The classical groups and K-theory:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1989
|
| Series: | Grundlehren der mathematischen Wissenschaften
291 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 576 S. |
| ISBN: | 3540177582 0387177582 |
Staff View
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| adam_text | Table of Contents
Foreword (by J. Dieudonne) vii
Acknowledgements ix
Introduction 1
Notation and Conventions 3
Chapter 1. General Linear Groups, Steinberg Groups, and K Groups . 5
1.1. Rings, Modules, and Groups 5
1.1A. Rings and Modules 5
LIB. Miscellaneous Group Theory 12
1.2. Linear Groups and Linear Transformations 18
1.2A. The General Linear Group GL(M) and Related Groups . . 18
1.2B. Residual and Fixed Modules 20
1.2C. Elementary Transvections and the Group En(R) 23
1.2D. Cartesian Squares 30
1.2E. The Linear Congruence Groups 33
1.3. The Stable Linear Groups and Kt 37
1.3A. Commutators of Linear Groups 37
1.3B. The Groups GL(K) and E{R), and K, 38
1.3C. The Normal Subgroups of GL(R) 43
1.4. The Linear Steinberg Groups 45
1.4A. The Groups Stn(R) and St(K) 45
1.4B. Comparing K2,n(R) and CenStn(«) 47
1.4C. Central Extensions of Groups 49
1.4D. St(R) as Universal Central Extension of E{R) 52
1.4E. The Groups Wn(R) and Hn(R) 55
1.4F. The General Steinberg Group 60
1.5. The K2 Groups 61
1.5A. Symbols in K2,n(R) 61
1.5B. A Ki K;, Exact Sequence 65
Chapter 2. Linear Groups over Division Rings 68
2.1. Basic Properties of the Linear Groups 68
xii Table of Contents
2.1 A. One Dimensional Transformations 69
2.IB. Generation Theorems for the Linear Groups 71
2.1C. Orders of the Finite Linear Groups 74
2.2. The Groups En{V) and Shn(V) 75
2.2A. The Dieudonne Determinant 75
2.2B. Iwasawa s Simplicity Criterion 78
2.2C. The Simplicity of the Group En(F)/CenEn(F) 79
2.2D*. Central Simple Algebras and the Norm One Group SLn(K) 81
2.2E*. Is SLn(V) = En(Vf. 85
2.3. Connections with K Theory 88
2.3A. A Bruhat Decomposition and Presentations of the
Linear Groups 88
2.3B*. The Theorems of Matsumoto and Merkurjev Suslin .... 92
Chapter 3. Isomorphism Theory for the Linear Groups 96
3.1. Basic Concepts and Facts 98
3.1A. The Standard Isomorphisms 98
3.1B. Rings with Division Rings of Quotients 101
3.1C. The Fundamental Theorem of Projective Geometry .... 104
3.2. Full Groups and Their Isomorphisms 106
3.2A. Full Groups 106
3.2B. More Properties of Linear Transformations 110
3.2C. Action of an Isomorphism on Projective Transvections . . 115
3.2D. The Isomorphism Theorems 119
3.3*. Results over More General Rings 127
3.3A*. Morita Theory and Isomorphisms of Matrix Rings 127
3.3B*. A Return to Domains 130
3.3C*. Description of Theorems and Proofs over More
General Rings 134
Chapter 4. Linear Groups over General Classes of Rings 139
4.1. The Stable Range Condition 141
4.1 A. Big Modules and the Stable Range Condition 141
4.1B. Examples of Rings with Stable Range Condition 143
4.2. The Normal Subgroup Structure of the Linear Groups 147
4.2A. Generalized Matrix Decompositions 147
4.2B. Linear Groups of Big Modules 150
4.2C. Commutators of Linear Groups 153
4.2D. Classification of Normal Subgroups 155
4.2E*. Stability for Kt and K2 160
4.3*. The Congruence Subgroup, Generation, and Presentation Problems 164
4.3A*. The Congruence Subgroup Problem 165
Table of Contents xiii
4.3B*. Generation by Elementary Matrices and Finite Generation 172
4.3C*. Presentations of the Linear Groups 177
Chapter 5. Unitary Groups, Unitary Steinberg Groups, and
Unitary K Groups 183
5.1. Sesquilinear, Hermitian, and Quadratic Forms 184
5.1A. Sesquilinear Forms 184
5.1B. Hermitian Forms 188
5.1C. Form Rings and Generalized Quadratic Forms 190
5.1D. Quadratic Modules over Form Rings 195
5.2. Unitary Groups and Unitary Transformations 200
5.2A. The Unitary Group of a Quadratic Module 201
5.2B. Special Cases of Unitary Groups and the Traditional
Classical Groups 204
5.2C. Unitary Transformations 212
5.2D. Ideals in Form Rings and Unitary Congruence Groups . . 215
5.3. The Hyperbolic Unitary Groups 221
5.3A. The Groups U2n(R,A) and EU2n(R,A) 221
