A Taste of Jordan Algebras

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November 2003



This 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 Zel'manov. Jordan algebras crop up in many surprising settings, and find application to a variety of mathematical areas. No knowledge is required beyond standard first-year graduate algebra courses.


0 A Colloquial Survey of Jordan Theory
0.1 Origin of the Species
0.2 The Jordan River
0.3 Links with Lie Algebras and Groups
0.4 Links with Differential Geometry
0.5 Links with the Real World
0.6 Links with the Complex World
0.7 Links with the Infinitely Complex World
0.8 Links with Projective Geometry

I A Historical Survey of Jordan Structure Theory

1 Jordan Algebras in Physical Antiquity
1.1 The Matrix Interpretation of Quantum Mechanics
1.2 The Jordan Program
1.3 The Jordan Operations
1.4 Digression on Linearization
1.5 Back to the Bullet
1.6 The Jordan Axioms
1.7 The First Example: Full Algebras
1.8 The Second Example: Hermitian Algebras
1.9 The Third Example: Spin Factors
1.1 Special and Exceptional
1.11 Classification

2 Jordan Algebras in the Algebraic Renaissance
2.1 Linear Algebras over General Scalars
2.2 Categorical Nonsense
2.3 Commutators and Associators
2.4 Lie and Jordan Algebras
2.5 The 3 Basic Examples Revisited
2.6 Jordan Matrix Algebras with Associative Coordinates
2.7 Jordan Matrix Algebras with Alternative Coordinates
2.8 The $n$-Squares Problem
2.9 Forms Permitting Composition
2.1 Composition Algebras
2.11 The Cayley--Dickson Construction and Process
2.12 Split Composition Algebras
2.13 Classification

3 Jordan Algebras in the Enlightenment
3.1 Forms of Algebras
3.2 Inverses and Isotopes
3.3 Nuclear Isotopes
3.4 Twisted involutions
3.5 Twisted Hermitian Matrices
3.6 Spin Factors
3.7 Quadratic factors
3.8 Cubic Factors
3.9 Reduced Cubic Factors
3.1 Classification

4 The Classical Theory
4.1 $U$-Operators
4.2 The Quadratic Program
4.3 The Quadratic Axioms
4.4 Justification
4.5 Inverses
4.6 Isotopes
4.7 Inner Ideals
4.8 Nondegeneracy
4.9 Radical remarks
4.1 i-Special and i-Exceptional
4.11 Artin--Wedderburn--Jacobson Structure Theorem

5 The Final Classical Formulation
5.1 Capacity
5.2 Classification

6 The Classical Methods
6.1 Peirce Decompositions
6.2 Coordinatization
6.3 The Coordinates
6.4 Minimum Inner Ideals
6.5 Capacity
6.6 Capacity Classification

7 The Russian Revolution: 1977--1983
7.1 The Lull Before the Storm
7.2 The First Tremors
7.3 The Main Quake
7.4 Aftershocks

8 Zel'manov's Exceptional Methods
8.1 I-Finiteness
8.2 Absorbers
8.3 Modular Inner Ideals
8.4 Primitivity
8.5 The Heart
8.6 Spectra
8.7 Comparing Spectra
8.8 Big Resolvents
8.9 Semiprimitive Imbedding
8.1 Ultraproducts
8.11 Prime Dichotomy

II The Classical Theory

1 The Category of Jordan Algebras
1.1 Categories
1.2 The Category of Linear Algebras
1.3 The Category of Unital Algebras
1.4 Unitalization
1.5 The Category of Algebras with Involution
1.6 Nucleus, Center, and Centroid
1.7 Strict Simplicity
1.8 The Category of Jordan Algebras
1.9 Problems for Chapter 1

2 The Category of Alternative Algebras
2.1 The Category of Alternative Algebras
2.2 Nuclear Involutions
2.3 Composition Algebras
2.4 Split Composition Algebras
2.5 The Cayley--Dickson Construction
2.6 The Hurwitz Theorem
2.7 Problems for Chapter 2

3 Three Special Examples
3.1 Full Type
3.2 Hermitian Type
3.3 Quadratic Form Type
3.4 Reduced Spin Factors
3.5 Problems for Chapter 3

4 Jordan Algebras of Cubic Forms
4.1 Cubic Maps
4.2 The General Construction
4.3 The Jordan Cubic Construction
4.4 The Freudenthal Construction
4.5 The Tits Constructions
4.6 Problems for Chapter 4

5 Two Basic Principles
5.1 The Macdonald and Shirshov--Cohn Principles
5.2 Funda


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.


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)
EAN: 9780387954479
ISBN: 0387954473
Untertitel: 2004. Auflage. Book. Sprache: Englisch.
Verlag: Springer
Erscheinungsdatum: November 2003
Seitenanzahl: 592 Seiten
Format: gebunden
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