A Taste of Jordan Algebras by Kevin McCrimmonA Taste of Jordan Algebras by Kevin McCrimmon

A Taste of Jordan Algebras

byKevin McCrimmon

Paperback | November 19, 2010

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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.
 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.
Title:A Taste of Jordan AlgebrasFormat:PaperbackDimensions:563 pages, 23.5 × 15.5 × 0.1 inPublished:November 19, 2010Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:1441930035

ISBN - 13:9781441930033


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 Fundamental Formulas 5.3 Nondegeneracy 5.4 Problems for Chapter 5   6 Inverses 6.1 Jordan Inverses 6.2 von Neumann and Nuclear Inverses 6.3 Problems for Chapter 6   7 Isotopes 7.1 Nuclear Isotopes 7.2 Jordan Isotopes 7.3 Quadratic Factor Isotopes 7.4 Cubic Factor Isotopes 7.5 Matrix Isotopes 7.6 Problems for Chapter 7   8 Peirce Decomposition 8.1 Peirce Decompositions 8.2 Peirce Multiplication Rules 8.3 Basic Examples of Peirce Decompositions 8.4 Peirce Identity Principle 8.5 Problems for Chapter 8   9 Off-Diagonal Rules 9.1 Peirce Specializations 9.2 Peirce Quadratic Forms 9.3 Problems for Chapter 9   10 Peirce Consequences 10.1 Diagonal Consequences 10.2 Diagonal Isotopes 10.3 Problems for Chapter 10   11 Spin Coordinatization 11.1 Spin Frames 11.2 Diagonal Spin Consequences 11.3 Strong Spin Coordinatization 11.4 Spin Coordinatization 11.5 Problems for Chapter 11   12 Hermitian Coordinatization 12.1 Cyclic Frames 12.2 Diagonal Hermitian Consequences 12.3 Strong Hermitian Coordinatization 12.4 Hermitian Coordinatization   13 Multiple Peirce Decompositions 13.1 Decomposition 13.2 Recovery 13.3 Multiplication 13.4 The Matrix Archetype 13.5 The Peirce Principle 13.6 Modular Digression 13.7 Problems for Chapter 13   14 Multiple Peirce Consequences 14.1 Jordan Coordinate Conditions 14.2 Peirce Specializations 14.3 Peirce Quadratic Forms 14.4 Connected Idempotents   15 Hermitian Symmetries 15.1 Hermitian Frames 15.2 Hermitian Symmetries 15.3 Problems for Chapter 15   16 The Coordinate Algebra 16.1 The Coordinate Triple   17 Jacobson Coordinatization 17.1 Strong Coordinatization 17.2 General Coordinatization   18 Von Neumann Regularity 18.1 vNr Pairing 18.2 Structural Pairing 18.3 Problems for Chapter 18   19 Inner Simplicity 19.1 Simple Inner Ideals 19.2 Minimal Inner Ideals 19.3 Problems for Chapter 19   20 Capacity 20.1 Capacity Existence 20.2 Connected Capacity 20.3 Problems for Chapter 20   21 Herstein--Kleinfeld--Osborn Theorem 21.1 Alternative Algebras Revisited 21.2 A Brief Tour of the Alternative Nucleus 21.3 Herstein--Kleinfeld--Osborn Theorem 21.4 Problems for Chapter 21   22 Osborn's Capacity 2 Theorem 22.1 Commutators 22.2 Capacity Two   23 Classical Classification 23.1 Capacity $n\geq3$   III Zel'manov's Exceptional Theorem   1 The Radical 1.1 Invertibility 1.2 Structurality 1.3 Quasi-Invertibility 1.4 Proper Quasi-Invertibility 1.5 Elemental Characterization 1.6 Radical Inheritance 1.7 Radical Surgery 1.8 Problems and Questions for Chapter 1   2 Begetting and Bounding Idempotents 2.1 I-gene 2.2 Algebraic Implies I-Genic 2.3 I-genic Nilness 2.4 I-Finiteness 2.5 Problems for Chapter 2   3 Bounded Spectra Beget Capacity 3.1 Spectra 3.2 Bigness 3.3 Evaporating Division Algebras 3.4 Spectral Bounds and Capacity 3.5 Problems for Chapter 3   4 Absorbers of Inner Ideals 4.1 Linear Absorbers 4.2 Quadratic Absorbers 4.3 Absorber Nilness 4.4 Problems for Chapter 4   5 Primitivity 5.1 Modularity 5.2 Primitivity 5.3 Semiprimitivity 5.4 Imbedding Nondegenerates in Semiprimitives 5.5 Problems for Chapter 5   6 The Primitive Heart 6.1 Hearts and Spectra 6.2 Primitive Hearts 6.3 Problems for Chapter 6   7 Filters and Ultrafilters 7.1 Filters in General 7.2 Filters from Primes 7.3 Ultimate Filters 7.4 Problems for Chapter 7   8 Ultraproducts 8.1 Ultraproducts 8.2 Examples 8.3 Problems for Chapter 8   9 The Final Argument 9.1 Dichotomy 9.2 The Prime Dichotomy 9.3 Problems for Chapter 9   IV Appendices   A Cohn's Special Theorems A.1 Free Gadgets A.2 Cohn Symmetry A.3 Cohn Speciality A.4 Problems for Appendix A   B Macdonald's Theorem B.1 The Free Jordan Algebra B.2 Identities B.3 Normal Form for Multiplications B.4 The Macdonald Principles B.5 Albert i-Exceptionality B.6 Problems for Appendix B   C Jordan Algebras of Degree 3 C.1 Jordan Matrix Algebras C.2 The general construction C.3 The Freudenthal Construction C.4 The Tits Construction C.5 Albert division algebras C.6 Problems for Appendix C   D The Jacobson--Bourbaki Density Theorem D.1 Semisimple Modules D.2 The Jacobson--Bourbaki Density Theorem   E Hints E.1 Hints for Part II E.2 Hints for Part III E.3 Hints for Part IV   V Indexes   A Index of Collateral Readings A.1 Foundational Readings A.2 Readings in Applications A.3 Historical Perusals   B Pronouncing Index of Names   C Index of Notations C.1 Elemental Products C.2 Multiplication operators C.3 Operations with Idempotents C.4 Alphabetical Constructs $A\rightarrow{\cal F}(A)$ C.5 Symbolic Constructs C.6 Elemental Constructs C.7 Filters on Index Set $X$ of Direct Product $\prod_{x\in X}{\rm A}_x$ C.8 Lists   D Index of Statements   E Index of Definitions

Editorial Reviews

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)