Jordan algebra

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In mathematics, a Jordan algebra is defined in abstract algebra as a (usually nonassociative) 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.

[(x .y) = ].

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.

A Jordan ring is a generalisation of Jordan algebras, requiring only that the Jordan ring be over general ring rather than a field. Alternatively one can define a Jordan ring as a commutative nonassociative ring that respects the Jordan identity.

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.) Because most other exceptional Jordan algebras are constructed using this one, it is often referred to as "the" exceptional Jordan algebra. It is also known as the Albert algebra.

A (possibly nonassociative) algebra over the real numbers is said to be formally real if it satisfies the property that a sum of n squares can only vanish if each one vanishes individually. In 1932, Pascual Jordan attempted to axiomatize quantum theory by saying that the algebra of observables of any quantum system should be a formally real algebra which is commutative ([xy = yx]) and power-associative (the associative law holds for any parenthesized string of [x]'s, so that powers of any element [x] are unambiguously defined). He proved that any such algebra is what we now call a Jordan algebra. Not every Jordan algebra is formally real, but in 1934, with Eugene Wigner and John von Neumann, Jordan classified the formally real Jordan algebras. Every formally real Jordan algebra can be written as a direct sum of so-called simple ones, which are not themselves direct sums in a nontrivial way. The simple formally real Jordan algebras come in 4 infinite families, together with one exceptional case:

• The Jordan algebra of [n\times n] self-adjoint real matrices, as above.
• The Jordan algebra of [n\times n] self-adjoint complex matrices, as above.
• The Jordan algebra of [n\times n] self-adjoint quaternionic matrices. as above.
• The Jordan algebra freely generated by [R^n] with the relations

[x^2 = \langle x, x\rangle ]

where the right-hand side is defined using the usual inner product on [R^n]. This is the so-called spin factor.

• The Jordan algebra of 3×3 self-adjoint octonionic matrices, as above - the exceptional Jordan algebra.

Of these possibilities, so far it appears that nature makes use only of the n×n complex matrices as algebra of observables. However, the spin factors play a role in special relativity, and all the formally real Jordan algebras are related to projective geometry.

References

• John Baez, The Octonions, Section 3: Projective Octonionic Geometry, [Bull. Amer. Math. Soc. 39 (2002), 145-205]. Online HTML version at http://math.ucr.edu/home/baez/octonions/node8.html.

 

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