Special and exceptional Jordan algebras

Definition

A Jordan algebra $J$ is special if it is isomorphic to a Jordan subalgebra of $A^+$ for some associative algebra $A$ , where

x\circ y=\frac12(xy+yx).

A Jordan algebra that is not special is exceptional.

What the definition does and does not say

“Special” means embeddable into a symmetrized associative algebra; it does not mean that the Jordan product itself is associative. Likewise, “exceptional” is a property, not the name of a unique algebra. Nevertheless, the phrase the exceptional Jordan algebra conventionally denotes the 27-dimensional real Albert algebra $\mathfrak h_3(\mathbb O)$ .

The embedding in the definition need not preserve units unless a unital embedding is explicitly required. This distinction is useful when Jordan algebras occur as matrix corners.

Fundamental examples

The self-adjoint part of an associative real or complex matrix algebra is special. This includes $\mathfrak h_n(\mathbb R)$ , $\mathfrak h_n(\mathbb C)$ , and $\mathfrak h_n(\mathbb H)$ with their symmetrized matrix product.

The real algebra $\mathfrak h_3(\mathbb O)$ is exceptional: it satisfies the Jordan identity but admits no embedding into $A^+$ for any associative algebra $A$ . Its existence is the exceptional case in the classification of finite-dimensional simple Euclidean Jordan algebras.

Identities

Every special Jordan algebra satisfies all polynomial identities inherited from symmetrized associative multiplication. Exceptional Jordan algebras are detected by identities that hold in all special Jordan algebras but fail in the exceptional case; the first classical example is the Glennie identity.

References

Pascual Jordan, John von Neumann, and Eugene Wigner, “On an Algebraic Generalization of the Quantum Mechanical Formalism,” Annals of Mathematics 35 (1934), 29–64. JSTOR record.
Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000. Publisher record.