symplectic groupoids	p. 223
7.5.1 Definition and basic properties	p. 223
7.5.2 Proper symplectic groupoids	p. 227
7.5.3 Hamiltonian actions of symplectic groupoids	p. 232
7.5.4 Some generalizations	p. 233
8 Lie Algebroids
8.1 Some basic definitions and properties	p. 235
8.1.1 Definition and some examples	p. 235
8.1.2 The Lie algebroid of a Lie groupoid	p. 237
8.1.3 Isotropy algebras	p. 238
8.1.4 Characteristic foliation of a Lie algebroid	p. 239
8.1.5 Lie pseudoalgebras	p. 239
8.2 Fiber-wise linear Poisson structures	p. 240
8.3 Lie algebroid morphisms	p. 242
8.4 Lie algebroid actions and connections	p. 243
8.5 Splitting theorem and transverse structures	p. 246
8.6 Cohomology of Lie algebroids	p. 249
8.7 Linearization of Lie algebroids	p. 252
8.8 Integrability of Lie brackets	p. 257
8.8.1 Reconstruction of groupoids from their algebroids	p. 257
8.8.2 Integrability criteria	p. 259
8.8.3 Integrability of Poisson manifolds	p. 262
Appendix
A.1 Moser's path method	p. 263
A.2 Division theorems	p. 269
A.3 Reeb stability	p. 271
A.4 Action-angle variables	p. 273
A.5 Normal forms of vector fields	p. 276
A.5.1 Poincaré-Dulac normal forms	p. 276
A.5.2 Birkhoff normal forms	p. 278
A.5.3 Toric characterization of normal forms	p. 280
A.5.4 Smooth normal forms	p. 282
A.6 Normal forms along a singular curve	p. 283
A.7 The neighborhood of a symplectic leaf	p. 286
A.7.1 Geometric data and coupling tensors	p. 286
A.7.2 Linear models	p. 290
A.8 Dirac structures	p. 292
A.9 Deformation quantization	p. 294
Bibliography	p. 299
Index	p. 317

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