Jordan Algebra (original) (raw)
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A nonassociative algebra named after physicist Pascual Jordan which satisfies
 |
(1) |
|---|
and
 |
(2) |
|---|
The latter is equivalent to the so-called Jordan identity
 |
(3) |
|---|
(Schafer 1996, p. 4). An associative algebra 
with associative product 
can be made into a Jordan algebra 
by the Jordan product
 |
(4) |
|---|
Division by 2 gives the nice identity 
, but it must be omitted in characteristic 
.
Unlike the case of a Lie algebra, not every Jordan algebra is isomorphic to a subalgebra of some 
. 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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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.
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Cite this as:
Weisstein, Eric W. "Jordan Algebra." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JordanAlgebra.html