Simple Euclidean Jordan algebra

Definition

A simple Euclidean Jordan algebra is a nonzero Euclidean Jordan algebra $J$ whose only Jordan ideals are $0$ and $J$ . A Jordan ideal is a linear subspace $I\subseteq J$ satisfying $J\circ I\subseteq I$ .

Classification

Up to isomorphism, the simple Euclidean Jordan algebras are:

the rank-one algebra $\mathbb R$ ;
$\mathfrak h_n(\mathbb R)$ , $n\geq3$ ;
$\mathfrak h_n(\mathbb C)$ , $n\geq3$ , regarded as a real algebra;
$\mathfrak h_n(\mathbb H)$ , $n\geq3$ ;
the spin factors $\mathbb R\oplus V$ with $\dim V\geq2$ ;
the exceptional Jordan algebra $\mathfrak h_3(\mathbb O)$ .

The restriction $n\geq3$ prevents duplication: the degree-two matrix algebras $\mathfrak h_2(\mathbb R)$ , $\mathfrak h_2(\mathbb C)$ , and $\mathfrak h_2(\mathbb H)$ are spin factors. So is $\mathfrak h_2(\mathbb O)$ , although octonionic Hermitian matrices do not produce a matrix family in arbitrary degree.

Decomposition theorem

Every Euclidean Jordan algebra decomposes uniquely up to permutation as an orthogonal direct sum of simple ideals. The one-dimensional factor is $\mathfrak h_1(\mathbb R)$ and is sometimes left implicit as the degenerate rank-one case. Conventions for the lowest-dimensional spin factors should therefore always be checked.

All factors except $\mathfrak h_3(\mathbb O)$ are special. The Albert algebra is the unique exceptional factor in the Euclidean classification.

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.
Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Publisher record.