Non-associative algebras

Jordan algebra

In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: 1. * (commutative law) 2. * (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 that a Jordan algebra is power-associative, meaning that is independent of how we parenthesize this expression. They also imply that 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 , the operations of multiplying by powers all commute. Jordan algebras were first introduced by Pascual Jordan 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 Abraham Adrian Albert, who began the systematic study of general Jordan algebras. (Wikipedia).

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