Spin factor J(C^1) - Knowlpedia

The spin-factor Jordan algebra $J(\mathbb C^{1})$ is $\mathbb C\oplus\mathbb C^{1}$ with product

(a,u)\circ(b,v)=(ab+\textstyle\sum_{i=1}^{1}u_iv_i,\;av+bu).

The unit is $(1,0)$ , and the dimension over $\mathbb C$ is $2$ . The displayed form is complex bilinear, with no complex conjugates.

Dimension parameter

The parameter $1$ counts the vector summand, not the entire algebra. The full dimension is one larger. The map $(a,u)\mapsto(a+u,a-u)$ identifies this algebra with $\mathbb C\times\mathbb C$ , so this case is not simple.

Product check

Write $B(u,v)=\sum_i u_iv_i$ . Then $(a,u)^2=(a^2+B(u,u),2au)$ . Substituting this expression into $x^2\circ(x\circ y)=x\circ(x^2\circ y)$ gives equal scalar and vector components using only symmetry and bilinearity of $B$ ; this verifies the Jordan identity. Unit-preserving and arbitrary Jordan homomorphisms therefore give separate catalogue categories on this object.

References

John C. Baez, The Octonions, Section 3.3, spin-factor product and Hermitian two-by-two identification.