Poisson structures and their normal forms:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2005
|
| Series: | Progress in mathematics
242 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 299 - 316 |
| Physical Description: | XV, 321 S. 24 cm |
| ISBN: | 9783764373344 3764373342 |
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MARC
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| 100 | 1 | |a Dufour, Jean-Paul |e Verfasser |0 (DE-588)130389730 |4 aut | |
| 245 | 1 | 0 | |a Poisson structures and their normal forms |c Jean-Paul Dufour ; Nguyen Tien Zung |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2005 | |
| 300 | |a XV, 321 S. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Progress in mathematics |v 242 | |
| 500 | |a Literaturverz. S. 299 - 316 | ||
| 650 | 7 | |a Geometria diferencial |2 larpcal | |
| 650 | 4 | |a Géométrie différentielle | |
| 650 | 4 | |a Géométrie symplectique | |
| 650 | 4 | |a Lie, Algèbres de | |
| 650 | 4 | |a Poisson, Variétés de | |
| 650 | 4 | |a Systèmes hamiltoniens | |
| 650 | 7 | |a Álgebras de lie |2 larpcal | |
| 650 | 4 | |a Geometry, Differential | |
| 650 | 4 | |a Hamiltonian systems | |
| 650 | 4 | |a Lagrange spaces | |
| 650 | 4 | |a Lie algebras | |
| 650 | 4 | |a Poisson manifolds | |
| 650 | 4 | |a Symplectic geometry | |
| 650 | 0 | 7 | |a Symplektische Geometrie |0 (DE-588)4194232-2 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Poisson-Mannigfaltigkeit |0 (DE-588)4231918-3 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Nguyen, Tien Zung |e Verfasser |0 (DE-588)130389765 |4 aut | |
| 830 | 0 | |a Progress in mathematics |v 242 |w (DE-604)BV000004120 |9 242 | |
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Record in the Search Index
| _version_ | 1804134681567821824 |
|---|---|
| adam_text | 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
Poisson
cohomology
............ 40
2.1.3
Poisson
cohomology versus
de Rham
cohomology
...... 41
2.1.4
Other versions of
Poisson
cohomology
............ 42
2.1.5
Computation of
Poisson
cohomology
............. 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
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
Poisson
germs
................. 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
Poisson
structures
............... 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
Poisson
structures
......... 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
Poisson
structures with a hyperbolic Reaction
....... 124
4.5.4
Transverse
Poisson
structures to coadjoint orbits
...... 125
4.5.5
Finite determinacy of
Poisson
structures
........... 126
5
Multiplicative and Quadratic
Poisson
Structures
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
Poisson
actions of Poisson-Lie groups
............ 139
5.4.2
Dressing transformations
................... 142
5.4.3
Momentum maps
........................ 144
5.5
r-matriccs and quadratic
Poisson
structures
............. 145
Contents
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
Dccomposability 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-local triviality
..................... 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
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
Poisson
manifolds
............... 262
Appendix
A.I 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
Poincarć-Dulac
normal forms
................. 276
A.
5.2
Birkhoff normal forms
..................... 278
A.
5.3
Toric characterization of 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
|
| adam_txt |
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
Poisson
cohomology
. 40
2.1.3
Poisson
cohomology versus
de Rham
cohomology
. 41
2.1.4
Other versions of
Poisson
cohomology
. 42
2.1.5
Computation of
Poisson
cohomology
. 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
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
Poisson
germs
. 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
Poisson
structures
. 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
Poisson
structures
. 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
Poisson
structures with a hyperbolic Reaction
. 124
4.5.4
Transverse
Poisson
structures to coadjoint orbits
. 125
4.5.5
Finite determinacy of
Poisson
structures
. 126
5
Multiplicative and Quadratic
Poisson
Structures
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
Poisson
actions of Poisson-Lie groups
. 139
5.4.2
Dressing transformations
. 142
5.4.3
Momentum maps
. 144
5.5
r-matriccs and quadratic
Poisson
structures
. 145
Contents
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
Dccomposability 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-local triviality
. 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
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
Poisson
manifolds
. 262
Appendix
A.I 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
Poincarć-Dulac
normal forms
. 276
A.
