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By: Kevin McCrimmon

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Hardcover | 15 December 2003

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592 Pages

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24.77 x 17.15 x 3.18

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The book describes the history of Jordan algebras, and describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zelmanov. To keep the exposition elementary, the structure theory is developed for linear Jordan algebras (where the scalar ring contains 1/2, avoiding the nuisancy distractions of characteristic 2), though the modern quadratic methods are used throughout. Both the quadratic methods and the Zelmanov results go beyond the previous textbooks on Jordan Theory, written in the 1960's and early 1980's before the theory reached its final form. The book is written to serve either as a text for a 2nd year graduate course, or for independent reading, for students who need or wish to know a bit about Jordan algebras. It is not primarily aimed at experts or students going on to do research in the area, and no knowledge is required beyond standard first-year graduate algebra courses. General students of algebra can profit from exposure to nonassociative algebras, and students or professional mathematicians working in areas such as Lie algebras, differential geometry (symmetric spaces or bounded symmetric domains), functional analysis (JB algebras and triples), or exceptional groups and geometry (related to the 27-dimensional Albert algebra) can also profit from acquaintance with the material. Jordan algebras crop up in many surprising settings, and find application to a variety of mathematical areas.

Industry Reviews

From the reviews:

"This is an excellent book, masterly written and very well organized, a real compendium of Jordan algebras offering all the relevant notions and results of the theory - and not only a 'taste'. ... is written as a direct mathematical conversation between the author and a reader, who has no knowledge of Jordan algebras. Thus more heuristic and explanatory comment is provided than is usual in graduate texts. ... An exceptional book!" (H. Mitsch, Monatshefte fuer Mathematik, Vol. 144 (3), 2005)

"As mentioned in the preface, 'this book tells the story of one aspect of Jordan structure theory ... . The author proceeds to tell this fascinating story with a lovely and lively style ... . concentrates explicitly on the structure theory of linear Jordan algebra ... . It can be used in many different ways to teach graduate courses and also for self-study ... . graduate students will have at their disposal a very well organized, motivating and engaging textbook." (Alberto Elduque, Zentralblatt MATH, Vol. 1044 (19), 2004)

"The book ... is intended, according to the author, to serve as an accompaniment to a graduate course in Jordan algebras. In fact the exposition goes far beyond this goal, resulting in a book much richer than the typical textbook. ... The book is well written, and I enjoyed reading it. ... The style is lively ... . In my opinion this book will be indispensable for all mathematicians ... . a great book and I believe it will serve the mathematical community well." (Plamen Koshlukov, Mathematical Reviews, 2004i)

"Read 'A Taste of Jordan Algebras' by K. McCrimmon, where, for the first time, a full account of both the mathematical development of Jordan algebra theory and its historical aspects is given. ... Thanks to the very clever organization of the book ... it is suited both to the very beginner and to the specialist ... . Unlike all other monographs on Jordan algebras ... McCrimmon'sbook will be the fundamental textbook in this domain for many years to come." (Wolfgang Bertram, SIAM Reviews, Vol. 47 (1), 2005)

"McCrimmon is a pioneer in the subject of Jordan algebras ... . 'The reader should see isomorphisms as cloning maps, isotopes as subtle rearrangements of an algebra's DNA ... . The book is written in this marvellous style ... very thorough, and very strong on (the right kind of) pedagogy. The reader will learn a lot of wonderful algebra well, if he takes care to follow McCrimmon's plan: read carefully, do the problems, meditate on what's going on, follow and absorb the analogies ... ." (Michael Berg, MAA online, November, 2004)

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Non-Fiction Mathematics Algebra Mathematical Foundation Set Theory Groups & Group Theory

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Published: 19th November 2010

