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A Taste of Jordan Algebras / Edition 1

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Overview

In this book, Kevin McCrimmon describes the history of Jordan Algebras and he describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. To keep the exposition elementary, the structure theory is developed for linear Jordan algebras, 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 1980's before the theory reached its final form.

This book is intended for graduate students and for individuals wishing to learn more about Jordan algebras. No previous knowledge is required beyond the standard first-year graduate algebra course. 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, functional analysis, or exceptional groups and geometry can also profit from acquaintance with the material. Jordan algebras crop up in many surprising settings and can be applied to a variety of mathematical areas.

Kevin McCrimmon introduced the concept of a quadratic Jordan algebra and developed a structure theory of Jordan algebras over an arbitrary ring of scalars. He is a Professor of Mathematics at the University of Virginia and the author of more than 100 research papers.

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Editorial Reviews

From the Publisher
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 für 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’s book 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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Product Details

  • ISBN-13: 9780387954479
  • Publisher: Springer New York
  • Publication date: 9/5/2007
  • Series: Universitext Series
  • Edition description: 2004
  • Edition number: 1
  • Pages: 592
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.31 (d)

Meet the Author

Kevin McCrimmon introduced the concept of a quadratic Jordan algebra and developed a structure theory of Jordan algebras over an arbitrary ring of scalars. He is a Professor of Mathematics at the University of Virginia and the author of more than 100 research papers.

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Table of Contents

Preface
Standard Notation
A Colloquial Survey of Jordan Theory 1
Pt. I A Historical Survey of Jordan Structure Theory
1 Jordan Algebras in Physical Antiquity 39
2 Jordan Algebras in the Algebraic Renaissance 51
3 Jordan Algebras in the Enlightenment 69
4 The Classical Theory 81
5 The Final Classical Formulation 95
6 The Classical Methods 99
7 The Russian Revolution 107
8 Zel'manov's Exceptional Methods 114
Pt. II The Classical Theory
1 The Category of Jordan Algebras 132
2 The Category of Alternative Algebras 153
3 Three Special Examples 168
4 Jordan Algebras of Cubic Forms 186
5 Two Basic Principles 199
6 Inverses 211
7 Isotopes 220
8 Peirce Decomposition 235
9 Off-Diagonal Rules 247
10 Peirce Consequences 253
11 Spin Coordinatization 259
12 Hermitian Coordinatization 268
13 Multiple Peirce Decompositions 278
14 Multiple Peirce Consequences 292
15 Hermitian Symmetries 301
16 The Coordinate Algebra 308
17 Jacobson Coordinatization 312
18 Von Neumann Regularity 318
19 Inner Simplicity 325
20 Capacity 330
21 Herstein - Kleinfeld - Osborn Theorem 335
22 Osborn's Capacity 2 Theorem 348
23 Classical Classification 356
Pt. III Zel'manov's Exceptional Theorem
1 The Radical 362
2 Begetting and Bounding Idempotents 381
3 Bounded Spectra Beget Capacity 388
4 Absorbers of Inner Ideals 397
5 Primitivity 410
6 The Primitive Heart 422
7 Filters and Ultrafilters 427
8 Ultraproducts 433
9 The Final Argument 440
Pt. IV Appendices
A Cohn's Special Theorems 447
B Macdonald's Theorem 455
C Jordan Algebras of Degree 3 476
D The Jacobson-Bourbaki Density Theorem 501
E Hints 506
Pt. V Indexes
A Index of Collateral Readings 524
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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