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Zung N.T. — Poisson Structures and their Normal Forms
Zung N.T. — Poisson Structures and their Normal Forms

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Название: Poisson Structures and their Normal Forms

Автор: Zung N.T.

Аннотация:

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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2005

Количество страниц: 304

Добавлена в каталог: 13.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\Gamma$-structure      205
$\omega$-condition      278
A-connection      244
A-path      243
action map      208
Action-angle variables      275
Affine connection      73
Algebroid cohomology      250
Almost Dirac structure      292
Almost Levi factor      85
Almost linearized      85
Anchor      204
Anchor map      13 41 235
Annulator distribution      166
Anti-Poisson map      12
Associated foliation      168
Associated singular foliation      163
Atlas      216
Basic function      240
Bi-degree      274
Bi-Hamiltonian system      34
Bimodule      212
Birkhoff normal form      279
Birkhoff normalization      279
Bisection      207
Bochner’s theorem      217
Book      176
Bundle of Lie algebras      237
Cabbage pile      176
Canonical coordinates      3 14 266
Cartan decomposition      112
Cartan involution      112
Cartan motion algebra      115
Cartan — Chevalley — Eilenberg construction      50
Cartan’s formula      2
Casimir element      89
Casimir functions      40
Characteristic distribution      162 239
Characteristic foliation      239
Characteristic space      13
Chart      216
Chevalley — Eilenberg complex      49
Classical r-matrix      133
Classical Yang — Baxter equation      33 133
Classical Yang — Mills — Higgs setup      26
Coboundary Poisson — Lie structure      132
Codimension      267
Coisotropic      12 13
Commuting singular foliations      164
Compact Levi factor      81
Compatible Poisson structures      33
Complete symplectic realization      262
Complete transversal      211
Completely integrable      168
Configuration space      4
Conservation of energy      4
Contravariant connection      245
Contravariant tensors      9
Convolution product      210
Cotangent algebroid      237
Coupling tensor      288
Courant bracket      292
Covariant derivation      73
Covariant derivative      244
Curl      70
Curl operator      70 73 178
Curl vector field      72
Curvature      245 261 286
CYBE      33 133
Darboux coordinates      3 266
Darboux theorem      3
de Rham complex      42 249
Decomposable      162 165
Deformation cohomology      251
Deformation quantization      295
Degree of resonance      280
density      71
Derivation      3
Diagonal quadratic Poisson structure      148
Differential operator      39
Differential star product      294
Dirac manifold      292
Dirac structure      292
Dirac’s formula      22
Dressing action      143
Dressing transformation      143
Dual pair      37 274
Dual Poisson — Lie algebra      137
Dual Poisson — Lie group      137
Ehresmann connection      286
Eigenvalue      276
Eigenvalues      148 150
Elliptic singular point      184
Equivariant momentum map      25 144
Euler equation      24
Euler top      24
Exact Poisson structure      41
Exact Poisson — Lie structure      132
Fiber-wise linear      241
Fiber-wise linear function      240
Fiber-wise linear p-vector fields      251
Fiber-wise linear Poisson cohomology      251
Fiber-wise linear Poisson structure      240
Fibered product      210
Filtration      54
Finite codimension      63
Finitely determined      126
Finitely generated distribution      19
First integral      4
Fixed point      205
Flat connection      246
Flow      263
Formal completion      78
Formal Levi factor      79
Formality theorem      296
Formally completely integrable      168
Free action      208
Frobenius integrability condition      166
Frobenius Lie algebra      120
Frobenius theorem      18 166
Frobenius with singularity      169
Fundamental groupoid      206
Fundamental identity      159
Gauge groupoid      207
gCYBE      133
Generalized classical Yang — Baxter equation      133
Generalized isomorphism      213
Generalized morphism      213
Geometric data      287
Germ      42
Godbillon — Vey algorithm      171
Graded anti-commutativity      28 32
Graded Jacobi identity      28 32
Graded Leibniz rule      28
Groupoid      203
Groupoid action      208
Groupoid morphism      204
Haar system      209
Hamiltonian action      25 232
Hamiltonian equation      3
Hamiltonian system      3
Hamiltonian vector field      2 3 159
Hochschild — Serre spectral sequence      57
Holonomy      271