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Preface	p. vii
Standard Notation	p. xiii
A Colloquial Survey of Jordan Theory	p. 1
Origin of the Species	p. 2
The Jordan River	p. 6
Links with Lie Algebras and Groups	p. 10
Links with Differential Geometry	p. 14
Links with the Real World	p. 16
Links with the Complex World	p. 22
Links with the Infinitely Complex World	p. 25
Links with Projective Geometry	p. 28
A Historical Survey of Jordan Structure Theory
Jordan Algebras in Physical Antiquity	p. 39
The Matrix Interpretation of Quantum Mechanics	p. 39
The Jordan Program	p. 40
The Jordan Operations	p. 40
Digression on Linearization	p. 41
Back to the Bullet	p. 42
The Jordan Axioms	p. 43
The First Example: Full Algebras	p. 45
The Second Example: Hermitian Algebras	p. 46
The Third Example: Spin Factors	p. 47
Special and Exceptional	p. 48
Classification	p. 48
Jordan Algebras in the Algebraic Renaissance	p. 51
Linear Algebras over General Scalars	p. 52
Categorical Nonsense	p. 53
Commutators and Associators	p. 55
Lie and Jordan Algebras	p. 57
The Three Basic Examples Revisited	p. 58
Jordan Matrix Algebras with Associative Coordinates	p. 59
Jordan Matrix Algebras with Alternative Coordinates	p. 60
The n-Squares Problem	p. 61
Forms Permitting Composition	p. 62
Composition Algebras	p. 63
The Cayley-Dickson Construction and Process	p. 64
Split Composition Algebras	p. 65
Classification	p. 67
Jordan Algebras in the Enlightenment	p. 69
Forms of Algebras	p. 69
Inverses and Isotopes	p. 70
Nuclear Isotopes	p. 71
Twisted Involutions	p. 72
Twisted Hermitian Matrices	p. 73
Spin Factors	p. 74
Quadratic Factors	p. 74
Cubic Factors	p. 76
Reduced Cubic Factors	p. 78
Classification	p. 79
The Classical Theory	p. 81
U-Operators	p. 81
The Quadratic Program	p. 82
The Quadratic Axioms	p. 83
Justification	p. 84
Inverses	p. 85
Isotopes	p. 86
Inner Ideals	p. 86
Nondegeneracy	p. 88
Radical Remarks	p. 89
i-Special and i-Exceptional	p. 90
Artin-Wedderburn-Jacobson Structure Theorem	p. 92
The Final Classical Formulation	p. 95
Algebras with Capacity	p. 95
Classification	p. 97
The Classical Methods	p. 99
Peirce Decompositions	p. 99
Coordinatization	p. 100
The Coordinates	p. 102
Minimal Inner Ideals	p. 102
Capacity	p. 104
Capacity Classification	p. 104
The Russian Revolution	p. 107
The Lull Before the Storm	p. 107
The First Tremors	p. 108
The Main Quake	p. 109
Aftershocks	p. 111
Zel'manov's Exceptional Methods	p. 114
I-Finiteness	p. 114
Absorbers	p. 116
Modular Inner Ideals	p. 117
Primitivity	p. 118
The Heart	p. 119
Spectra	p. 120
Comparing Spectra	p. 122
Big Resolvents	p. 123
Semiprimitive Imbedding	p. 124
Ultraproducts	p. 125
Prime Dichotomy	p. 127
The Classical Theory
The Category of Jordan Algebras	p. 132
Categories	p. 132
The Category of Linear Algebras	p. 133
The Category of Unital Algebras	p. 136
Unitalization	p. 137
The Category of Algebras with Involution	p. 139
Nucleus, Center, and Centroid	p. 141
Strict Simplicity	p. 144
The Category of Jordan Algebras	p. 146
Problems for Chapter 1	p. 150
The Category of Alternative Algebras	p. 153
The Category of Alternative Algebras	p. 153
Nuclear Involutions	p. 154
Composition Algebras	p. 155
Split Composition Algebras	p. 157
The Cayley-Dickson Construction	p. 160
The Hurwitz Theorem	p. 164
Problems for Chapter 2	p. 167
Three Special Examples	p. 168
Full Type	p. 168
Hermitian Type	p. 171
Quadratic Form Type	p. 176
Reduced Spin Factors	p. 180
Problems for Chapter 3	p. 183
Jordan Algebras of Cubic Forms	p. 186
Cubic Maps	p. 187
The General Construction	p. 188
The Jordan Cubic Construction	p. 191
The Freudenthal Construction	p. 193
The Tits Constructions	p. 195
Problems for Chapter 4	p. 197
Two Basic Principles	p. 199
The Macdonald and Shirshov-Cohn Principles	p. 199
Fundamental Formulas	p. 200
Nondegeneracy	p. 205
Problems for Chapter 5	p. 209
Inverses	p. 211
Jordan Inverses	p. 211
von Neumann and Nuclear Inverses	p. 217
Problems for Chapter 6	p. 219
Isotopes	p. 220
Nuclear Isotopes	p. 220
Jordan Isotopes	p. 222
Quadratic Factor Isotopes	p. 224
Cubic Factor Isotopes	p. 226
Matrix Isotopes	p. 228
Problems for Chapter 7	p. 232
Peirce Decomposition	p. 235
Peirce Decompositions	p. 235
Peirce Multiplication Rules	p. 239
Basic Examples of Peirce Decompositions	p. 240
Peirce Identity Principle	p. 245
Problems for Chapter 8	p. 246
Off-Diagonal Rules	p. 247
Peirce Specializations	p. 247
Peirce Quadratic Forms	p. 250
Problems for Chapter 9	p. 252
