Spin factor J(R^3) - Knowlpedia

The spin-factor Jordan algebra $J(\mathbb R^{3})$ is $\mathbb R\oplus\mathbb R^{3}$ with product

(a,u)\circ(b,v)=(ab+\textstyle\sum_{i=1}^{3}u_iv_i,\;av+bu).

The unit is $(1,0)$ , and the dimension over $\mathbb R$ is $4$ .

Dimension parameter

The parameter $3$ counts the vector summand, not the entire algebra. The full dimension is one larger. It is isomorphic to $\operatorname{Herm}_2(\mathbb C)$ . Writing a Hermitian matrix as $\left(\begin{smallmatrix}a+t&z\\\overline z&a-t\end{smallmatrix}\right)$ gives the spin coordinates $(a,(t,z))$ .

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.