A taste of Jordan algebras:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
2004
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXV, 562 S. |
| ISBN: | 0387954473 |
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| 245 | 1 | 0 | |a A taste of Jordan algebras |c Kevin McCrimmon |
| 264 | 1 | |a New York u.a. |b Springer |c 2004 | |
| 300 | |a XXV, 562 S. | ||
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Record in the Search Index
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| adam_text | Contents
Preface vii
Standard Notation xiii
0 A Colloquial Survey of Jordan Theory 1
0.1 Origin of the Species 2
0.2 The Jordan River 6
0.3 Links with Lie Algebras and Groups 10
0.4 Links with Differential Geometry 14
0.5 Links with the Real World 16
0.6 Links with the Complex World 22
0.7 Links with the Infinitely Complex World 25
0.8 Links with Projective Geometry 28
Part I A Historical Survey of Jordan Structure Theory
1 Jordan Algebras in Physical Antiquity 39
1.1 The Matrix Interpretation of Quantum Mechanics 39
1.2 The Jordan Program 40
1.3 The Jordan Operations 40
1.4 Digression on Linearization 41
1.5 Back to the Bullet 42
1.6 The Jordan Axioms 43
1.7 The First Example: Full Algebras 45
1.8 The Second Example: Hermitian Algebras 46
1.9 The Third Example: Spin Factors 47
1.10 Special and Exceptional 48
1.11 Classification 48
xviii Contents
2 Jordan Algebras in the Algebraic Renaissance 51
2.1 Linear Algebras over General Scalars 52
2.2 Categorical Nonsense 53
2.3 Commutators and Associators 55
2.4 Lie and Jordan Algebras 57
2.5 The Three Basic Examples Revisited 58
2.6 Jordan Matrix Algebras with Associative Coordinates 59
2.7 Jordan Matrix Algebras with Alternative Coordinates 60
2.8 The n Squares Problem 61
2.9 Forms Permitting Composition 62
2.10 Composition Algebras 63
2.11 The Cayley Dickson Construction and Process 64
2.12 Split Composition Algebras 65
2.13 Classification 67
3 Jordan Algebras in the Enlightenment 69
3.1 Forms of Algebras 69
3.2 Inverses and Isotopes 70
3.3 Nuclear Isotopes 71
3.4 Twisted Involutions 72
3.5 Twisted Hermitian Matrices 73
3.6 Spin Factors 74
3.7 Quadratic Factors 74
3.8 Cubic Factors 76
3.9 Reduced Cubic Factors 78
3.10 Classification 79
4 The Classical Theory 81
4.1 [/ Operators 81
4.2 The Quadratic Program 82
4.3 The Quadratic Axioms 83
4.4 Justification 84
4.5 Inverses 85
4.6 Isotopes 86
4.7 Inner Ideals 86
4.8 Nondegeneracy 88
4.9 Radical Remarks 89
4.10 i Special and i Exceptional 90
4.11 Artin Wedderburn Jacobson Structure Theorem 92
5 The Final Classical Formulation 95
5.1 Algebras with Capacity 95
5.2 Classification 97
Contents xix
6 The Classical Methods 99
6.1 Peirce Decompositions 99
6.2 Coordinatization 100
6.3 The Coordinates 102
6.4 Minimal Inner Ideals 102
6.5 Capacity 104
6.6 Capacity Classification 104
7 The Russian Revolution 107
7.1 The Lull Before the Storm 107
7.2 The First Tremors 108
7.3 The Main Quake 109
7.4 Aftershocks Ill
8 Zel manov s Exceptional Methods 114
8.1 I Finiteness 114
8.2 Absorbers 116
8.3 Modular Inner Ideals 117
8.4 Primitivity 118
8.5 The Heart 119
8.6 Spectra 120
8.7 Comparing Spectra 122
8.8 Big Resolvents 123
8.9 Semiprimitive Imbedding 124
8.10 Ultraproducts 125
8.11 Prime Dichotomy 127
Part II The Classical Theory
1 The Category of Jordan Algebras 132
1.1 Categories 132
1.2 The Category of Linear Algebras 133
1.3 The Category of Unital Algebras 136
1.4 Unitalization 137
1.5 The Category of Algebras with Involution 139
1.6 Nucleus, Center, and Centroid 141
1.7 Strict Simplicity 144
1.8 The Category of Jordan Algebras 146
1.9 Problems for Chapter 1 150
2 The Category of Alternative Algebras 153
2.1 The Category of Alternative Algebras 153
2.2 Nuclear Involutions 154