5.2
Birkhoff normal forms
. 278
A.
5.3
Toric characterization of 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 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Dufour, Jean-Paul Nguyen, Tien Zung |
| author_GND | (DE-588)130389730 (DE-588)130389765 |
| author_facet | Dufour, Jean-Paul Nguyen, Tien Zung |
| author_role | aut aut |
| author_sort | Dufour, Jean-Paul |
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| dewey-full | 516.3/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
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| discipline | Mathematik Romanistik |
| discipline_str_mv | Mathematik Romanistik |
| format | Book |
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| id | DE-604.BV021236516 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T13:29:53Z |
| indexdate | 2024-07-09T20:28:29Z |
| institution | BVB |
| isbn | 9783764373344 3764373342 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014279275 |
| oclc_num | 61478855 |
| open_access_boolean | |
| owner | DE-824 DE-384 DE-11 DE-188 DE-20 |
| owner_facet | DE-824 DE-384 DE-11 DE-188 DE-20 |
| physical | XV, 321 S. 24 cm |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Dufour, Jean-Paul Verfasser (DE-588)130389730 aut Poisson structures and their normal forms Jean-Paul Dufour ; Nguyen Tien Zung Basel [u.a.] Birkhäuser 2005 XV, 321 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 242 Literaturverz. S. 299 - 316 Geometria diferencial larpcal Géométrie différentielle Géométrie symplectique Lie, Algèbres de Poisson, Variétés de Systèmes hamiltoniens Álgebras de lie larpcal Geometry, Differential Hamiltonian systems Lagrange spaces Lie algebras Poisson manifolds Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Poisson-Mannigfaltigkeit (DE-588)4231918-3 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 s Poisson-Mannigfaltigkeit (DE-588)4231918-3 s DE-604 Nguyen, Tien Zung Verfasser (DE-588)130389765 aut Progress in mathematics 242 (DE-604)BV000004120 242 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014279275&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dufour, Jean-Paul Nguyen, Tien Zung Poisson structures and their normal forms Progress in mathematics Geometria diferencial larpcal Géométrie différentielle Géométrie symplectique Lie, Algèbres de Poisson, Variétés de Systèmes hamiltoniens Álgebras de lie larpcal Geometry, Differential Hamiltonian systems Lagrange spaces Lie algebras Poisson manifolds Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd Poisson-Mannigfaltigkeit (DE-588)4231918-3 gnd |
| subject_GND | (DE-588)4194232-2 (DE-588)4231918-3 |
| title | Poisson structures and their normal forms |
| title_auth | Poisson structures and their normal forms |
| title_exact_search | Poisson structures and their normal forms |
| title_exact_search_txtP | Poisson structures and their normal forms |
| title_full | Poisson structures and their normal forms Jean-Paul Dufour ; Nguyen Tien Zung |
| title_fullStr | Poisson structures and their normal forms Jean-Paul Dufour ; Nguyen Tien Zung |
| title_full_unstemmed | Poisson structures and their normal forms Jean-Paul Dufour ; Nguyen Tien Zung |
| title_short | Poisson structures and their normal forms |
| title_sort | poisson structures and their normal forms |
| topic | Geometria diferencial larpcal Géométrie différentielle Géométrie symplectique Lie, Algèbres de Poisson, Variétés de Systèmes hamiltoniens Álgebras de lie larpcal Geometry, Differential Hamiltonian systems Lagrange spaces Lie algebras Poisson manifolds Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd Poisson-Mannigfaltigkeit (DE-588)4231918-3 gnd |
| topic_facet | Geometria diferencial Géométrie différentielle Géométrie symplectique Lie, Algèbres de Poisson, Variétés de Systèmes hamiltoniens Álgebras de lie Geometry, Differential Hamiltonian systems Lagrange spaces Lie algebras Poisson manifolds Symplectic geometry Symplektische Geometrie Poisson-Mannigfaltigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014279275&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000004120 |
| work_keys_str_mv | AT dufourjeanpaul poissonstructuresandtheirnormalforms AT nguyentienzung poissonstructuresandtheirnormalforms |