A Taste of Jordan Algebras / Edition 1

A Taste of Jordan Algebras / Edition 1

by Kevin McCrimmon
     
 

ISBN-10: 0387954473

ISBN-13: 9780387954479

Pub. Date: 09/05/2007

Publisher: Springer New York

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 Zel'manov. 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

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Overview

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 Zel'manov. 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 primaily 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.

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Product Details

ISBN-13:
9780387954479
Publisher:
Springer New York
Publication date:
09/05/2007
Series:
Universitext Series
Edition description:
2004
Pages:
592
Product dimensions:
9.21(w) x 6.14(h) x 1.31(d)

Related Subjects

Table of Contents

Preface
Standard Notation
A Colloquial Survey of Jordan Theory1
Pt. IA Historical Survey of Jordan Structure Theory
1Jordan Algebras in Physical Antiquity39
2Jordan Algebras in the Algebraic Renaissance51
3Jordan Algebras in the Enlightenment69
4The Classical Theory81
5The Final Classical Formulation95
6The Classical Methods99
7The Russian Revolution107
8Zel'manov's Exceptional Methods114
Pt. IIThe Classical Theory
1The Category of Jordan Algebras132
2The Category of Alternative Algebras153
3Three Special Examples168
4Jordan Algebras of Cubic Forms186
5Two Basic Principles199
6Inverses211
7Isotopes220
8Peirce Decomposition235
9Off-Diagonal Rules247
10Peirce Consequences253
11Spin Coordinatization259
12Hermitian Coordinatization268
13Multiple Peirce Decompositions278
14Multiple Peirce Consequences292
15Hermitian Symmetries301
16The Coordinate Algebra308
17Jacobson Coordinatization312
18Von Neumann Regularity318
19Inner Simplicity325
20Capacity330
21Herstein - Kleinfeld - Osborn Theorem335
22Osborn's Capacity 2 Theorem348
23Classical Classification356
Pt. IIIZel'manov's Exceptional Theorem
1The Radical362
2Begetting and Bounding Idempotents381
3Bounded Spectra Beget Capacity388
4Absorbers of Inner Ideals397
5Primitivity410
6The Primitive Heart422
7Filters and Ultrafilters427
8Ultraproducts433
9The Final Argument440
Pt. IVAppendices
ACohn's Special Theorems447
BMacdonald's Theorem455
CJordan Algebras of Degree 3476
DThe Jacobson-Bourbaki Density Theorem501
EHints506
Pt. VIndexes
AIndex of Collateral Readings524
BPronouncing Index of Names531
C: Index of Notations536
D: Index of Statements545
E: Index of Definitions555

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