2.3 Composition Algebras 155
xx Contents
2.4 Split Composition Algebras 157
2.5 The Cayley Dickson Construction 160
2.6 The Hurwitz Theorem 164
2.7 Problems for Chapter 2 167
3 Three Special Examples 168
3.1 Full Type 168
3.2 Hermitian Type 171
3.3 Quadratic Form Type 176
3.4 Reduced Spin Factors 180
3.5 Problems for Chapter 3 183
4 Jordan Algebras of Cubic Forms 186
4.1 Cubic Maps 187
4.2 The General Construction 188
4.3 The Jordan Cubic Construction 191
4.4 The Freudenthal Construction 193
4.5 The Tits Constructions 195
4.6 Problems for Chapter 4 197
5 Two Basic Principles 199
5.1 The Macdonald and Shirshov Cohn Principles 199
5.2 Fundamental Formulas 200
5.3 Nondegeneracy 205
5.4 Problems for Chapter 5 209
6 Inverses 211
6.1 Jordan Inverses 211
6.2 von Neumann and Nuclear Inverses 217
6.3 Problems for Chapter 6 219
7 Isotopes 220
7.1 Nuclear Isotopes 220
7.2 Jordan Isotopes 222
7.3 Quadratic Factor Isotopes 224
7.4 Cubic Factor Isotopes 226
7.5 Matrix Isotopes 228
7.6 Problems for Chapter 7 232
8 Peirce Decomposition 235
8.1 Peirce Decompositions 235
8.2 Peirce Multiplication Rules 239
8.3 Basic Examples of Peirce Decompositions 240
8.4 Peirce Identity Principle 245
8.5 Problems for Chapter 8 246
Contents xxi
9 Off Diagonal Rules 247
9.1 Peirce Specializations 247
9.2 Peirce Quadratic Forms 250
9.3 Problems for Chapter 9 252
10 Peirce Consequences 253
10.1 Diagonal Consequences 253
10.2 Diagonal Isotopes 255
10.3 Problems for Chapter 10 258
11 Spin Coordinatization 259
11.1 Spin Frames 259
11.2 Diagonal Spin Consequences 262
11.3 Strong Spin Coordinatization 263
11.4 Spin Coordinatization 265
11.5 Problems for Chapter 11 267
12 Hermitian Coordinatization 268
12.1 Cyclic Frames 268
12.2 Diagonal Hermitian Consequences 270
12.3 Strong Hermitian Coordinatization 272
12.4 Hermitian Coordinatization 274
13 Multiple Peirce Decompositions 278
13.1 Decomposition 278
13.2 Recovery 282
13.3 Multiplication 282
13.4 The Matrix Archetype 284
13.5 The Peirce Principle 286
13.6 Modular Digression 288
13.7 Problems for Chapter 13 290
14 Multiple Peirce Consequences 292
14.1 Jordan Coordinate Conditions 292
14.2 Peirce Specializations 294
14.3 Peirce Quadratic Forms 295
14.4 Connected Idempotents 296
15 Hermitian Symmetries 301
15.1 Hermitian Frames 301
15.2 Hermitian Symmetries 303
15.3 Problems for Chapter 15 307
16 The Coordinate Algebra 308
16.1 The Coordinate Triple 308
xxii Contents
17 Jacobson Coordinatization 312
17.1 Strong Coordinatization 312
17.2 General Coordinatization 315
18 Von Neumann Regularity 318
18.1 vNr Pairing 318
18.2 Structural Pairing 321
18.3 Problems for Chapter 18 324
19 Inner Simplicity 325
19.1 Simple Inner Ideals 325
19.2 Minimal Inner Ideals 327
19.3 Problems for Chapter 19 329
20 Capacity 330
20.1 Capacity Existence 330
20.2 Connected Capacity 331
20.3 Problems for Chapter 20 334
21 Herstein Kleinfeld Osborn Theorem 335
21.1 Alternative Algebras Revisited 335
21.2 A Brief Tour of the Alternative Nucleus 338
21.3 Herstein Kleinfeld Osborn Theorem 341
21.4 Problems for Chapter 21 345
22 Osborn s Capacity 2 Theorem 348
22.1 Commutators 348
22.2 Capacity Two 351
23 Classical Classification 356
23.1 Capacity n 3 356
Part III Zel manov s Exceptional Theorem
1 The Radical 362
1.1 Invertibility 363
1.2 Structurally 364
1.3 Quasi Invertibility 366
1.4 Proper Quasi Invertibility 369
1.5 Elemental Characterization 374
1.6 Radical Inheritance 375
1.7 Radical Surgery 376
1.8 Problems for Chapter 1 379
Contents xxiii
2 Begetting and Bounding Idempotents 381
2.1 I gene 381
2.2 Algebraic Implies I Genic 383
2.3 I genic Nilness 384
2.4 I Finiteness 384
2.5 Problems for Chapter 2 387
3 Bounded Spectra Beget Capacity 388
3.1 Spectra 388
3.2 Bigness 390
