Poisson structures:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2013
|
| Series: | Grundlehren der mathematischen Wissenschaften
347 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIV, 461 S. graph. Darst. |
| ISBN: | 9783642310898 9783642310904 |
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MARC
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Record in the Search Index
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| adam_text | IMAGE 1
CONTENTS
PART I THEORETICAL BACKGROUND 1 POISSON STRUCTURES: BASIC DEFINITIONS 3
1.1 POISSON ALGEBRAS AND THEIR MORPHISMS 4
1.1.1 POISSON ALGEBRAS 4
1.1.2 MORPHISMS O F POISSON ALGEBRAS 5
1.1.3 HAMILTONIAN DERIVATIONS 6
1.2 POISSON VARIETIES 7
1.2.1 AFFINE POISSON VARIETIES AND THEIR MORPHISMS 8
1.2.2 THE POISSON MATRIX 10
1.2.3 THE RANK OF A POISSON STRUCTURE 13
1.3 POISSON MANIFOLDS 15
1.3.1 BIVECTOR FIELDS ON MANIFOLDS 16
1.3.2 POISSON MANIFOLDS AND POISSON MAPS 19
1.3.3 LOCAL STRUCTURE: WEINSTEIN S SPLITTING THEOREM 23
1.3.4 GLOBAL STRUCTURE: THE SYMPLECTIC FOLIATION 26
1.4 POISSON STRUCTURES ON A VECTOR SPACE 32
1.5 EXERCISES 34
1.6 NOTES 35
2 POISSON STRUCTURES: BASIC CONSTRUCTIONS 37
2.1 THE TENSOR PRODUCT O F POISSON ALGEBRAS AND THE PRODUCT O F POISSON
MANIFOLDS 38
2.1.1 THE TENSOR PRODUCT OF POISSON ALGEBRAS 38
2.1.2 THE PRODUCT OF POISSON MANIFOLDS 4 2
2.2 POISSON IDEALS AND POISSON SUBMANIFOLDS 4 6
2.2.1 POISSON IDEALS AND POISSON SUBVARIETIES 4 6
2.2.2 POISSON SUBMANIFOLDS 49
2.3 REAL AND HOLOMORPHIC POISSON STRUCTURES 52
IX
HTTP://D-NB.INFO/1022401890
IMAGE 2
X C O N T E N T S
2.3.1 REAL POISSON STRUCTURES ASSOCIATED TO HOLOMORPHIC
POISSON STRUCTURES 52
2.3.2 HOLOMORPHIC POISSON STRUCTURES ON SMOOTH AFFINE POISSON VARIETIES
55
2.4 OTHER CONSTRUCTIONS 56
2.4.1 FIELD EXTENSION 56
2.4.2 LOCALIZATION 57
2.4.3 GERMIFICATION 59
2.5 EXERCISES 60
2.6 NOTES 61
3 MULTI-DERIVATIONS AND KAHLER FORMS 63
3.1 MULTI-DERIVATIONS AND MULTIVECTOR FIELDS 64
3.1.1 MULTI-DERIVATIONS 64
3.1.2 BASIC OPERATIONS ON MULTI-DERIVATIONS 65
3.1.3 MULTIVECTOR FIELDS 67
3.2 KAHLER FORMS AND DIFFERENTIAL FORMS 68
3.2.1 KAHLER DIFFERENTIALS 69
3.2.2 KAHLER FORMS 7 0
3.2.3 ALGEBRA AND COALGEBRA STRUCTURE 71
3.2.4 THE DE RHAM DIFFERENTIAL AND COHOMOLOGY 72
3.2.5 DIFFERENTIAL FORMS 74
3.3 THE SCHOUTEN BRACKET 75
3.3.1 INTERNAL PRODUCTS 7 6
3.3.2 THE SCHOUTEN BRACKET 79
3.3.3 THE ALGEBRAIC SCHOUTEN BRACKET 84
3.3.4 THE (GENERALIZED) LIE DERIVATIVE 85
3.4 EXERCISES 87
