The Arithmetics of Quadratic Jordan Algebras
The Arithmetics of Quadratic Jordan Algebras

Author: Michel L. Racine

Publisher: American Mathematical Soc.

Published: 1973

Total Pages: 134

ISBN-13: 0821818368

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The first step in obtaining an arithmetic theory for finite dimensional quadratic Jordan algebras over the quotient field of a Dedekind ring is the determination of maximal orders. This is the main concern of this paper. Jordan analogues of some of the first theorems in classical associative arithmetic are obtained. For special quadratic Jordan algebras, the problem of determining maximal orders is reduced to arithmetic questions in quadratic forms and associative algebras with involution. The number of isomorphism classes of maximal orders is computed for most central simple quadratic Jordan algebras over a local field. In the process, previous results of Knebusch are obtained in a uniform fashion and are extended to the case of algebras over fields of characteristic 2 and 3.

A Taste of Jordan Algebras
A Taste of Jordan Algebras

Author: Kevin McCrimmon

Publisher: Springer Science & Business Media

Published: 2006-05-29

Total Pages: 584

ISBN-13: 0387217967

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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. Jordan algebras crop up in many surprising settings, and find application to a variety of mathematical areas. No knowledge is required beyond standard first-year graduate algebra courses.

Structure and Representations of Jordan Algebras
Structure and Representations of Jordan Algebras

Author: Nathan Jacobson

Publisher: American Mathematical Soc.

Published: 1968-12-31

Total Pages: 464

ISBN-13: 082184640X

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The theory of Jordan algebras has played important roles behind the scenes of several areas of mathematics. Jacobson's book has long been the definitive treatment of the subject. It covers foundational material, structure theory, and representation theory for Jordan algebras. Of course, there are immediate connections with Lie algebras, which Jacobson details in Chapter 8. Of particular continuing interest is the discussion of exceptional Jordan algebras, which serve to explain the exceptional Lie algebras and Lie groups. Jordan algebras originally arose in the attempts by Jordan, von Neumann, and Wigner to formulate the foundations of quantum mechanics. They are still useful and important in modern mathematical physics, as well as in Lie theory, geometry, and certain areas of analysis.

The Minnesota Notes on Jordan Algebras and Their Applications
The Minnesota Notes on Jordan Algebras and Their Applications

Author: Max Koecher

Publisher: Springer

Published: 2006-11-14

Total Pages: 180

ISBN-13: 3540484027

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This volume contains a re-edition of Max Koecher's famous Minnesota Notes. The main objects are homogeneous, but not necessarily convex, cones. They are described in terms of Jordan algebras. The central point is a correspondence between semisimple real Jordan algebras and so-called omega-domains. This leads to a construction of half-spaces which give an essential part of all bounded symmetric domains. The theory is presented in a concise manner, with only elementary prerequisites. The editors have added notes on each chapter containing an account of the relevant developments of the theory since these notes were first written.

Jordan Algebras and Algebraic Groups
Jordan Algebras and Algebraic Groups

Author: Tonny A. Springer

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 181

ISBN-13: 3642619703

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From the reviews: "This book presents an important and novel approach to Jordan algebras. [...] Springer's work will be of service to research workers familiar with linear algebraic groups who find they need to know something about Jordan algebras and will provide Jordan algebraists with new techniques and a new approach to finite-dimensional algebras over fields." American Scientist