Jordan algebra homomorphism

Definition

Let $J$ and $K$ be Jordan algebras over the same field. A Jordan algebra homomorphism is a linear map $\varphi:J\to K$ such that

\varphi(x\circ y)=\varphi(x)\circ\varphi(y) \qquad(x,y\in J).

If both algebras are unital, $\varphi$ is unital when $\varphi(e_J)=e_K$ .

Units are not automatic

Product preservation does not make an arbitrary homomorphism unital. For example, inclusion of a matrix corner sends the corner unit to a proper idempotent in the larger algebra. By contrast, a bijective homomorphism between unital Jordan algebras necessarily sends unit to unit, since the image of the source unit acts as a unit on every element of the target.

Kernels and images

The kernel is a Jordan ideal: if $x\in\ker\varphi$ and $y\in J$ , then $\varphi(x\circ y)=0$ . The image is a Jordan subalgebra of $K$ . An injective homomorphism is a Jordan embedding, while a bijective one is a Jordan isomorphism.

Relation to associative maps

Every homomorphism of associative algebras induces a Jordan homomorphism between their symmetrized Jordan algebras. The converse fails: a Jordan map preserves $xy+yx$ , not necessarily the ordered product $xy$ . On matrix algebras, transpose-type operations furnish standard Jordan symmetries that are not associative-algebra homomorphisms.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994. Publisher record.
Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, 2000. Publisher record.