In mathematics, a Jordan algebra is defined in abstract algebra as an algebra over a field with multiplication satisfying the following axioms:

1. xy = yx (commutative law)
2. (xy)(xx) = x(y(xx)) (Jordan identity)

Jordan algebras were first introduced by Pascual Jordan in quantum mechanics.

Given an associative algebra A (not of characteristic 2), one can construct a Jordan algebra A ⁺ with the same underlying addition, and a new multiplication (x.y) as follows.

.

If A has an involution, then the involution fixes elements of the form

(xy + yx) / 2.

Thus the set of all elements fixed by the involution form a subalgebra of A ⁺ .

A Jordan algebra that is isomorphic to an algebra of the form A ⁺ is known as a special Jordan algebra. Otherwise it is an exceptional Jordan algebra.

Examples

• The set of self-adjoint real, complex, or quaternionic matrices with multiplication

(xy + yx) / 2

form a special Jordan algebra.

• The set of 3×3 self-adjoint matrices over the octonions again with multiplication

(xy + yx) / 2.

Despite the similarity to the previous example, this is an exceptional Jordan algebra. (The octonions are not an associative algebra.)