Lie groups and algebraic groups:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1990
|
| Schriftenreihe: | Springer series in Soviet mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Aus dem Russ. übers. |
| Beschreibung: | XVII, 328 S. graph. Darst. |
| ISBN: | 3540506144 0387506144 |
Internformat
MARC
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| 100 | 1 | |a Oniščik, Arkadij L. |d 1933-2019 |e Verfasser |0 (DE-588)112427359 |4 aut | |
| 240 | 1 | 0 | |a Seminar po gruppam Li i algebraičeskim gruppam |
| 245 | 1 | 0 | |a Lie groups and algebraic groups |c A. L. Onishchik ; E. B. Vinberg |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1990 | |
| 300 | |a XVII, 328 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer series in Soviet mathematics | |
| 500 | |a Aus dem Russ. übers. | ||
| 650 | 4 | |a Lie-Gruppe - Affine algebraische Gruppe | |
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Datensatz im Suchindex
| _version_ | 1804118448948641792 |
|---|---|
| adam_text | Table of Contents
Commonly Used Symbols xix
Chapter 1. Lie Groups 1
§ 1. Background 1
1°. Lie Groups 1
2°. Lie Subgroups 2
3°. Homomorphisms, Linear Representations and
Actions of Lie Groups 3
4°. Operations on Linear Representations 5
5°. Orbits and Stabilizers 6
6°. The Image and the Kernel of a Homomorphism 8
7°. Coset Manifolds and Quotient Groups 9
8°. Theorems on Transitive Actions and Epimorphisms 11
9°. Homogeneous Spaces 12
10°. Inverse Image of a Lie Subgroup with Respect
to a Homomorphism 14
11°. Semidirect Product 15
Exercises 16
Hints to Problems 17
§ 2. Tangent Algebra 19
1°. Definition of the Tangent Algebra 19
2°. Tangent Homomorphism 21
3°. The Tangent Algebra of a Stabilizer 22
4°. The Adjoint Representation and the Jacobi Identity 23
5°. Differential Equations for Paths on a Lie Group 25
6°. Uniqueness Theorem for Lie Group Homomorphisms 27
7°. Exponential Map 28
8°. Existence Theorem for Lie Group Homomorphisms 30
9°. Virtual Lie Subgroups 33
10°. Automorphisms and Derivations 35
11°. The Tangent Algebra of a Semidirect Product of Lie Groups ... 36
Exercises 38
Hints to Problems 39
XII Table of Contents
§ 3. Connectedness and Simple Connectedness 42
1°. Connectedness 43
2°. Covering Homomorphisms 44
3°. Simply Connected Covering Lie Groups 45
4°. Exact Homotopy Sequence 46
Exercises 48
Hints to Problems 48
§ 4. The Derived Algebra and the Radical 50
1°. The Commutator Group and the Derived Algebra 50
2°. Malcev Closures 51
3°. Existence of Virtual Lie Subgroups 52
4°. Solvable Lie Groups 53
5°. Lie s Theorem 54
6°. The Radical. Semisimple Lie Groups 55
7°. Complexification 56
Exercises 56
Hints to Problems 57
Chapter 2. Algebraic Varieties 59
§ 1. Affine Algebraic Varieties 59
1°. Embedded Affine Varieties 59
2°. Morphisms 61
3°. Zariski Topology 62
4°. The Direct Product 64
5°. Homomorphism Extension Theorems 65
6°. The Image of a Dominant Morphism 66
7°. Hilbert s Nullstellensatz 67
8°. Rational Functions 68
9°. Rational Maps 69
10°. Factorization of a Morphism 70
Exercises 71
Hints to Problems 72
§ 2. Projective and Quasiprojective Varieties 74
1°. Graded Algebras 74
2°. Embedded Projective Algebraic Varieties 75
3°. Sheaves of Functions 77
4°. Sheaves of Algebras of Rational Functions 79
5°. Quasiprojective Varieties 79
6°. The Direct Product 80
7°. Flag Varieties 83
Exercises 84
Hints to Problems 85
Table of Contents XIII
§ 3. Dimension and Analytic Properties of Algebraic Varieties 87
1 °. Definition of the Dimension and its Main Properties 87
2°. Derivations of the Algebra of Functions 88
3°. Simple Points 89
4°. The Analytic Structure of Complex and
Real Algebraic Varieties 90
5°. Realification of Complex Algebraic Varieties 91
6°. Forms of Vector Spaces and Algebras 92
7°. Real Forms of Complex Algebraic Varieties 93
Exercises 94
Hints to Problems 96
Chapter 3. Algebraic Groups 98
§ 1. Background 98
1°. Main Definitions 98
2°. Complex and Real Algebraic Groups 100
3°. Semidirect Products 102
4°. Certain Theorems on Subgroups and Homomorphisms
of Algebraic Groups 102
5°. Actions of Algebraic Groups 103
6°. Existence of a Faithful Linear Representation 104
7°. The Coset Variety and the Quotient Group 106
Exercises 108
