Spin-factor Jordan algebra

Definition

Let $V$ be a finite-dimensional real inner-product space. The spin factor $J(V)$ is the real vector space $\mathbb R\oplus V$ with product

(\lambda,u)\circ(\mu,v) =\bigl(\lambda\mu+\langle u,v\rangle,\lambda v+\mu u\bigr).

Its unit is $e=(1,0)$ . With inner product $\langle(\lambda,u),(\mu,v)\rangle_J =\lambda\mu+\langle u,v\rangle$ , it is a Euclidean Jordan algebra.

Spectral data

For $u\neq0$ , the element $x=(\lambda,u)$ has decomposition

x=(\lambda+\|u\|)c_+ +(\lambda-\|u\|)c_-, \qquad c_\pm=\frac12\left(1,\pm\frac{u}{\|u\|}\right).

Thus every nontrivial spin factor has rank $2$ , Jordan trace $2\lambda$ , and determinant $\lambda^2-\|u\|^2$ . Its cone of squares is the Lorentz cone $\{(\lambda,u):\lambda\geq\|u\|\}$ .

Simplicity and symmetries

The spin factor is simple when $\dim V\geq2$ . For $\dim V=1$ , it is isomorphic to $\mathbb R\oplus\mathbb R$ with coordinatewise product and is not simple. Every orthogonal transformation of $V$ extends to an automorphism, and all automorphisms arise this way. Hence $\operatorname{Aut}(J(V))\cong O(V)$ .

Hermitian degree-two models

\mathfrak h_2(\mathbb R)\cong J(\mathbb R^2),\quad \mathfrak h_2(\mathbb C)\cong J(\mathbb R^3),\quad \mathfrak h_2(\mathbb H)\cong J(\mathbb R^5),\quad \mathfrak h_2(\mathbb O)\cong J(\mathbb R^9).

These algebras have dimensions $3,4,6,10$ , respectively.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Publisher record.
Pascual Jordan, John von Neumann, and Eugene Wigner, “On an Algebraic Generalization of the Quantum Mechanical Formalism,” Annals of Mathematics 35 (1934), 29–64. JSTOR record.