3.5 NOTES 88
4 POISSON (CO)HOMOLOGY 91
4.1 LIE ALGEBRA AND POISSON COHOMOLOGY 92
4.1.1 LIE ALGEBRA COHOMOLOGY 92
4.1.2 POISSON COHOMOLOGY 94
4.2 LIE ALGEBRA AND POISSON HOMOLOGY 97
4.2.1 LIE ALGEBRA HOMOLOGY 98
4.2.2 POISSON HOMOLOGY 99
4.3 OPERATIONS IN HOMOLOGY AND COHOMOLOGY 101
4.4 THE MODULAR CLASS OF A POISSON MANIFOLD 102
4.4.1 THE MODULAR VECTOR FIELD 103
4.4.2 THE MODULAR CLASS 104
4.4.3 THE DIVERGENCE OF THE POISSON BRACKET 107
4.4.4 UNIMODULAR POISSON MANIFOLDS 109
4.5 EXERCISES 110
4.6 NOTES I L L
IMAGE 3
CONTENTS XI
5 REDUCTION 113
5.1 LIE GROUPS AND (THEIR) LIE ALGEBRAS 114
5.1.1 LIE GROUPS AND LIE ALGEBRAS 114
5.1.2 LIE GROUP ACTIONS 116
5.1.3 ADJOINT AND COADJOINT ACTION 119
5.1.4 INVARIANT BILINEAR FORMS AND INVARIANT FUNCTIONS 121
5.2 POISSON REDUCTION 123
5.2.1 ALGEBRAIC POISSON REDUCTION 123
5.2.2 POISSON REDUCTION FOR POISSON MANIFOLDS 127
5.3 POISSON-DIRAC REDUCTION 134
5.3.1 ALGEBRAIC POISSON-DIRAC REDUCTION 134
5.3.2 POISSON-DIRAC REDUCTION FOR POISSON MANIFOLDS 137
5.3.3 THE TRANSVERSE POISSON STRUCTURE 143
5.4 POISSON STRUCTURES AND GROUP ACTIONS 148
5.4.1 POISSON ACTIONS 148
5.4.2 POISSON ACTIONS AND QUOTIENT SPACES 149
5.4.3 FIXED POINT SETS AS POISSON-DIRAC SUBMANIFOLDS 150
5.4.4 REDUCTION WITH RESPECT TO A MOMENTUM MAP 153
5.5 EXERCISES 155
5.6 NOTES 157
PART II EXAMPLES
6 CONSTANT POISSON STRUCTURES, REGULAR AND SYMPLECTIC MANIFOLDS . . . .
161 6.1 CONSTANT POISSON STRUCTURES 162
6.2 REGULAR POISSON MANIFOLDS 165
6.3 SYMPLECTIC MANIFOLDS 166
6.3.1 SYMPLECTIC VECTOR SPACES 167
6.3.2 SYMPLECTIC MANIFOLDS 168
6.3.3 SYMPLECTIC REDUCTION 172
6.3.4 QUOTIENTS BY FINITE GROUPS OF SYMPLECTOMORPHISMS 174 6.3.5 EXAMPLE
1: COTANGENT BUNDLES 175
6.3.6 EXAMPLE 2: KAHLER MANIFOLDS 177
6.4 NOTES 178
7 LINEAR POISSON STRUCTURES AND LIE ALGEBRAS 179
7.1 THE LIE-POISSON STRUCTURE ON G* 180
7.2 THE LIE-POISSON STRUCTURE ON G 182
7.3 PROPERTIES OF THE LIE-POISSON STRUCTURE 185
7.3.1 THE SYMPLECTIC FOLIATION: COADJOINT ORBITS 185
7.3.2 THE COHOMOLOGY O F LIE-POISSON STRUCTURES 190
7.3.3 THE MODULAR CLASS OF A LIE-POISSON STRUCTURE 192
7.4 AFFINE POISSON STRUCTURES 193
7.5 THE LINEARIZATION OF POISSON STRUCTURES 196
7.6 NOTES 203
IMAGE 4
XII C O N T E N T S
8 HIGHER DEGREE POISSON STRUCTURES 205
8.1 POLYNOMIAL AND (WEIGHT) HOMOGENEOUS POISSON STRUCTURES 206 8.1.1
POLYNOMIAL POISSON STRUCTURES 206
8.1.2 HOMOGENEOUS POISSON STRUCTURES 207
8.1.3 WEIGHT HOMOGENEOUS POISSON STRUCTURES 211