Hints to Problems 108
§ 2. Commutative and Solvable Algebraic Groups 110
1°. The Jordan Decomposition of a Linear Operator 110
2°. Commutative Unipotent Algebraic Linear Groups Ill
3°. Algebraic Tori and Quasitori 113
4°. The Jordan Decomposition in an Algebraic Group 115
5°. The Structure of Commutative Algebraic Groups 116
6°. Borel s Theorem 116
7°. The Splitting of a Solvable Algebraic Group 117
8°. Semisimple Elements of a Solvable Algebraic Group 118
9°. Borel Subgroups 118
Exercises 119
Hints of Problems 120
§3. The Tangent Algebra 122
1°. Connectedness of Irreducible Complex Algebraic Groups 122
2°. The Rational Structure on the Tangent Algebra of a Torus 123
3°. Algebraic Subalgebras 123
4°. The Algebraic Structure on Certain Complex Lie Groups 124
5°. Engel s Theorem 125
XIV Table of Contents
6°. Unipotent Algebraic Linear Groups 126
7°. The Jordan Decomposition in the Tangent Algebra
of an Algebraic Group 126
8°. The Tangent Algebra of a Real Algebraic Group 127
9°. The Union of Borel Subgroups and the Centralizers of Tori 127
Exercises 128
Hints to Problems 129
§4. Compact Linear Groups 130
1°. A Fixed Point Theorem 130
2°. Complete Reducibility 131
3°. Separating Orbits with the Help of Invariants 132
4°. Algebraicity 133
Exercises 134
Hints to Problems 134
Chapter 4. Complex Semisimple Lie Groups 136
§ 1. Preliminaries 136
1°. Invariant Scalar Products 136
2°. Algebraicity 138
3°. Normal Subgroups 139
4°. Weight and Root Decompositions 140
5°. Root Decompositions and Root Systems
of Classical Lie Algebras 144
6°. Three Dimensional Subalgebras 147
Exercises 149
Hints to Problems 151
§2. Root Systems 153
1°. Principal Definitions and Examples 153
2°. Weyl Chambers and Simple Roots 156
3°. Borel Subgroups and Maximal Tori 160
4°. Weyl Group 162
5°. Dynkin Diagrams 164
6°. Cartan Matrices 167
7°. Classification 169
8°. Root and Weight Lattices 172
Exercises 174
Hints to Problems 177
§ 3. Existence and Uniqueness Theorems 181
1°. Free Lie Algebras, Generators and Defining Relations 182
2°. Uniqueness Theorems 183
Table of Contents XV
3°. Existence Theorems 186
4°. The Linearity of a Connected Complex Semisimple Lie Group .. 190
5°. The Center and the Fundamental Group 191
6°. Classification of Connected Semisimple Lie Groups 192
7°. Classification of Irreducible Representations 193
Exercises 194
Hints to Problems 199
§ 4. Automorphisms 202
1°. The Group of Outer Automorphisms 202
2°. Semisimple Automorphisms 203
3°. Characters and Automorphisms of Quasi Tori 205
4°. Affine Root Decomposition 207
5°. Affine Weyl Group 209
6°. Affine Roots of a Simple Lie Algebra 211
7°. Classification of Unitary Automorphisms of Simple Lie Algebras . 212
8°. Fixed Points of Semisimple Automorphisms of a
Simply Connected Group 214
Exercises 216
Hints to Problems 217
Chapter 5. Real Semisimple Lie Groups 221
§ 1. Real Forms of Complex Semisimple Lie Groups and Algebras 221
1°. Real Structures and Real Forms 221
2°. Real Forms of Classical Lie Groups and Algebras 225
3°. The Compact Real Form 227
4°. Real Forms and Involutive Automorphisms 229
5°. Involutive Automorphisms of Complex Simple Lie Algebras .... 232
6°. Classification of Real Simple Lie Algebras 233
Exercises 234
Hints to Problems 236
§2. Compact Lie Groups and Reductive Algebraic Groups 238
1°. Polar Decomposition 238
2°. Lie Groups with Compact Tangent Algebras 241
3°. Compact Real Forms of Reductive Algebraic Groups 244
4°. Linearity of Compact Lie Groups 245
5°. Correspondence Between Compact Lie Groups and
Reductive Algebraic Groups 246
6°. Complete Reducibility of Linear Representations 247
7°. Maximal Tori in Compact Lie Groups 249
Exercises 250
Hints to Problems 252
XVI Table of Contents
§ 3. Cartan Decomposition 254
1°. Cartan Decomposition of a Semisimple Lie Algebra 254
2°. Cartan Decomposition of a Semisimple Lie Group 256
3°. Conjugacy of Maximal Compact Subgroups 258
4°. Canonically Embedded Subalgebras 261
5°. Classification of Connected Semisimple Lie Groups 262
6°. Linearizer 264
Exercises 265
Hints to Problems 266
§ 4. Real Root Decomposition 268
1°. Maximal R Diagonalizable Subalgebras 268
2°. Real Root Systems 270
3°. Satake Diagram 272
4°. Split Semisimple Lie Algebras 274
5°. Iwasawa Decomposition 275
