Topology of transitive transformation groups:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | Undetermined |
| Veröffentlicht: |
Leipzig u.a.
Barth
1994
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 285 - 294 |
| Beschreibung: | XV, 300 S. |
| ISBN: | 3335003551 |
Internformat
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| 245 | 1 | 0 | |a Topology of transitive transformation groups |c by Arkadi L. Onishchik |
| 264 | 1 | |a Leipzig u.a. |b Barth |c 1994 | |
| 300 | |a XV, 300 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 500 | |a Literaturverz. S. 285 - 294 | ||
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Datensatz im Suchindex
| _version_ | 1804122714434174976 |
|---|---|
| adam_text | Table of contents
List of notations XIV
Chapter 1. Lie groups and homogeneous spaces 1
§1. Lie groups and their actions on manifolds 2
1. Basic conventions about manifolds 2
2. Lie groups and their homomorphisms 5
3. Actions of Lie groups 6
4. Lie subgroups and coset manifolds 9
5. The structure of an orbit 12
6. General morphisms of actions 13
7. Coverings of actions 14
§2. Infinitesimal study of Lie groups and their actions 16
1. Flows and one parameter subgroups 16
2. The tangent algebra of a Lie group 18
3. Lie subgroups and Lie subalgebras 20
4. Infinitesimal study of actions 23
5. Lie transformation groups 25
6. Compact Lie groups and Lie algebras 27
7. Complexification 29
8. Examples of compact and reductive complex Lie groups 30
§3. Compact Lie groups, their subgroups and homomorphisms 37
1. Maximal tori 37
2. Characters, weights and roots 39
3. Weyl chambers, simple roots, Weyl group 41
4. Dynkin diagrams and the classification of compact Lie algebras 43
5. The classification of connected compact Lie groups 44
6. Automorphisms 48
7. Linear representations 49
8. The character and the dimension of a representation 51
9. On the classification of connected Lie subgroups of simple compact Lie groups 55
10. Indices of homomorphisms and subgroups 58
11. Subgroups of maximal rank 61
12. Centralizers of tori 64
13. Parabolic subgroups 65
§4. Homogeneous spaces 67
1. The group model 67
2. The main problems 70
X Table of contents
3. The isotropy representation 72
4. The group of automorphisms 73
5. The group of autosimilitudes 74
6. Invariant tensor fields 76
7. Averaging operators 78
8. Invariant Riemannian structures 80
9. Symmetric homogeneous spaces 82
10. Some homotopy properties of homogeneous spaces 83
§5. Factorizations of Lie groups 85
1. Enlargements of transitive actions and factorizations of groups 85
2. Factorizations of Lie groups and Lie algebras 87
3. Some examples of inclusions between transitive actions 90
4. Factorizations of compact Lie groups and Lie algebras 94
Notes 95
Chapter 2. Graded algebras and cohomology 97
§6. Graded algebras 97
1. Preliminaries about graded vector spaces 97
2. Preliminaries about graded algebras 101
3. Generators of a canonical graded algebra 104
4. Derivations 106
5. A uniqueness theorem for the tensor product decomposition 108
6. Graded coalgebras 110
7. Graded bialgebras 113
8. Primitive elements 115
9. Hopf s and Samelson s theorems 117
10. Filtered vector spaces and algebras 119
§7. Complexes and differential graded algebras 121
1. Complexes 121
2. Differential graded algebras 122
3. Bicomplexes 124
4. Actions of graded Lie algebras 126
5. Homotopies 128
6. Acyclic and contractible differential graded algebras 129
7. Minimal differential graded algebras 131
8. The minimal model of a direct product 133
§8. Cartan algebras 134
1. Koszul algebras and Cartan algebras 134
2. Regular sequences 136
3. The Koszul formula 137
4. The minimal model of a Cartan algebra 139
Table of contents XI
5. The classification of Cartan algebras 141
6. Deficiences of a polynomial ideal 143
7. Formal Cartan algebras 144
8. A class of indecomposable minimal Cartan algebras 146
Notes 146
Chapter 3. Real topology of compact Lie groups and their homo¬
geneous spaces 148
§9. Invariant exterior forms 149
1. Preliminaries 149
2. The main theorem 150
3. Right invariant exterior forms on a Lie group 151
4. The chain complex of a Lie algebra 153
5. Bi invariant forms 156
6. Invariant forms on a locally direct product 158
7. The cohomology bialgebra of a compact Lie group 160
8. The tangent algebra interpretation 163
9. The description of primitive elements 164
10. Invariant forms on homogeneous spaces 166
11. The cohomology of symmetric homogeneous spaces 167
§10. Weil algebras 168
1. The construction of the Weil algebra 168
2. The invariants 171
3. The vanishing of the cohomology 171
4. The transgression 172
5. The primitive elements and the transgression 173
6. The structure of symmetric invariants 175
7. The inverse of the transgression 176
8. The action of a homomorphism 178
9. The Weil algebra of a direct product 179
10. An explicit expression for the transgression 179
§11. Symmetric invariants 183
1. The reduction to invariants of the Weyl group 183
2. Computations for classical groups 186
3. A survey of fundamental properties of invariants of the Weyl group 189
4. On polynomial ideals generated by invariants 191
5. Simple subgroups with a big Coxeter number 193
6. Computations for exceptional groups 194
7. The homomorphism associated with a linear representation 196
§12. Cartan s theorem 198
1. A generalization of the Weil algebra 198
XII Table of contents
2. The Cartan algebra 201
3. Cartan s theorem 202
