Jordan algebra

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)

The product on a Jordan algebra is also denoted $x\bullet y$, particularly to avoid confusion with the product of a related associative algebra.

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

Special 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 $x\bullet y$ is as follows:

$x\bullet y={xy+yx \over 2}.$

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.

Hermitian Jordan algebras

If (A, σ) is an associative algebra with an involution σ, then the involution fixes elements in A of the form

$(xy+yx)/2.$

Thus the set of all elements fixed by the involution (i.e. the hermitian elements) form a subalgebra of $A^{+}$ which is denoted by H(A,σ).

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.) Since over the real numbers this is the only exceptional Jordan algebras, it is often referred to as "the" exceptional Jordan algebra. It was the first example of an 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.

Classification

Not every Jordan algebra is formally real, but in 1934, with Eugene Wigner and John von Neumann, Jordan classified the finite dimensional 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. In finite dimensions, 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.

Infinite dimensions

In 1979, Efim Zelmanov classified infinite dimensional simple (and prime) Jordan algebras. They are:

• Hermitian type
• Clifford type
• Albert algebras, which are the only exceptional ones (in particular, are finite dimensional: 27).

Jordan rings

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

References

• Kevin McCrimmon, A Taste of Jordan Algebras

• John C. 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.