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**A taste of Jordan algebras.**
*(English)*
Zbl 1044.17001

Universitext. New York, NY: Springer (ISBN 0-387-95447-3/hbk). xxvi, 562 p. (2004).

As mentioned in the preface, “this book tells the story of one aspect of Jordan structure theory: the origin of the theory in an attempt by quantum physicists to find algebraic systems more general than Hermitian matrices, and ending with the surprising proof by Efim Zel’manov that there is really only one such system, the 27-dimensional Albert algebra, much too small to accommodate quantum mechanics”. The author proceeds to tell this fascinating story with a lovely and lively style, as if he were talking to the students.

Anyone curious about Jordan algebras will enjoy the Introduction, entitled A Colloquial Survey of Jordan Theory, where Jordan algebras are introduced and put in perspective, stressing their links with Lie Algebras and Groups, with Geometry and Analysis.

The main body of the book consists of three parts which can be read independently. Part I is devoted to A Historical Survey of Jordan Structure Theory, where an overview of this theory is given in its historical context, starting with its origin in the work of Jordan, von Neumann and Wigner, following then with the work in the late 40s by Albert, Jacobson and others on finite-dimensional Jordan algebras, the theory of Jordan algebras with minimum condition on inner ideals in the 60s and the classification of nondegenerate Jordan algebras with capacity, and ending with a survey of the results obtained by the School of Novosibirsk in the late 70s and early 80s, with a sketch of the methods used by Zel’manov. No complete proofs are given in this part.

Part II is devoted to the Classical Theory. Here a detailed treatment is given of Jacobson’s classical structure theory for nondegenerate Jordan algebras with capacity. The author sticks to linear Jordan algebras over rings of scalars containing \({1\over 2}\), but the quadratic point of view is present everywhere.

Part III gives a full treatment of Zel’manov’s results asserting that the only i-exceptional prime Jordan algebras are forms of Albert algebras, stressing the importance of primitive algebras over big algebraically closed fields and the use of ultraproducts. It is the first place where all this material appears in such detail.

The book also includes five appendices: Cohn’s Special Theorems, Macdonald’s Theorem, Jordan Algebras of Degree \(3\), The Jacobson-Bourbaki Density Theorem, and Hints (to selected exercises).

This book does not serve as a substitute for previous monographs on the subject, as all of them contain material not covered here. Instead, it concentrates explicitly on the structure theory of linear Jordan algebras and it does a wonderful job at this. It can be used in many different ways to teach graduate courses and also for self-study, but it is much more than a textbook. Researchers in this area will find here a detailed exposition, mathematicians in other areas will get the opportunity to have a gentle introduction to the main ideas and results and, of course, graduate students will have at their disposal a very well organized, motivating and engaging textbook.

Anyone curious about Jordan algebras will enjoy the Introduction, entitled A Colloquial Survey of Jordan Theory, where Jordan algebras are introduced and put in perspective, stressing their links with Lie Algebras and Groups, with Geometry and Analysis.

The main body of the book consists of three parts which can be read independently. Part I is devoted to A Historical Survey of Jordan Structure Theory, where an overview of this theory is given in its historical context, starting with its origin in the work of Jordan, von Neumann and Wigner, following then with the work in the late 40s by Albert, Jacobson and others on finite-dimensional Jordan algebras, the theory of Jordan algebras with minimum condition on inner ideals in the 60s and the classification of nondegenerate Jordan algebras with capacity, and ending with a survey of the results obtained by the School of Novosibirsk in the late 70s and early 80s, with a sketch of the methods used by Zel’manov. No complete proofs are given in this part.

Part II is devoted to the Classical Theory. Here a detailed treatment is given of Jacobson’s classical structure theory for nondegenerate Jordan algebras with capacity. The author sticks to linear Jordan algebras over rings of scalars containing \({1\over 2}\), but the quadratic point of view is present everywhere.

Part III gives a full treatment of Zel’manov’s results asserting that the only i-exceptional prime Jordan algebras are forms of Albert algebras, stressing the importance of primitive algebras over big algebraically closed fields and the use of ultraproducts. It is the first place where all this material appears in such detail.

The book also includes five appendices: Cohn’s Special Theorems, Macdonald’s Theorem, Jordan Algebras of Degree \(3\), The Jacobson-Bourbaki Density Theorem, and Hints (to selected exercises).

This book does not serve as a substitute for previous monographs on the subject, as all of them contain material not covered here. Instead, it concentrates explicitly on the structure theory of linear Jordan algebras and it does a wonderful job at this. It can be used in many different ways to teach graduate courses and also for self-study, but it is much more than a textbook. Researchers in this area will find here a detailed exposition, mathematicians in other areas will get the opportunity to have a gentle introduction to the main ideas and results and, of course, graduate students will have at their disposal a very well organized, motivating and engaging textbook.

Reviewer: Alberto Elduque (Zaragoza)

### MSC:

17-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras |

17Cxx | Jordan algebras (algebras, triples and pairs) |