Poisson Structures and Their Normal Forms

331 Pages·2005·2.169 MB·English

by  Jean-Paul Dufour, Nguyen Tien Zung (auth.), H. Bass, J. Oesterlé, A. Weinstein (eds.)

Progress in Mathematics
Volume 242
Series Editor
H. Bass
J. Oesterlé
A. Weinstein
Jean-Paul Dufour
Nguyen Tien Zung
Poisson Structures
and Their
Normal Forms
Birkhäuser Verlag
Basel (cid:1) Boston (cid:1) Berlin
Authors:
Jean-Paul Dufour Nguyen Tien Zung
Département de mathématique Laboratoire Émile Picard,
Université de Montpellier 2 UMR 5580 CNRS
place Eugène Bataillon Institut de Mathématiques
34095 Montpellier Université Paul Sabatier
France 118 route de Narbonne
e-mail: [email protected] 31062 Toulouse
France
e-mail: [email protected]
2000 Mathematics Subject Classifi cation 53Dxx, 53Bxx, 85Kxx, 70Hxx
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ISBN 3-7643-7334-2 Birkhäuser Verlag, Basel – Boston – Berlin
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Generalities on Poisson Structures
1.1 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Poisson tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Poisson morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Local canonical coordinates . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Singular symplectic foliations . . . . . . . . . . . . . . . . . . . . . 16
1.6 Transverse Poisson structures . . . . . . . . . . . . . . . . . . . . . 21
1.7 Group actions and reduction . . . . . . . . . . . . . . . . . . . . . 23
1.8 The Schouten bracket . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8.1 Schouten bracket of multi-vector fields . . . . . . . . . . . . 27
1.8.2 Schouten bracket on Lie algebras . . . . . . . . . . . . . . . 31
1.8.3 Compatible Poisson structures . . . . . . . . . . . . . . . . 33
1.9 Symplectic realizations . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 Poisson Cohomology
2.1 Poisson cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1.1 Definition of Poisson cohomology . . . . . . . . . . . . . . . 39
2.1.2 Interpretation of Poissoncohomology . . . . . . . . . . . . 40
2.1.3 Poissoncohomology versus de Rham cohomology . . . . . . 41
2.1.4 Other versions of Poisson cohomology . . . . . . . . . . . . 42
2.1.5 Computation of Poissoncohomology . . . . . . . . . . . . . 43
2.2 Normal forms of Poisson structures . . . . . . . . . . . . . . . . . . 44
2.3 Cohomology of Lie algebras . . . . . . . . . . . . . . . . . . . . . . 49
2.3.1 Chevalley–Eilenberg complexes . . . . . . . . . . . . . . . . 49
2.3.2 Cohomology of linear Poisson structures . . . . . . . . . . . 51
2.3.3 Rigid Lie algebras . . . . . . . . . . . . . . . . . . . . . . . 53
2.4 Spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4.1 Spectral sequence of a filtered complex . . . . . . . . . . . . 54
2.4.2 Leray spectral sequence . . . . . . . . . . . . . . . . . . . . 56
2.4.3 Hochschild–Serre spectral sequence . . . . . . . . . . . . . . 57
viii Contents
2.4.4 Spectral sequence for Poisson cohomology . . . . . . . . . . 59
2.5 Poisson cohomology in dimension 2 . . . . . . . . . . . . . . . . . . 60
2.5.1 Simple singularities. . . . . . . . . . . . . . . . . . . . . . . 61
2.5.2 Cohomology of Poissongerms . . . . . . . . . . . . . . . . . 63
2.5.3 Some examples and remarks. . . . . . . . . . . . . . . . . . 68
2.6 The curl operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.6.1 Definition of the curl operator. . . . . . . . . . . . . . . . . 69
2.6.2 Schouten bracket via curl operator . . . . . . . . . . . . . . 71
2.6.3 The modular class . . . . . . . . . . . . . . . . . . . . . . . 72
2.6.4 The curl operator of an affine connection . . . . . . . . . . 73
2.7 Poisson homology. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Levi Decomposition
3.1 Formal Levi decomposition . . . . . . . . . . . . . . . . . . . . . . 78
3.2 Levi decomposition of Poissonstructures . . . . . . . . . . . . . . . 81
3.3 Construction of Levi decomposition. . . . . . . . . . . . . . . . . . 84
3.4 Normed vanishing of cohomology . . . . . . . . . . . . . . . . . . . 88
3.5 Proof of analytic Levi decomposition theorem . . . . . . . . . . . . 92
3.6 The smooth case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4 Linearization of Poisson Structures
4.1 Nondegenerate Lie algebras . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Linearization of low-dimensional Poissonstructures . . . . . . . . . 107
