Symmetric n-by-n complex Jordan algebra

For integers $n\geq1$ , The Symmetric n-by-n complex Jordan algebra is the $\mathbb C$ -vector space

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

with unit $I_{n}$ . 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

There are $n$ diagonal scalar coordinates and $n(n-1)/2$ independent coefficient-algebra entries. Thus the dimension over $\mathbb C$ is $n+1n(n-1)/2$ . Here the coefficient involution is the identity, so the condition is $X^T=X$ , not $\overline X^T=X$ with ordinary complex conjugation.

Jordan identity and range

The fixed subspace is closed under $X\circ Y$ , because $(XY)^*=Y^*X^*$ . Associativity of the ambient matrix algebra proves the Jordan identity, so the construction works in every positive size.

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.