4. The minimal model and the ranks of the homotopy groups 205
5. The exterior grading 208
6. The deficiencies, the Samelson subalgebra and the formality 210
7. The homomorphism associated with the orbit mapping 212
8. The case when the stabilizer is not necessarily connected 213
§13. Some special cases and examples 214
1. Some sufficient formality conditions 214
2. Hopf homogeneous manifolds 216
3. The Euler characteristic 217
4. The homogeneous spaces defined by characters of a maximal torus 218
Notes 222
Chapter 4. Inclusions between transitive transformation groups 224
§14. Factorizations of compact Lie groups 224
1. Topological properties of factorizations 224
2. Factorizations of simple compact Lie groups 226
3. Factorizations of arbitrary compact Lie groups 229
4. Compact enlargements of transitive actions of simple groups 231
5. The ordered set of transitive actions 234
6. Factorizations with a discrete intersection 235
§15. Compact complex homogeneous spaces 238
1. Flag manifolds 238
2. Projective homogeneous spaces 240
3. Tits fibering 242
4. The connected automorphism group of a flag manifold 243
5. The group of biholomorphic transformations of a flag manifold 245
§16. The group of isometries of Riemannian homogeneous spaces 246
1. The simplest consequences of the classification of enlargements 246
2. The group of isometries of the natural Riemannian structure 248
3. Auxiliary lemmas 248
4. Proof of Theorem 3 250
Notes 253
Chapter 5. On the classification of transitive actions 254
§17. Some general properties of transitive actions 255
1. An estimate for the length of a transitive group 255
2. The topological meaning of the Dynkin index 255
3. Homogeneous spaces of simple compact Lie groups 258
Table of contents XIII
4. The splitting of transitive actions on highly connected manifolds 259
5. Some remarks concerning the decomposability 261
§18. Homogeneous spaces of rank 1 or 2 263
1. Homogeneous spaces of rank 1 263
2. The list of all homogeneous spaces of rank 1 264
3. The classification of homogeneous spaces of rank 1 265
4. The list of homogeneous spaces of rank 2: the case of a simple group 267
5. The list of homogeneous spaces of rank 2: the case of a group of length 2 269
6. Transitive actions on a product of two spheres 273
7. Some examples 275
§19. Homogeneous spaces of positive Euler characteristic 276
1. Derivations of the cohomology algebra 276
2. The canonical decomposition 279
3. Transitive actions on the complex and the quaternion manifolds of flags 280
4. Transitive actions on the coset manifold modulo the maximal torus 281
5. The classification of homogeneous spaces of positive Euler characteristic 282
Notes 283
Bibliography 285
Index 295
|
| any_adam_object | 1 |
| author | Oniščik, Arkadij L. 1933-2019 |
| author_GND | (DE-588)112427359 |
| author_facet | Oniščik, Arkadij L. 1933-2019 |
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| author_variant | a l o al alo |
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| discipline | Mathematik |
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| id | DE-604.BV008320054 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:18:16Z |
| institution | BVB |
| isbn | 3335003551 |
| language | Undetermined |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005497153 |
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| physical | XV, 300 S. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
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| publisher | Barth |
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| spelling | Oniščik, Arkadij L. 1933-2019 Verfasser (DE-588)112427359 aut Topology of transitive transformation groups by Arkadi L. Onishchik Leipzig u.a. Barth 1994 XV, 300 S. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 285 - 294 Transformationsgruppe (DE-588)4127386-2 gnd rswk-swf Kompakte Lie-Gruppe (DE-588)4164846-8 gnd rswk-swf Homogener Raum (DE-588)4025787-3 gnd rswk-swf Kompakte Lie-Gruppe (DE-588)4164846-8 s Homogener Raum (DE-588)4025787-3 s DE-604 Transformationsgruppe (DE-588)4127386-2 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005497153&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Oniščik, Arkadij L. 1933-2019 Topology of transitive transformation groups Transformationsgruppe (DE-588)4127386-2 gnd Kompakte Lie-Gruppe (DE-588)4164846-8 gnd Homogener Raum (DE-588)4025787-3 gnd |
| subject_GND | (DE-588)4127386-2 (DE-588)4164846-8 (DE-588)4025787-3 |
| title | Topology of transitive transformation groups |
| title_auth | Topology of transitive transformation groups |
| title_exact_search | Topology of transitive transformation groups |
| title_full | Topology of transitive transformation groups by Arkadi L. Onishchik |
| title_fullStr | Topology of transitive transformation groups by Arkadi L. Onishchik |
| title_full_unstemmed | Topology of transitive transformation groups by Arkadi L. Onishchik |
| title_short | Topology of transitive transformation groups |
| title_sort | topology of transitive transformation groups |
| topic | Transformationsgruppe (DE-588)4127386-2 gnd Kompakte Lie-Gruppe (DE-588)4164846-8 gnd Homogener Raum (DE-588)4025787-3 gnd |
| topic_facet | Transformationsgruppe Kompakte Lie-Gruppe Homogener Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005497153&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT oniscikarkadijl topologyoftransitivetransformationgroups |