8.2 QUADRATIC POISSON STRUCTURES 213
8.2.1 MULTIPLICATIVE POISSON STRUCTURES 215
8.2.2 THE MODULAR VECTOR FIELD O F A QUADRATIC POISSON STRUCTURE . 2 1 8
8.3 NAMBU-POISSON STRUCTURES 222
8.4 TRANSVERSE POISSON STRUCTURES TO ADJOINT ORBITS 227
8.4.1 S-TRIPLES IN SEMI-SIMPLE LIE ALGEBRAS 227
8.4.2 WEIGHTS ASSOCIATED TO AN S-TRIPLE 228
8.4.3 TRANSVERSE POISSON STRUCTURES TO NILPOTENT ORBITS 229
8.5 NOTES 232
9 POISSON STRUCTURES IN DIMENSIONS TWO AND THREE 233
9.1 POISSON STRUCTURES IN DIMENSION TWO 234
9.1.1 GLOBAL POINT O F VIEW 234
9.1.2 LOCAL POINT O F VIEW 235
9.1.3 CLASSIFICATION IN DIMENSION TWO 237
9.2 POISSON STRUCTURES IN DIMENSION THREE 250
9.2.1 POISSON STRUCTURES ON F 3 2 5 0
9.2.2 POISSON MANIFOLDS O F DIMENSION THREE 254
9.2.3 QUADRATIC POISSON STRUCTURES ON F 3 2 5 8
9.2.4 POISSON SURFACES IN C 3 AND DU VAL SINGULARITIES 263
9.3 NOTES 267
10 /?-BRACKETS AND R -BRACKETS 269
10.1 LINEAR /^-BRACKETS 270
10.1.1 /?-MATRICES AND THE YANG-BAXTER EQUATION 270
10.1.2 /^-MATRICES AND LIE ALGEBRA SPLITTINGS 272
10.1.3 LINEAR /^-BRACKETS ON 9* AND ON G 273
10.2 LINEAR R-BRACKETS 275
10.2.1 COBOUNDARY LIE BIALGEBRAS AND R-MATRICES 275
10.2.2 LINEAR R-BRACKETS ON G 278
10.2.3 FROM R-MATRICES TO /?-MATRICES 280
10.3 /^-BRACKETS AND R-BRACKETS O F HIGHER DEGREE 283
10.4 NOTES 289
11 POISSON-LIE GROUPS 291
11.1 MULTIPLICATIVE POISSON STRUCTURES AND POISSON-LIE GROUPS 292 11.1.1
THE CONDITION OF MULTIPLICATIVITY 292
11.1.2 BASIC PROPERTIES O F POISSON-LIE GROUPS 294
11.1.3 POISSON-LIE SUBGROUPS 296
11.1.4 LINEAR MULTIPLICATIVE POISSON STRUCTURES 297
IMAGE 5
CONTENTS XIII
11.1.5 COBOUNDARY POISSON-LIE GROUPS 298
11.1.6 MULTIPLICATIVE POISSON STRUCTURES ON VECTOR SPACES 299 11.2 LIE
BIALGEBRAS 302
11.2.1 LIE BIALGEBRAS 303
11.2.2 LIE SUB-BIALGEBRAS 306
11.2.3 DUALITY FOR LIE BIALGEBRAS 307
11.2.4 COBOUNDARY LIE BIALGEBRAS AND R-MATRICES 308
11.2.5 MANIN TRIPLES 310
11.2.6 LIE BIALGEBRAS AND POISSON STRUCTURES 313
11.3 POISSON-LIE GROUPS AND LIE BIALGEBRAS 314
11.3.1 THE LIE BIALGEBRA OF A POISSON-LIE GROUP 314
11.3.2 COBOUNDARY POISSON-LIE GROUPS AND LIE BIALGEBRAS 316 11.3.3 THE
INTEGRATION O F LIE BIALGEBRAS 318
11.4 DRESSING ACTIONS AND SYMPLECTIC LEAVES 321
11.5 NOTES 325
PART III APPLICATIONS
12 LIOUVILLE INTEGRABLE SYSTEMS 329
12.1 FUNCTIONS IN INVOLUTION 330
12.2 CONSTRUCTIONS O F FUNCTIONS IN INVOLUTION 332
12.2.1 POISSON S THEOREM 332
12.2.2 NOETHER S THEOREM 333
12.2.3 BI-HAMILTONIAN VECTOR FIELDS 333
