Jordan Algebra
A nonassociative algebra named after physicist Pascual Jordan which satisfies
| xy=yx |
(1)
|
and
| (xx)(xy)=x((xx)y)). |
(2)
|
The latter is equivalent to the so-called Jordan identity
| (xy)x^2=x(yx^2) |
(3)
|
(Schafer 1996, p. 4). An associative algebra A with associative product xy can be made into a Jordan algebra A^+ by the Jordan product
| x·y=1/2(xy+yx). |
(4)
|
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 AlgebraExplore with Wolfram|Alpha
References
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.Referenced on Wolfram|Alpha
Jordan AlgebraCite this as:
Weisstein, Eric W. "Jordan Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/JordanAlgebra.html