Arvid Ehrlén: Jordan Algebras: Definitions and examples

Abstract:

A Jordan algebra is a nonassociative, commutative algebra that satisfies a weaker form of associativity known as the Jordan identity. We go through some basic properties and look at of some the most important classes of Jordan algebras: full, Hermitian, and quadratic factors. We formalize the notion of composition algebras which appear naturally as coordinates of certain Jordan matrix algebras. We state Macdonald’s theorem and explore some of its important consequences, and give a brief exposition of Peirce decompositions used in studying the structure of Jordan algebras. Finally we sketch the development of the structure theory for Jordan algebras since their inception in 1933.