12.2.4 POISSON MAPS AND THIMM S METHOD 334
12.2.5 LAX EQUATIONS 336
12.2.6 THE ADLER-KOSTANT-SYMES THEOREM 337
12.3 THE LIOUVILLE THEOREM AND THE ACTION-ANGLE THEOREM 341
12.3.1 LIOUVILLE INTEGRABLE SYSTEMS AND LIOUVILLE S THEOREM 341 12.3.2
FOLIATION BY STANDARD LIOUVILLE TORI 343
12.3.3 PERIOD NORMALIZATION 345
12.3.4 THE EXISTENCE O F ACTION-ANGLE COORDINATES 346
12.4 NOTES 350
13 DEFORMATION QUANTIZATION 353
13.1 DEFORMATIONS O F ASSOCIATIVE PRODUCTS 354
13.1.1 FORMAL AND K - I H ORDER DEFORMATIONS 354
13.1.2 HOCHSCHILD COHOMOLOGY 357
13.1.3 DEFORMATIONS AND COHOMOLOGY 359
13.1.4 THE MAURER-CARTAN EQUATION 361
13.1.5 STAR PRODUCTS 362
13.1.6 A STAR PRODUCT FOR CONSTANT POISSON STRUCTURES 363
13.1.7 A STAR PRODUCT FOR SYMPLECTIC MANIFOLDS 365
13.2 DEFORMATIONS OF POISSON STRUCTURES 367
13.2.1 FORMAL AND &-TH ORDER DEFORMATIONS 367
IMAGE 6
X I V
C O N T E N T S
13.2.2 DEFORMATIONS AND COHOMOLOGY 368
13.2.3 THE MAURER-CARTAN EQUATION 371
13.3 DIFFERENTIAL GRADED LIE ALGEBRAS 371
13.3.1 THE SYMMETRIC ALGEBRA O F A GRADED VECTOR SPACE 371
13.3.2 DIFFERENTIAL GRADED LIE ALGEBRAS 372
13.3.3 THE MAURER-CARTAN EQUATION 376
13.3.4 GAUGE EQUIVALENCE 378
13.3.5 PATH EQUIVALENCE 380
13.3.6 APPLICATIONS TO DEFORMATION THEORY 385
13.4 LOO-MORPHISMS OF DIFFERENTIAL GRADED LIE ALGEBRAS 386
13.4.1 N IS WEIL-DEFINED 389
13.4.2 SURJECTIVITY OF 2& 392
13.4.3 INJECTIVITY O F C2 395
13.5 KONTSEVICH S FORMALITY THEOREM AND ITS CONSEQUENCES 397
13.5.1 KONTSEVICH S FORMALITY THEOREM 398
13.5.2 KONTSEVICH S FORMALITY THEOREM FOR W 1 399
13.5.3 A FEW CONSEQUENCES OF KONTSEVICH S FORMALITY THEOREM . . 406 13.6
NOTES 408
A MULTILINEAR ALGEBRA 411
A.L TENSOR ALGEBRA 411
A.2 EXTERIOR AND SYMMETRIC ALGEBRA 415
A.3 ALGEBRAS AND GRADED ALGEBRAS 418
A.4 COALGEBRAS AND GRADED COALGEBRAS 422
A.5 GRADED DERIVATIONS AND CODERIVATIONS 425
B REAL AND COMPLEX DIFFERENTIAL GEOMETRY 427
B. 1 REAL AND COMPLEX MANIFOLDS 427
B.2 THE TANGENT SPACE 429
B.3 VECTOR FIELDS 433
B.4 THE FLOW OF A VECTOR FIELD 435
B.5 THE FROBENIUS THEOREM 436
REFERENCES 439
INDEX 449
LIST O F NOTATIONS 459
|
| any_adam_object | 1 |
| author | Laurent-Gengoux, Camille 1976- Pichereau, Anne 1979- Vanhaecke, Pol 1963- |
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| dewey-full | 512.48 |
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512.48 |
| dewey-search | 512.48 |
| dewey-sort | 3512.48 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV040427282 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:23:50Z |
