Ideal in a Jordan algebra

Definition

Let $J$ be a Jordan algebra. An ideal of $J$ is a linear subspace $I\subseteq J$ such that

x\circ y\in I \qquad\text{for every }x\in J\text{ and }y\in I.

Equivalently, every multiplication operator $L_x:y\mapsto x\circ y$ preserves $I$ . Because the Jordan product is commutative, no separate left- and right-ideal conditions are needed.

Quotients and kernels

The quotient vector space $J/I$ has the well-defined Jordan product

(x+I)\circ(y+I)=x\circ y+I.

The quotient map $J\to J/I$ is a Jordan homomorphism. Conversely, the kernel of every Jordan homomorphism is an ideal. These facts make ideals the subobjects appropriate to quotient constructions; an arbitrary Jordan subalgebra need not be an ideal.

Units and simplicity

If $J$ has unit $1$ and an ideal $I$ contains $1$ , then $x=x\circ1\in I$ for every $x\in J$ , so $I=J$ . A nonzero Jordan algebra is simple when it has no ideals other than $0$ and $J$ , together with the usual convention excluding the one-dimensional zero-product degeneracy.

When $J$ is a direct sum of ideals, multiplication between distinct summands vanishes. In particular, the simple-factor decomposition of a Euclidean Jordan algebra is a decomposition by ideals, not merely by subalgebras.

Warning about quadratic Jordan theory

This definition uses the bilinear Jordan-algebra convention in characteristic different from $2$ . Over more general base rings, quadratic Jordan algebras use a stronger ideal condition involving their quadratic operators; that notion should not be silently identified with the one used here.

References

Nathan Jacobson, Structure and Representations of Jordan Algebras, American Mathematical Society, 1968, Chapters I and V. Publisher record.
Kevin McCrimmon, A Taste of Jordan Algebras, Springer, 2004, §§1.5 and 4.2. Publisher record.