Exercises 277
Hints to Problems 279
Chapter 6. Levi Decomposition 282
1°. Levi s Theorem 282
2°. Existence of a Lie Group with the Given Tangent Algebra 283
3°. Malcev s Theorem 284
4°. Algebraic Levi Decomposition 285
Exercises 287
Hints to Problems 288
Reference Chapter 289
§ 1. Useful Formulae 289
1°. Weyl Groups and Exponents 289
2°. Linear Representations of Complex Semisimple Lie Algebras.... 290
3°. Linear Representations of Real Semisimple Lie Algebras 290
§2. Tables 292
Table 1. Weights and Roots 292
Table 2. Matrices Inverse to Cartan Matrices 295
Table 3. Centers, Outer Automorphisms and Bilinear Invariants.... 297
Table 4. Exponents 299
Table 5. Decomposition of Tensor Products and Dimensions
of Certain Representations 299
Table 6. Affine Dynkin Diagrams 305
Table 7. Involutive Automorphisms of Complex Simple
Lie Algebras 307
Table 8. Matrix Realizations of Classical Real Lie Algebras 309
Table of Contents XVII
Table 9. Real Simple Lie Algebras 312
Table 10. Centers and Linearizers of Simply Connected Real
Simple Lie Groups 318
Bibliography 322
Index 325
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T16:10:28Z |
| institution | BVB |
| isbn | 3540506144 0387506144 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002645507 |
| oclc_num | 246768011 |
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| physical | XVII, 328 S. graph. Darst. |
| publishDate | 1990 |
| publishDateSearch | 1990 |
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| publisher | Springer |
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| series2 | Springer series in Soviet mathematics |
| spelling | Oniščik, Arkadij L. 1933-2019 Verfasser (DE-588)112427359 aut Seminar po gruppam Li i algebraičeskim gruppam Lie groups and algebraic groups A. L. Onishchik ; E. B. Vinberg Berlin [u.a.] Springer 1990 XVII, 328 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in Soviet mathematics Aus dem Russ. übers. Lie-Gruppe - Affine algebraische Gruppe Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Algebraische Mannigfaltigkeit (DE-588)4128509-8 gnd rswk-swf Affine algebraische Gruppe (DE-588)4141561-9 gnd rswk-swf Algebraische Gruppe (DE-588)4001164-1 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 s Affine algebraische Gruppe (DE-588)4141561-9 s DE-604 Algebraische Mannigfaltigkeit (DE-588)4128509-8 s Algebraische Gruppe (DE-588)4001164-1 s Vinberg, Ėrnest B. 1937-2020 Verfasser (DE-588)115668063 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002645507&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Oniščik, Arkadij L. 1933-2019 Vinberg, Ėrnest B. 1937-2020 Lie groups and algebraic groups Lie-Gruppe - Affine algebraische Gruppe Lie-Gruppe (DE-588)4035695-4 gnd Algebraische Mannigfaltigkeit (DE-588)4128509-8 gnd Affine algebraische Gruppe (DE-588)4141561-9 gnd Algebraische Gruppe (DE-588)4001164-1 gnd |
| subject_GND | (DE-588)4035695-4 (DE-588)4128509-8 (DE-588)4141561-9 (DE-588)4001164-1 |
| title | Lie groups and algebraic groups |
| title_alt | Seminar po gruppam Li i algebraičeskim gruppam |
| title_auth | Lie groups and algebraic groups |
| title_exact_search | Lie groups and algebraic groups |
| title_full | Lie groups and algebraic groups A. L. Onishchik ; E. B. Vinberg |
| title_fullStr | Lie groups and algebraic groups A. L. Onishchik ; E. B. Vinberg |
| title_full_unstemmed | Lie groups and algebraic groups A. L. Onishchik ; E. B. Vinberg |
| title_short | Lie groups and algebraic groups |
| title_sort | lie groups and algebraic groups |
| topic | Lie-Gruppe - Affine algebraische Gruppe Lie-Gruppe (DE-588)4035695-4 gnd Algebraische Mannigfaltigkeit (DE-588)4128509-8 gnd Affine algebraische Gruppe (DE-588)4141561-9 gnd Algebraische Gruppe (DE-588)4001164-1 gnd |
| topic_facet | Lie-Gruppe - Affine algebraische Gruppe Lie-Gruppe Algebraische Mannigfaltigkeit Affine algebraische Gruppe Algebraische Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002645507&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT oniscikarkadijl seminarpogruppamliialgebraiceskimgruppam AT vinbergernestb seminarpogruppamliialgebraiceskimgruppam AT oniscikarkadijl liegroupsandalgebraicgroups AT vinbergernestb liegroupsandalgebraicgroups |