Peirce Consequences	p. 253
Diagonal Consequences	p. 253
Diagonal Isotopes	p. 255
Problems for Chapter 10	p. 258
Spin Coordinatization	p. 259
Spin Frames	p. 259
Diagonal Spin Consequences	p. 262
Strong Spin Coordinatization	p. 263
Spin Coordinatization	p. 265
Problems for Chapter 11	p. 267
Hermitian Coordinatization	p. 268
Cyclic Frames	p. 268
Diagonal Hermitian Consequences	p. 270
Strong Hermitian Coordinatization	p. 272
Hermitian Coordinatization	p. 274
Multiple Peirce Decompositions	p. 278
Decomposition	p. 278
Recovery	p. 282
Multiplication	p. 282
The Matrix Archetype	p. 284
The Peirce Principle	p. 286
Modular Digression	p. 288
Problems for Chapter 13	p. 290
Multiple Peirce Consequences	p. 292
Jordan Coordinate Conditions	p. 292
Peirce Specializations	p. 294
Peirce Quadratic Forms	p. 295
Connected Idempotents	p. 296
Hermitian Symmetries	p. 301
Hermitian Frames	p. 301
Hermitian Symmetries	p. 303
Problems for Chapter 15	p. 307
The Coordinate Algebra	p. 308
The Coordinate Triple	p. 308
Jacobson Coordinatization	p. 312
Strong Coordinatization	p. 312
General Coordinatization	p. 315
Von Neumann Regularity	p. 318
vNr Pairing	p. 318
Structural Pairing	p. 321
Problems for Chapter 18	p. 324
Inner Simplicity	p. 325
Simple Inner Ideals	p. 325
Minimal Inner Ideals	p. 327
Problems for Chapter 19	p. 329
Capacity	p. 330
Capacity Existence	p. 330
Connected Capacity	p. 331
Problems for Chapter 20	p. 334
Herstein-Kleinfeld-Osborn Theorem	p. 335
Alternative Algebras Revisited	p. 335
A Brief Tour of the Alternative Nucleus	p. 338
Herstein-Kleinfeld-Osborn Theorem	p. 341
Problems for Chapter 21	p. 345
Osborn's Capacity 2 Theorem	p. 348
Commutators	p. 348
Capacity Two	p. 351
Classical Classification	p. 356
Capacity n ≥ 3	p. 356
Zel'manov's Exceptional Theorem
The Radical	p. 362
Invertibility	p. 363
Structurality	p. 364
Quasi-Invertibility	p. 366
Proper Quasi-Invertibility	p. 369
Elemental Characterization	p. 374
Radical Inheritance	p. 375
Radical Surgery	p. 376
Problems for Chapter 1	p. 379
Begetting and Bounding Idempotents	p. 381
I-gene	p. 381
Algebraic Implies I-Genic	p. 383
I-genic Nilness	p. 384
I-Finiteness	p. 384
Problems for Chapter 2	p. 387
Bounded Spectra Beget Capacity	p. 388
Spectra	p. 388
Bigness	p. 390
Evaporating Division Algebras	p. 392
Spectral Bounds and Capacity	p. 392
Problems for Chapter 3	p. 395
Absorbers of Inner Ideals	p. 397
Linear Absorbers	p. 397
Quadratic Absorbers	p. 400
Absorber Nilness	p. 403
Problems for Chapter 4	p. 409
Primitivity	p. 410
Modularity	p. 410
Primitivity	p. 413
Semiprimitivity	p. 415
Imbedding Nondegenerates in Semiprimitives	p. 416
Problems for Chapter 5	p. 420
The Primitive Heart	p. 422
Hearts and Spectra	p. 422
Primitive Hearts	p. 424
Problems for Chapter 6	p. 425
Filters and Ultrafilters	p. 427
Filters in General	p. 427
Filters from Primes	p. 428
Ultimate Filters	p. 430
Problems for Chapter 7	p. 432
Ultraproducts	p. 433
Ultraproducts	p. 433
Examples	p. 435
Problems for Chapter 8	p. 439
The Final Argument	p. 440
Dichotomy	p. 440
The Prime Dichotomy	p. 441
Problems for Chapter 9	p. 443
Appendices
Cohn's Special Theorems	p. 447
Free Gadgets	p. 447
Cohn Symmetry	p. 449
Cohn Speciality	p. 450
Problems for Appendix A	p. 454
Macdonald's Theorem	p. 455
The Free Jordan Algebra	p. 455
Identities	p. 458
Normal Form for Multiplications	p. 461
The Macdonald Principles	p. 466
Albert i-Exceptionality	p. 469
Nonvanishing of G₉	p. 471
Nonvanishing of G₈	p. 472
Nonvanishing of T₁₁	p. 473
Problems for Appendix B	p. 475
Jordan Algebras of Degree 3	p. 476
Jordan Matrix Algebras	p. 476
The General Construction	p. 480
The Freudenthal Construction	p. 487
The Tits Constructions	p. 490
Albert Division Algebras	p. 498
Problems for Appendix C	p. 500
The Jacobson-Bourbaki Density Theorem	p. 501
Semisimple Modules	p. 501
The Jacobson-Bourbaki Density Theorem	p. 504
Hints	p. 506
Hints for Part II	p. 506
Hints for Part III	p. 517
E.3	p. 521
Indexes
Index of Collateral Readings	p. 524
Foundational Readings	p. 524
Readings in Applications	p. 527
Historical Perusals	p. 529
Pronouncing Index of Names	p. 531
Index of Notations	p. 536
Index of Statements	p. 545
Index of Definitions	p. 555
Table of Contents provided by Publisher. All Rights Reserved.

Universitext

At a Glance

Hardcover

Industry Reviews

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Defective items

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