3.3 Evaporating Division Algebras 392
3.4 Spectral Bounds and Capacity 392
3.5 Problems for Chapter 3 395
4 Absorbers of Inner Ideals 397
4.1 Linear Absorbers 397
4.2 Quadratic Absorbers 400
4.3 Absorber Nilness 403
4.4 Problems for Chapter 4 409
5 Primitivity 410
5.1 Modularity 410
5.2 Primitivity 413
5.3 Semiprimitivity 415
5.4 Imbedding Nondegenerates in Semiprimitives 416
5.5 Problems for Chapter 5 420
6 The Primitive Heart 422
6.1 Hearts and Spectra 422
6.2 Primitive Hearts 424
6.3 Problems for Chapter 6 425
7 Filters and Ultrafilters 427
7.1 Filters in General 427
7.2 Filters from Primes 428
7.3 Ultimate Filters 430
7.4 Problems for Chapter 7 432
8 Ultraproducts 433
8.1 Ultraproducts 433
8.2 Examples 435
8.3 Problems for Chapter 8 439
xxiv Contents
9 The Final Argument 440
9.1 Dichotomy 440
9.2 The Prime Dichotomy 441
9.3 Problems for Chapter 9 443
Part IV Appendices
A Cohn s Special Theorems 447
A.I Free Gadgets 447
A.2 Cohn Symmetry 449
A.3 Cohn Speciality 450
A.4 Problems for Appendix A 454
B Macdonald s Theorem 455
B.I The Free Jordan Algebra 455
B.2 Identities 458
B.3 Normal Form for Multiplications 461
B.4 The Macdonald Principles 466
B.5 Albert i Exceptionality 469
B.5.1 Nonvanishing of G9 471
B.5.2 Nonvanishing of G8 472
B.5.3 Nonvanishing of Tn 473
B.6 Problems for Appendix B 475
C Jordan Algebras of Degree 3 476
C.I Jordan Matrix Algebras 476
C.2 The General Construction 480
C.3 The Freudenthal Construction 487
C.4 The Tits Constructions 490
C.5 Albert Division Algebras 498
C.6 Problems for Appendix C 500
D The Jacobson Bourbaki Density Theorem 501
D.I Semisimple Modules 501
D.2 The Jacobson Bourbaki Density Theorem 504
E Hints 506
E.I Hints for Part II 506
E.2 Hints for Part III 517
E.3 Hints for Part IV 521
Contents xxv
Part V Indexes
A Index of Collateral Readings 524
A.I Foundational Readings 524
A.2 Readings in Applications 527
A.3 Historical Perusals 529
B Pronouncing Index of Names 531
C Index of Notations 536
D Index of Statements 545
E Index of Definitions 555
|
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| id | DE-604.BV014807502 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:07:24Z |
| institution | BVB |
| isbn | 0387954473 |
| language | English |
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| spelling | McCrimmon, Kevin 1941- Verfasser (DE-588)12870618X aut A taste of Jordan algebras Kevin McCrimmon New York u.a. Springer 2004 XXV, 562 S. txt rdacontent n rdamedia nc rdacarrier Universitext Includes bibliographical references and index Jordan, Algèbres de Jordan algebras Jordan-Algebra (DE-588)4162770-2 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Jordan-Algebra (DE-588)4162770-2 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010019641&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | McCrimmon, Kevin 1941- A taste of Jordan algebras Jordan, Algèbres de Jordan algebras Jordan-Algebra (DE-588)4162770-2 gnd |
| subject_GND | (DE-588)4162770-2 (DE-588)4123623-3 |
| title | A taste of Jordan algebras |
| title_auth | A taste of Jordan algebras |
| title_exact_search | A taste of Jordan algebras |
| title_full | A taste of Jordan algebras Kevin McCrimmon |
| title_fullStr | A taste of Jordan algebras Kevin McCrimmon |
| title_full_unstemmed | A taste of Jordan algebras Kevin McCrimmon |
| title_short | A taste of Jordan algebras |
| title_sort | a taste of jordan algebras |
| topic | Jordan, Algèbres de Jordan algebras Jordan-Algebra (DE-588)4162770-2 gnd |
| topic_facet | Jordan, Algèbres de Jordan algebras Jordan-Algebra Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010019641&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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