| institution | BVB |
| isbn | 9783642310898 9783642310904 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025279848 |
| oclc_num | 812384880 |
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| owner_facet | DE-20 DE-384 DE-83 DE-19 DE-BY-UBM DE-824 DE-188 DE-11 DE-29T |
| physical | XXIV, 461 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Laurent-Gengoux, Camille 1976- Verfasser (DE-588)1028283644 aut Poisson structures Camille Laurent-Gengoux ; Anne Pichereau ; Pol Vanhaecke Berlin [u.a.] Springer 2013 XXIV, 461 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 347 Poisson-Mannigfaltigkeit (DE-588)4231918-3 gnd rswk-swf Poisson-Algebra (DE-588)4309234-2 gnd rswk-swf Poisson-Klammer (DE-588)4332536-1 gnd rswk-swf Poisson-Algebra (DE-588)4309234-2 s Poisson-Mannigfaltigkeit (DE-588)4231918-3 s Poisson-Klammer (DE-588)4332536-1 s DE-604 Pichereau, Anne 1979- Verfasser (DE-588)1028284632 aut Vanhaecke, Pol 1963- Verfasser (DE-588)1028305303 aut Grundlehren der mathematischen Wissenschaften 347 (DE-604)BV000000395 347 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025279848&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Laurent-Gengoux, Camille 1976- Pichereau, Anne 1979- Vanhaecke, Pol 1963- Poisson structures Grundlehren der mathematischen Wissenschaften Poisson-Mannigfaltigkeit (DE-588)4231918-3 gnd Poisson-Algebra (DE-588)4309234-2 gnd Poisson-Klammer (DE-588)4332536-1 gnd |
| subject_GND | (DE-588)4231918-3 (DE-588)4309234-2 (DE-588)4332536-1 |
| title | Poisson structures |
| title_auth | Poisson structures |
| title_exact_search | Poisson structures |
| title_full | Poisson structures Camille Laurent-Gengoux ; Anne Pichereau ; Pol Vanhaecke |
| title_fullStr | Poisson structures Camille Laurent-Gengoux ; Anne Pichereau ; Pol Vanhaecke |
| title_full_unstemmed | Poisson structures Camille Laurent-Gengoux ; Anne Pichereau ; Pol Vanhaecke |
| title_short | Poisson structures |
| title_sort | poisson structures |
| topic | Poisson-Mannigfaltigkeit (DE-588)4231918-3 gnd Poisson-Algebra (DE-588)4309234-2 gnd Poisson-Klammer (DE-588)4332536-1 gnd |
| topic_facet | Poisson-Mannigfaltigkeit Poisson-Algebra Poisson-Klammer |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025279848&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT laurentgengouxcamille poissonstructures AT pichereauanne poissonstructures AT vanhaeckepol poissonstructures |