Holonomy groupoid      206
Homogeneous foliation      177
Homogeneous Lie algebroid      252
Homogeneous part      45
Homotopy operator      89
Horizontal lifting      245
Horizontal subbundle      286
Identity section      204
Induced symplectic form      14
Infinitesimal automorphism      10 241
Infinitesimally strongly rigid Lie algebra      107
Inner product      30 69
Integrability condition      166
Integrable differential form      166
Integrable distribution      17
Integrable in generalized Liouville sense      273
Integrable Poisson manifold      227
Integral submanifold      17
Interpolation inequality      102
Invariant density      72
Inversion map      204
Involutive distribution      18
Isochore vector field      148
Isolated singularity      63
Isotropic      12
Isotropy algebra      125 239
Isotropy group      205
Iwasawa decomposition      112
Iwasawa Poisson — Lie structure      124 138
Jacobi identity      31
Jacobian determinant      7
k-determinant      267
Kirillov — Kostant — Souriau form      20
Kupka’s phenomenon      179
Lagrangian      12
Leaves      16
Left action      23
Left dressing action      143
Left-invariant vector field      237
Leray spectral sequence      56
Levi decomposition      77 81 255
Levi factor      77
Levi normal form      81 255
Levi — Malcev Theorem      77
Lichnerowicz complex      39
Lie algebra action      23
Lie algebroid      235
Lie algebroid action      244
Lie algebroid morphism      242
Lie bialgebra      137
Lie groupoid      206
Lie groupoid morphism      206
Lie pseudoalgebra      239
Lie — Poisson structure      20
Lie-linear foliation      177
Lie’s third theorem      257
Linear A-module      244
Linear connection      73
Linear holonomy      246
Linear Nambu structure      171
Linear part      45 182 219 253
Linear Poisson structure      8
Linear representation      208 244
Liouville 1-form      4
Liouville form      72
Liouville integrable      273
Liouville torus      273 274
Liouville — Mineur — Arnold theorem      275
Local foliation property      16
Manin triple      138
Marked symplectic realization      35
Marsden — Ratiu reduction      26
Marsden — Weinstein — Meyer reduction      26
Maurer — Cartan equation      141 144
Maurer — Cartan form      144
Maximal invariant distribution      167
Maximal invariant foliation      168
Mayer — Vietoris sequence      44
mCYBE      134
Mineur — Arnold formula      276
Modified classical Yang — Baxter equation      134
Modular class      72
Modular vector field      72
Moduli space      205
Moment map      25 208 244
Momentum map      25 124 144 208 244 273
Morita equivalent      211
Moser’s path method      263
Moyal product      296
Multi-derivation      6
Multi-vector      5
Multiplicative Poisson structure      146
Multiplicative tensor field      129
Multiplicity      63
n-ary Lie algebra      177
Nambu bracket      159
Nambu structure      159
Nambu tensor      160
Nambu-linear foliation      177
Non-equivariant momentum map      25
Noncommutative integrability      273
Nondegenerate      183
Nondegenerate Lie algebra      105
Nondegenerate module      255
Nonresonant      280
Nonresonant Poisson structure      150
Nonresonant quadratic Poisson structure      148
Normal form      45
Normalization      45
Normed vanishing of cohomology      90
Obstructions to formal deformation      40
Orbifold      216
Orbifold groupoid      216
Orbispace      222
Orbit      205
Orbit space      205
Orbit-like foliation      164
Pair groupoid      204
Pairing      5
Parallel transport      245
Pencil of Poisson structures      33
Phase space      4
Poincare — Dulac normal form      277
Poincare — Dulac normalization      277
Poisson action      124 139 141
Poisson algebra      24
Poisson bracket      1
Poisson cohomology      39
Poisson groupoid      234
Poisson harmonic form      75
Poisson homogeneous space      140
Poisson homology      74
Poisson isomorphism      10
Poisson manifold      1
Poisson map      9
Poisson morphism      9
Poisson sigma model      297 298
Poisson structure      1 6
Poisson tensor      6
Poisson theorem      4
Poisson vector field      10
Poisson — Lie algebra      136
Poisson — Lie group      132
Poisson — Lie structure      132
Poisson — Lie tensor      122
Pro-finite Lie algebra      78
Pro-solvable radical      78
Product map      203
Product Poisson structure      10
Proper action      24 213 214
Proper Lie groupoid      214
Proper map      24 213
Proper symplectic groupoid      227
Quasi-homogeneous      46
Quasi-homogeneous part      46
Quasi-homogenization      46
1 2
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