A Taste of Jordan Algebras / Edition 1

A Taste of Jordan Algebras / Edition 1

by Kevin McCrimmon
     
 

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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

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.

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

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

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