A Taste of Jordan Algebras

1. The Classical Theory

   1. Off-Diagonal Rules

      Pages 247-252

   2. Peirce Consequences

      Pages 253-258

   3. Spin Coordinatization

      Pages 259-267

   4. Hermitian Coordinatization

      Pages 268-277

   5. Multiple Peirce Decompositions

      Pages 278-291

   6. Multiple Peirce Consequences

      Pages 292-300

   7. Hermitian Symmetries

      Pages 301-307

   8. The Coordinate Algebra

      Pages 308-311

   9. Jacobson Coordinatization

      Pages 312-317

   10. Von Neumann Regularity

       Pages 318-324

   11. Inner Simplicity

       Pages 325-329

   12. Capacity

       Pages 330-334

   13. Herstein-Kleinfeld-Osborn Theorem

       Pages 335-347

   14. Osborn’s Capacity 2 Theorem

       Pages 348-355

   15. Classical Classification

       Pages 356-358

2. Zel’manov’s Exceptional Theorem

   1. Front Matter

      Pages 359-361

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   2. The Radical

      Pages 362-380

   3. Begetting and Bounding Idempotents

      Pages 381-387

   4. Bounded Spectra Beget Capacity

      Pages 388-396

   5. Absorbers of Inner Ideals

      Pages 397-409

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.

• algebra
• associative algebra
• lie algebra
• linear algebra
• matrices
• matrix
• quadratic form

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)

• Department of Mathematics, University of Virginia, Charlottesville, USA

  Kevin McCrimmon

 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.

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