Primitive Jordan idempotent

Definition

A nonzero Jordan idempotent $e$ is primitive if there do not exist nonzero orthogonal idempotents $e_1,e_2$ such that

e=e_1+e_2.

For finite-dimensional Euclidean Jordan algebras, “primitive” is also often called minimal.

Rank-one meaning

In a simple Euclidean Jordan algebra, every idempotent is a sum of mutually orthogonal primitive idempotents. The number of summands is its rank. Thus the primitive idempotents are exactly the rank-one idempotents.

In $H_n(\mathbb K)$ , for $\mathbb K=\mathbb R,\mathbb C,\mathbb H$ , and also for the allowed octonionic case $H_3(\mathbb O)$ , the primitive idempotents have matrix trace $1$ . This trace criterion depends on the standard normalization and should not be taken as the definition in an arbitrary Jordan algebra.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994, Chapter III, §§1–2. Publisher record.
Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000, §5.8. Publisher record.