Jordan superalgebra

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

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 that a Jordan algebra is power-associative and satisfies the following generalization of the Jordan identity: ${\displaystyle (x^{m}y)x^{n}=x^{m}(yx^{n})}$ for all positive integers m and n.

Jordan algebras were first introduced by Pascual Jordan (1933) to formalize the notion of an algebra of observables in quantum mechanics. They were originally called "r-number systems", but were renamed "Jordan algebras" by Albert (1946), who began the systematic study of general Jordan algebras.

Given an associative algebra A (not of characteristic 2), one can construct a Jordan algebra A⁺ using the same underlying addition vector space. Notice first that an associative algebra is a Jordan algebra if and only if it is commutative. If it is not commutative we can define a new multiplication on A to make it commutative, and in fact make it a Jordan algebra. The new multiplication x ∘ y is the Jordan product:

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