A taste of Jordan algebras
type: Item
instance of: scholarly article
title: A taste of Jordan algebras
zbMATH Open document ID: 1044.17001
author: Kevin McCrimmon
published in: Universitext
publication date: 2004-02-17T00:00:00Z
review text: 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 \textit{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 \textit{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 for…
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 17-01
Mathematics Subject Classification ID: 17Cxx
zbMATH DE Number: 2043992
zbMATH Keywords: Jordan algebra
zbMATH Keywords: special Jordan algebras
zbMATH Keywords: exceptional Jordan algebras
zbMATH Keywords: nondegenerate Jordan algebras
zbMATH Keywords: prime Jordan algebras
zbMATH Keywords: simple Jordan algebras
Wikidata QID: Q29303919
MaRDI profile type: Publication
full work available at URL: b97489
OpenAlex ID: W1598049090
DOI: 10.1007/B97489
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description: scientific article; zbMATH DE number 2043992