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V - Jordan algebras

Published online by Cambridge University Press:  07 November 2024

Skip Garibaldi
Affiliation:
Institute for Defense Analyses, USA
Holger P. Petersson
Affiliation:
FernUniversität in Hagen
Michel L. Racine
Affiliation:
University of Ottawa
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Summary

This chapter will develop from scratch the elementary theory of (quadratic) Jordan algebras over commutative rings. After a brief account of linear Jordan algebras and their most rudimentary properties over rings in which 2 is invertible, we proceed to para-quadratic algebras, which play the same role in the quadratic setting as is played by ordinary nonassociative algebras in the linear setting. Quadratic Jordan algebras are introduced. We derive a wide range of useful identities and acquaint the reader with the standard examples of special Jordan algebras, namely the Jordan algebra constructed from a unital associative algebra, from an associative algebra with involution, or from a pointed quadratic module. After a brief interlude concerning a peculiar class of two-variable identities, we investigate what are arguably the most important concepts of the theory: invertibility, isotopy, and the structure group. The chapter concludes with a concise description of the Peirce decomposition relative to an idempotent, and also relative to a complete orthogonal system of idempotents.

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Albert Algebras over Commutative Rings
The Last Frontier of Jordan Systems
, pp. 252 - 316
Publisher: Cambridge University Press
Print publication year: 2024

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