4.2.1 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . 107
4.2.2 Three-dimensional case . . . . . . . . . . . . . . . . . . . . 108
4.2.3 Four-dimensional case . . . . . . . . . . . . . . . . . . . . . 110
4.3 Poisson geometry of real semisimple Lie algebras . . . . . . . . . . 112
4.4 Nondegeneracy of aff(n) . . . . . . . . . . . . . . . . . . . . . . . . 117
4.5 Some other linearization results . . . . . . . . . . . . . . . . . . . . 122
4.5.1 Equivariant linearization. . . . . . . . . . . . . . . . . . . . 122
4.5.2 Linearization of Poisson–Lie tensors . . . . . . . . . . . . . 122
4.5.3 Poissonstructures with a hyperbolic Rk-action . . . . . . . 124
4.5.4 Transverse Poissonstructures to coadjoint orbits . . . . . . 125
4.5.5 Finite determinacy of Poissonstructures . . . . . . . . . . . 126
5 Multiplicative and Quadratic PoissonStructures
5.1 Multiplicative tensors . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Poisson–Lie groups and r-matrices . . . . . . . . . . . . . . . . . . 132
5.3 The dual and the double of a Poisson-Lie group . . . . . . . . . . . 136
5.4 Actions of Poisson–Lie groups . . . . . . . . . . . . . . . . . . . . . 139
5.4.1 Poissonactions of Poisson–Lie groups . . . . . . . . . . . . 139
5.4.2 Dressing transformations . . . . . . . . . . . . . . . . . . . 142
5.4.3 Momentum maps . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5 r-matrices and quadratic Poissonstructures . . . . . . . . . . . . . 145
Contents ix
5.6 Linear curl vector fields . . . . . . . . . . . . . . . . . . . . . . . . 147
5.7 Quadratization of Poisson structures . . . . . . . . . . . . . . . . . 150
5.8 Nonhomogeneous quadratic Poisson structures . . . . . . . . . . . 156
6 Nambu Structures and Singular Foliations
6.1 Nambu brackets and Nambu tensors . . . . . . . . . . . . . . . . . 159
6.2 Integrable differential forms . . . . . . . . . . . . . . . . . . . . . . 165
6.3 Frobenius with singularities . . . . . . . . . . . . . . . . . . . . . . 168
6.4 Linear Nambu structures . . . . . . . . . . . . . . . . . . . . . . . 171
6.5 Kupka’s phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.6 Linearization of Nambu structures . . . . . . . . . . . . . . . . . . 182
6.6.1 Decomposability of ω . . . . . . . . . . . . . . . . . . . . . 184
6.6.2 Formal linearization of the associated foliation . . . . . . . 185
6.6.3 The analytic case . . . . . . . . . . . . . . . . . . . . . . . . 188
6.6.4 Formal linearization of Λ . . . . . . . . . . . . . . . . . . . 188
6.6.5 The smooth elliptic case . . . . . . . . . . . . . . . . . . . . 190
6.7 Integrable 1-forms with a non-zero linear part . . . . . . . . . . . . 192
6.8 Quadratic integrable 1-forms . . . . . . . . . . . . . . . . . . . . . 197
6.9 Poisson structures in dimension 3 . . . . . . . . . . . . . . . . . . . 199
7 Lie Groupoids
7.1 Some basic notions on groupoids . . . . . . . . . . . . . . . . . . . 203
7.1.1 Definitions and first examples . . . . . . . . . . . . . . . . . 203
7.1.2 Lie groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.1.3 Germs and slices of Lie groupoids. . . . . . . . . . . . . . . 208
7.1.4 Actions of groupoids . . . . . . . . . . . . . . . . . . . . . . 208
7.1.5 Haar systems . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.2 Morita equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.3 Proper Lie groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.3.1 Definition and elementary properties . . . . . . . . . . . . . 213
7.3.2 Source-localtriviality . . . . . . . . . . . . . . . . . . . . . 215
7.3.3 Orbifold groupoids . . . . . . . . . . . . . . . . . . . . . . . 216
7.4 Linearization of Lie groupoids . . . . . . . . . . . . . . . . . . . . . 217
7.4.1 Linearization of Lie group actions . . . . . . . . . . . . . . 217
7.4.2 Local linearization of Lie groupoids. . . . . . . . . . . . . . 218
7.4.3 Slice theorem for Lie groupoids . . . . . . . . . . . . . . . . 222
7.5 Symplectic groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.5.1 Definition and basic properties . . . . . . . . . . . . . . . . 223
7.5.2 Proper symplectic groupoids . . . . . . . . . . . . . . . . . 227
7.5.3 Hamiltonian actions of symplectic groupoids . . . . . . . . 232
