Taste of Jordan Algebras

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.

• 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.

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)