Jordan algebra

In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: In abstract algebra, a Jordan algebra is a 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: ( x m y ) x n = x m ( y x n ) {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 Abraham Adrian 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: This defines a Jordan algebra A+, and we call these Jordan algebras, as well as any subalgebras of these Jordan algebras, special Jordan algebras. All other Jordan algebras are called exceptional Jordan algebras. The Shirshov–Cohn theorem states that any Jordan algebra with two generators is special. Related to this, Macdonald's theorem states that any polynomial in three variables, which has degree one in one of the variables, and which vanishes in every special Jordan algebra, vanishes in every Jordan algebra. If (A, σ) is an associative algebra with an involution σ, then if σ(x)=x and σ(y)=y it follows that Thus the set of all elements fixed by the involution (sometimes called the hermitian elements) form a subalgebra of A+ which is sometimes denoted H(A,σ). 1. The set of self-adjoint real, complex, or quaternionic matrices with multiplication