Jordan frame - Knowlpedia

Definition

Let $J$ be a finite-dimensional unital Euclidean Jordan algebra. A Jordan frame is a tuple $(e_1,\ldots,e_r)$ of primitive idempotents such that

e_i\circ e_j=0\quad(i\ne j), \qquad e_1+\cdots+e_r=1_J.

If the order is irrelevant, the same data may be regarded as the set $\{e_1,\ldots,e_r\}$ . The number $r$ is the rank of $J$ .

Matrix example

In the Hermitian Jordan algebra $H_n(\mathbb K)$ , the diagonal matrix units

E_{11},E_{22},\ldots,E_{nn}

form the standard Jordan frame. A frame is the Jordan-algebraic analogue of an orthonormal basis of rank-one projections.

Why labels matter

An automorphism can fix every $e_i$ , or merely preserve the frame as an unordered set while permuting its members. These are different stabilizer conventions. For the Albert algebra, the pointwise stabilizer of a labelled frame is $\mathrm{Spin}(8)$ , whereas the setwise stabilizer can additionally permute the three idempotents; see the frame-stabilizer theorem.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994, Chapter IV, §2. Publisher record.
Kevin McCrimmon, A Taste of Jordan Algebras, Springer, 2004, §13.1. Publisher record.