7.5.4 Some generalizations . . . . . . . . . . . . . . . . . . . . . . 233
x Contents
8 Lie Algebroids
8.1 Some basic definitions and properties . . . . . . . . . . . . . . . . . 235
8.1.1 Definition and some examples . . . . . . . . . . . . . . . . . 235
8.1.2 The Lie algebroid of a Lie groupoid . . . . . . . . . . . . . 237
8.1.3 Isotropy algebras . . . . . . . . . . . . . . . . . . . . . . . . 238
8.1.4 Characteristic foliation of a Lie algebroid . . . . . . . . . . 239
8.1.5 Lie pseudoalgebras . . . . . . . . . . . . . . . . . . . . . . . 239
8.2 Fiber-wise linear Poisson structures. . . . . . . . . . . . . . . . . . 240
8.3 Lie algebroid morphisms . . . . . . . . . . . . . . . . . . . . . . . . 242
8.4 Lie algebroid actions and connections. . . . . . . . . . . . . . . . . 243
8.5 Splitting theorem and transverse structures . . . . . . . . . . . . . 246
8.6 Cohomology of Lie algebroids . . . . . . . . . . . . . . . . . . . . . 249
8.7 Linearization of Lie algebroids. . . . . . . . . . . . . . . . . . . . . 252
8.8 Integrability of Lie brackets . . . . . . . . . . . . . . . . . . . . . . 257
8.8.1 Reconstruction of groupoids from their algebroids . . . . . 257
8.8.2 Integrability criteria . . . . . . . . . . . . . . . . . . . . . . 259
8.8.3 Integrability of Poissonmanifolds . . . . . . . . . . . . . . . 262
Appendix
A.1 Moser’s path method . . . . . . . . . . . . . . . . . . . . . . . . . . 263
A.2 Division theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
A.3 Reeb stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
A.4 Action-angle variables . . . . . . . . . . . . . . . . . . . . . . . . . 273
A.5 Normal forms of vector fields . . . . . . . . . . . . . . . . . . . . . 276
A.5.1 Poincar´e–Dulacnormal forms . . . . . . . . . . . . . . . . . 276
A.5.2 Birkhoff normal forms . . . . . . . . . . . . . . . . . . . . . 278
A.5.3 Toric characterizationof normal forms . . . . . . . . . . . . 280
A.5.4 Smooth normal forms . . . . . . . . . . . . . . . . . . . . . 282
A.6 Normal forms along a singular curve . . . . . . . . . . . . . . . . . 283
A.7 The neighborhood of a symplectic leaf . . . . . . . . . . . . . . . . 286
A.7.1 Geometric data and coupling tensors . . . . . . . . . . . . . 286
A.7.2 Linear models . . . . . . . . . . . . . . . . . . . . . . . . . . 290
A.8 Dirac structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
A.9 Deformation quantization . . . . . . . . . . . . . . . . . . . . . . . 294
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Preface
“Il ne semblait pas que cette importante th´eorie puˆt encore ˆetre perfectionn´ee,
lorsque les deux g´eom`etres qui ont le plus contribu´e `a la rendre compl`ete, en ont
fait de nouveau le sujet de leurs m´editations...”. By these words, Sim´eon Denis
Poissonannouncedin1809[293]thathehadfoundanimprovementinthetheoryof
Lagrangianmechanics,which wasbeing developedby Joseph-LouisLagrangeand
Pierre-Simon Laplace. In that pioneering paper, Poisson introduced (we slightly
modernize his writing) the notation
(cid:2) (cid:3)
(cid:1)n
∂a ∂b ∂a ∂b
(a,b)= − , (0.1)
∂q ∂p ∂p ∂q
i i i i
i=1
whereaandbaretwofunctionsofthecoordinatesq andthe conjugatequantities
i
p = ∂R for a mechanical system with Lagrangianfunction R. He proved that, if
i ∂q˙i
a and b are first integrals of the system then (a,b) also is. This (a,b) is nowadays
denoted by {a,b} and called the Poisson bracket of a and b. Mathematicians of
the 19thcentury alreadyrecognizedthe importance of this bracket.In particular,
William Hamilton used it extensively to express his equations in an essay in 1835
[168]onwhatwenowcallHamiltoniandynamics.CarlJacobiinhis“Vorlesungen
u¨ber Dynamik” around 1842 (see [185]) showed that the Poisson bracket satisfies
the famous Jacobi identity:
{{a,b},c}+{{b,c},a}+{{c,a},b}=0. (0.2)
This same identity is satisfied by Lie algebras, which are infinitesimal versions of
Liegroups,firststudiedbySophusLieandhiscollaboratorsintheendofthe19th
century [213].
In our modern language, a Poisson structure on a manifold M is a 2-vector
field Π (Poisson tensor) on M, such that the corresponding bracket (Poisson
bracket) on the space of functions on M, defined by
{f,g}:=(cid:1)df ∧dg,Π(cid:3) , (0.3)
satisfiesthe Jacobiidentity. (M,Π) is then calleda Poissonmanifold. This notion
of Poissonmanifolds generalizes both symplectic manifolds and Lie algebras.The