5.3B. Basic Properties of the Group EU2n(R, A) 228
5.3C. The Homomorphisms T_, T + , H, and F 235
5.3D. The Congruence Groups U2n(a,T) and EU2n(a,T) 238
5.4. The Stable Unitary Groups and KUX 241
5.4A. Commutators of Unitary Groups 241
5.4B. The Stable Unitary Groups U(R,A) and EU{R,A) 243
5.4C. Unitary Kj 248
5.4D. The Normal Subgroups of U{R,A) 250
5.5. The Unitary Steinberg Groups 256
5.5A. The Groups StU2n(R,A) and StU(R,/l) 256
5.5B. The Hyperbolic and Forgetful Maps 259
5.5C. Comparing KU2,2n(R, A) and Cen StU2n(R,/i) 261
5.5D. En(R) Homomorphisms onto An and AJn 263
5.5E. StU(K, A) as Universal Central Extension of EU(#, A). . . 269
5.5F*. The Groups WU2n(R,A) and UU2n(R, A) 273
5.6*. The KU2 Groups 277
5.6A*. Symbols in KU2j2n(K,^l) 277
5.6B*. Grothendieck Groups, Witt Groups, and L Groups .... 281
5.6C*. Sharpe s Version of the Unitary Steinberg Group 286
5.6D*. The Exact Sequence of Sharpe 289
Chapter 6. Unitary Groups over Division Rings 292
6.1. Forms over Division Rings 294
6.1 A. Form Parameters in Division Rings 294
xiv Table of Contents
6.1B. ./ Forms on Vector Spaces 295
6.1C. Quadratic Spaces 300
6.ID. Quadratic Spaces over Finite Form Rings 302
6.2. Basic Properties of the Unitary Groups 307
6.2A. Residual Spaces of Unitary Transformations 308
6.2B. A Canonical J Form on the Residual Space 311
6.2C. Witt s Theorems and the Witt Index 314
6.2D. Generation Theorems for the Unitary Groups 317
6.2E. The Finite Unitary Groups 322
6.3. The Group EUn(F) for Isotropic V 325
6.3A. Isotropic Transvections in EUn(K) 326
6.3B. The Equality EUB(7) = EUj(F) for Hyperbolic V 328
6.3C. The Centralizer of EUn(F) 329
6.3D. The Action of EUn(F) on Isotropic Lines 331
6.3E. The Simplicity of the Group EUn(K)/Cen EUB(K) 333
6.4. The Groups Un+(F), Un(F), and SUB(F) 338
6.4A. The Spinor Norm 0 339
6.4B. The Group Un+(F) 344
6.4C. The Spinorial Kernel J n(V) 349
6.4D. Applications to EUn(F) 354
6.4E*. The Unitary Norm One Group SUn(F) 359
6.4F*. A Refinement of the Spinor Norm 362
6.4G*. Unitary Groups over Special Fields 364
6.5*. Connections with Unitary K Theory 369
6.5A*. The Groups K JU2n(R,A) 369
6.5B*. A Bruhat Decomposition and Presentations of the
Hyperbolic Unitary Groups 377
6.5C*. KU2 2n(R,A) over Fields and the Exact Diagram
of Merkurjev Suslin Sharpe 377
Chapter 7. Clifford Algebras and Orthogonal Groups over
Commutative Rings 381
7.1. The Clifford Algebra of a Quadratic Module 382
7.1 A. Definition, Existence, and Basic Properties 382
7.1B. Gradings and Tensor Products 387
7.1C. The Clifford Algebra of a Free Quadratic Module 393
7.ID. The Generalized Quaternion Algebra 398
7.IE. Centers and Graded Centers 401
7.2. Clifford, Spin, and Related Orthogonal Groups 405
7.2A. The Groups CL(M), CL + (M), and Spin(M) 406
7.2B. The Groups Epin£(M), KSpini 2n(K), and KSpin2,2n(K)
in the Hyperbolic Case 410
7.2C*. Bass Theory of the Spinor Norm 417
Table of Contents xv
7.3. Isomorphisms Between Classical Groups of Small Rank 425
7.3A. The Rank 3 Situation 427
7.3B. The Rank 4 Situation 430
7.3C*. The Situations of Rank 5, 6, and 8 435
Chapter 8. Isomorphism Theory for the Unitary Groups 441
8.1. Basic Properties of Quadratic Spaces 443
8.1 A. Some Elementary Concepts and Facts 443
8.IB. The Fundamental Theorem of Projective Geometry .... 446
8.1C. The Geometry of Totally Isotropic Subspaces 452
8.2. Full Orthogonal Groups and Their Isomorphisms 457
8.2A. Full Orthogonal Groups 458
8.2B. Elementary Abelian p Groups in Orthogonal Groups . . . 462
8.2C. More Properties of Eichler Transformations 465
8.2D. Centralizers and Double Centralizers 472
8.2E. Action of an Isomorphism on Projective Eichler
Transformations 476
8.2F. The Isomorphisms of Full Groups in Dimensions Not 8 . . 482
8.2G*. Cayley Algebras and the Isomorphisms of Full Groups in
Dimension 8 491
8.2H*. Isomorphism Theory for Saturated Orthogonal Groups . . 496
8.3*. Non Orthogonal Full Groups and Their Isomorphisms 500
8.3A*. The Isomorphisms of Non Orthogonal Full Unitary Groups 501
8.3B*. Non Existence of Isomorphisms Between Full Groups
of Different Types 505
