Herm_1(O) complexification

The Herm_1(O) complexification is the $\mathbb C$ -vector space

\operatorname{Herm}_{1}((\mathbb O\otimes_{\mathbb R}\mathbb C))=\{X\in M_{1}((\mathbb O\otimes_{\mathbb R}\mathbb C)):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. That conjugation is extended complex-linearly, so it fixes the newly adjoined complex scalars.

Coordinates and dimension

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

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 $\mathbb C1$ , so $[a]\mapsto a$ identifies this one-dimensional Jordan algebra with $\mathbb C$ .

Scalar convention

This is the complexification of the corresponding real Hermitian Jordan algebra. Its complex-bilinear product and real restriction give different category views; it is not the space of ordinary self-adjoint matrices on a complex Hilbert space.

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.