Jordan Algebra

A nonassociative algebra named after physicist Pascual Jordan which satisfies




The latter is equivalent to the so-called Jordan identity


(Schafer 1996, p. 4). An associative algebra A with associative product xy can be made into a Jordan algebra A^+ by the Jordan product


Division by 2 gives the nice identity x·x=xx, but it must be omitted in characteristic p=2.

Unlike the case of a Lie algebra, not every Jordan algebra is isomorphic to a subalgebra of some A^+. Jordan algebras which are isomorphic to a subalgebra are called special Jordan algebras, while those that are not are called exceptional Jordan algebras.

See also

Anticommutator, Nonassociative Algebra

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Jacobson, N. Structure and Representations of Jordan Algebras. Providence, RI: Amer. Math. Soc., 1968.Jordan, P. "Über eine Klasse nichtassoziativer hyperkomplexer Algebren." Nachr. Ges. Wiss. Göttingen, 569-575, 1932.Schafer, R. D. An Introduction to Nonassociative Algebras. New York: Dover, pp. 4-5, 1996.

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Jordan Algebra

Cite this as:

Weisstein, Eric W. "Jordan Algebra." From MathWorld--A Wolfram Web Resource.

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