PlanetMath: Jordan algebra

(more info)

Math for the people, by the people.

Main Menu

Jordan algebra

(Definition)

Let be a commutative ring with $1\neq 0$ . An -algebra with multiplication not assumed to be associative is called a (commutative) Jordan algebra if

for any $a,b\in A$ .

The above can be restated as

is a Jordan algebra, a subset $B\subseteq A$ is called a Jordan subalgebra if $BB\subseteq B$ . Let

and

be two Jordan algebras. A Jordan algebra homomorphism, or simply Jordan homomorphism, from

is an algebra homomorphism that respects the above two laws. A Jordan algebra isomorphism is just a bijective Jordan algebra homomorphism.

Remarks.

If is a Jordan algebra such that $\operatorname{char}(A)\neq2$ , then is power-associative.
If in addition $2=1+1\neq\operatorname{char}(A)$ , then by replacing with in the Jordan identity and simplifying, is flexible.
Given any associative algebra , we can define a Jordan algebra $A^{+}$ . To see this, let be an associative algebra with associative multiplication $\cdot$ and suppose is invertible in . Define a new multiplication given by

$\displaystyle ab=\frac{1}{2}(a\cdot b+b\cdot a).$ (1)

It is readily checked that this new multiplication satisifies both the commutative law and the Jordan identity. Thus with the new multiplication is a Jordan algebra and we denote it by $A^{+}$ . However, unlike Lie algebras, not every Jordan algebra is embeddable in an associative algebra. Any Jordan algebra that is isomorphic to a Jordan subalgebra of $A^{+}$ for some associative algebra is called a special Jordan algebra. Otherwise, it is called an exceptional Jordan algebra. As a side note, the right hand side of Equation (1) is called the Jordan product.
An example of an exceptional Jordan algebra is $H_3(\mathbb{O})$ , the algebra of $3\times3$ Hermitian matrices over the octonions.