Jordan algebra: Difference between revisions

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(explained more conceptual way to define Jordan algebras)
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The product of two elements ''x'' and ''y'' in a Jordan algebra is also denoted ''x'' ∘ ''y'', particularly to avoid confusion with the product of a related [[associative algebra]].
 
The axioms imply<ref>Jacobson (1968), p.35–36, specifically remark before (56) and theorem 8.</ref> that a Jordan algebra is [[power-associative]], meaning that <math>x^n = x \cdots x</math> is independent of how we parenthesize this expression. They also imply<ref>Jacobson (1968), p.35–36, specifically remark before (56) and theorem 8.</ref> that <math>x^m (x^n y) = x^n(x^m y)</math> for all positive integers ''m'' and ''n''. Thus, we may equivalently define a Jordan algebra to be a commutative, power-associative algebra such that for any element ''x'', the operations of multiplying by powers <math>x^n</math> all commute.
Jordan algebras were first introduced by {{harvs|txt|authorlink=Pascual Jordan|first=Pascual |last=Jordan|year=1933}} to formalize the notion of an algebra of [[observable]]s in [[quantum mechanics]]. They were originally called "r-number systems", but were renamed "Jordan algebras" by {{harvs|txt|authorlink=Abraham Adrian Albert|last=Albert|first=Abraham Adrian|year=1946}}, who began the systematic study of general Jordan algebras.