A Taste of Jordan Algebras

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.

Autorentext

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.

Klappentext

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.

Inhalt

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