Editing Jordan algebra (section)

{{Short description|1=Not-necessarily-associative commutative algebra satisfying (𝑥𝑦)𝑥²=𝑥(𝑦𝑥²)}}
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In [[abstract algebra]], a '''Jordan algebra''' is a [[nonassociative algebra]] [[algebra over a field|over a field]] whose [[Product (mathematics)|multiplication]] satisfies the following axioms:

# <math>xy = yx</math> ([[commutative]] law)
# <math>(xy)(xx) = x(y(xx))</math> ({{visible anchor|Jordan identity}}).

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 name=Jacobson68p35>{{harvnb|Jacobson|1968|pp=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 name=Jacobson68p35/> 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 <math>x</math>, the operations of multiplying by powers <math>x^n</math> all commute.
 
Jordan algebras were introduced by {{harvs|txt|authorlink=Pascual Jordan|first=Pascual |last=Jordan|year=1933}} in an effort to formalize the notion of an algebra of [[observable]]s in [[quantum electrodynamics]]. It was soon shown that the algebras were not useful in this context, however they have since found many applications in mathematics.<ref>{{Cite journal |last=Dahn |first=Ryan |date=2023-01-01 |title=Nazis, émigrés, and abstract mathematics |journal=Physics Today |volume=76 |issue=1 |pages=44–50 |doi= 10.1063/PT.3.5158|issn=|doi-access=free |bibcode=2023PhT....76a..44D }}</ref> The algebras 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.