A Taste of Jordan Algebras

On several occasions I and colleagues have found ourselves teaching a o- semester course for students at the second year of graduate study in ma- ematics who want to gain a general perspective on Jordan algebras, their structure, and their role in mathematics, or want to gain direct experience with nonassociative algebra. These students typically have a solid grounding in ?rst-year graduate algebra and the Artin-Wedderburn theory of assoc- tive algebras, and a few have been introduced to Lie algebras (perhaps even Cayley algebras, in an o?hand way), but otherwise they have not seen any nonassociative algebras. Most of them will not go on to do research in non- sociative algebra, so the course is not primarily meant to be a training or breeding ground for research, though the instructor often hopes that one or two will be motivated to pursue the subject further. This text is meant to serve as an accompaniment to such a course. It is designed ?rst and foremost to be read by students on their own without assistance by a teacher. It is a direct mathematical conversation between the author and a reader whose mind (as far as nonassociative algebra goes) is a tabula rasa. In keeping with the tone of a private conversation, I give more heuristicandexplanatorycommentthanisusualingraduatetextsatthislevel (pep talks, philosophical pronouncements on the proper way to think about certain concepts, historical anecdotes, mention of some mathematicians who have contributed to our understanding of Jordan algebras, etc.

List of contents

A Colloquial Survey of Jordan Theory.- A Colloquial Survey of Jordan Theory.- A Historical Survey of Jordan Structure Theory.- Jordan Algebras in Physical Antiquity: The Search for an Exceptional Setting for Quantum Mechanics.- Jordan Algebras in the Algebraic Renaissance: Finite-Dimensional Jordan Algebras over Algebraically Closed Fields.- Jordan Algebras in the Enlightenment: Finite-Dimensional Jordan Algebras over General Fields.- The Classical Theory: Jordan Algebras with Minimum Condition.- The Final Classical Formulation: Algebras with Capacity.- The Classical Methods: Cherchez les Division Idempotents.- The Russian Revolution: 1977-1983.- Zel'manov's Exceptional Methods.- The Classical Theory.- The Category of Jordan Algebras.- The Category of Alternative Algebras.- Three Special Examples.- Jordan Algebras of Cubic Forms.- Two Basic Principles.- Inverses.- Isotopes.- Peirce Decomposition.- Off-Diagonal Rules.- Peirce Consequences.- Spin Coordinatization.- Hermitian Coordinatization.- Multiple Peirce Decompositions.- Multiple Peirce Consequences.- Hermitian Symmetries.- The Coordinate Algebra.- Jacobson Coordinatization.- Von Neumann Regularity.- Inner Simplicity.- Capacity.- Herstein-Kleinfeld-Osborn Theorem.- Osborn's Capacity 2 Theorem.- Classical Classification.- Zel'manov's Exceptional Theorem.- The Radical.- Begetting and Bounding Idempotents.- Bounded Spectra Beget Capacity.- Absorbers of Inner Ideals.- Primitivity.- The Primitive Heart.- Filters and Ultrafilters.- Ultraproducts.- The Final Argument.

About the author

 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.

Summary

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. 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.

Additional text

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’sbook 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)

Report

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'sbook 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)

Product details

• Authors: Kevin Mccrimmon
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 27.10.2010
• EAN: 9781441930033
• ISBN: 978-1-4419-3003-3
• No. of pages: 563
• Illustrations: XXV, 563 p.
• Series: Universitext; Universitext
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra; Algebra, C, Matrix, Mathematics and Statistics, Linear Algebra, matrices, quadratic form