Herm_1(O_s) Jordan algebra

The Herm_1(O_s) Jordan algebra is the $\mathbb R$ -vector space

\operatorname{Herm}_{1}(\mathbb O_s)=\{X\in M_{1}(\mathbb O_s):X^*=X\}, \qquad X\circ Y=\tfrac12(XY+YX),

with unit $I_{1}$ . Here $X^*=\overline X^T$ , using the standard conjugation of the coefficient algebra.

Coordinates and dimension

The diagonal contributes $1$ scalar coordinates, and the off-diagonal pairs contribute $0$ , giving dimension $1$ over $\mathbb R$ .

Jordan identity and range

Sizes one, two, and three satisfy the Jordan identity. The symbolic family in this catalogue is restricted to those sizes; ordinary Hermitian octonionic matrices in arbitrary size are not a uniform Jordan-algebra family. Conjugation-fixed coefficients are exactly $\mathbb R1$ , so $[a]\mapsto a$ identifies this one-dimensional Jordan algebra with $\mathbb R$ .

References

Holger P. Petersson, A survey on Albert algebras, Sections 2.2, 2.5–2.7 and 7.8; printed pages 2–3 and 19.