Chapter 9. Unitary Groups over General Classes of Form Rings .... 508
9.1*. The Normal Subgroup Structure of the Unitary Groups 508
9.1A*. Elementary Subgroups of the Unitary Groups 509
9.1B*. Classification of Normal Subgroups 514
9.1C*. Stability for KUt and KU2 520
9.2*. The Congruence Subgroup, Generation, and Presentation Problems 528
9.2A*. The Congruence Subgroup Problem 529
9.2B*. Generation by Elementary Matrices and Finite Generation 533
9.2C*. Presentations of Symplectic and Orthogonal Groups .... 539
Concluding Remarks 543
Bibliography 545
Index of Concepts 567
Index of Symbols 574
|
| any_adam_object | 1 |
| author | Hahn, Alexander J. 1943- O'Meara, Onorato T. 1928-2018 |
| author_GND | (DE-588)120421909 (DE-588)121509559 |
| author_facet | Hahn, Alexander J. 1943- O'Meara, Onorato T. 1928-2018 |
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| author_sort | Hahn, Alexander J. 1943- |
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| building | Verbundindex |
| bvnumber | BV002238070 |
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| classification_tum | MAT 200f MAT 552f |
| ctrlnum | (OCoLC)260163757 (DE-599)BVBBV002238070 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV002238070 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:42:34Z |
| institution | BVB |
| isbn | 3540177582 0387177582 |
| language | English |
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| physical | XV, 576 S. |
| publishDate | 1989 |
| publishDateSearch | 1989 |
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| publisher | Springer |
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| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Hahn, Alexander J. 1943- Verfasser (DE-588)120421909 aut The classical groups and K-theory Alexander J. Hahn ; O. Timothy O'Meara Berlin [u.a.] Springer 1989 XV, 576 S. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 291 Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Klassische Gruppe (DE-588)4164040-8 gnd rswk-swf Algebraische Gruppe (DE-588)4001164-1 gnd rswk-swf K-Theorie (DE-588)4033335-8 gnd rswk-swf Lineare algebraische Gruppe (DE-588)4295326-1 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 s DE-604 K-Theorie (DE-588)4033335-8 s Klassische Gruppe (DE-588)4164040-8 s Algebraische Gruppe (DE-588)4001164-1 s Lineare algebraische Gruppe (DE-588)4295326-1 s O'Meara, Onorato T. 1928-2018 Verfasser (DE-588)121509559 aut Grundlehren der mathematischen Wissenschaften 291 (DE-604)BV000000395 291 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001470875&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hahn, Alexander J. 1943- O'Meara, Onorato T. 1928-2018 The classical groups and K-theory Grundlehren der mathematischen Wissenschaften Gruppentheorie (DE-588)4072157-7 gnd Klassische Gruppe (DE-588)4164040-8 gnd Algebraische Gruppe (DE-588)4001164-1 gnd K-Theorie (DE-588)4033335-8 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4164040-8 (DE-588)4001164-1 (DE-588)4033335-8 (DE-588)4295326-1 |
| title | The classical groups and K-theory |
| title_auth | The classical groups and K-theory |
| title_exact_search | The classical groups and K-theory |
| title_full | The classical groups and K-theory Alexander J. Hahn ; O. Timothy O'Meara |
| title_fullStr | The classical groups and K-theory Alexander J. Hahn ; O. Timothy O'Meara |
| title_full_unstemmed | The classical groups and K-theory Alexander J. Hahn ; O. Timothy O'Meara |
| title_short | The classical groups and K-theory |
| title_sort | the classical groups and k theory |
| topic | Gruppentheorie (DE-588)4072157-7 gnd Klassische Gruppe (DE-588)4164040-8 gnd Algebraische Gruppe (DE-588)4001164-1 gnd K-Theorie (DE-588)4033335-8 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd |
| topic_facet | Gruppentheorie Klassische Gruppe Algebraische Gruppe K-Theorie Lineare algebraische Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001470875&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT hahnalexanderj theclassicalgroupsandktheory AT omearaonoratot theclassicalgroupsandktheory |