Jordan algebra

In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms:

1. ${\displaystyle xy=yx}$ (commutative law)
2. ${\displaystyle (xy)(xx)=x(y(xx))}$ (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^[1] that a Jordan algebra is power-associative, meaning that ${\displaystyle x^{n}=x\cdots x}$ is independent of how we parenthesize this expression. They also imply^[1] that ${\displaystyle x^{m}(x^{n}y)=x^{n}(x^{m}y)}$ 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 ${\displaystyle x}$, the operations of multiplying by powers ${\displaystyle x^{n}}$ all commute.

Jordan algebras were introduced by Pascual Jordan (1933) in an effort to formalize the notion of an algebra of observables 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.^[2] The algebras were originally called "r-number systems", but were renamed "Jordan algebras" by Abraham Adrian Albert (1946), who began the systematic study of general Jordan algebras.