Jordan algebra
Let $R$ be a commutative ring with $1\ne 0$. An $R$algebra^{} $A$ with multiplication not assumed to be associative is called a (commutative^{}) Jordan algebra^{} if

1.
$A$ is commutative: $ab=ba$, and

2.
$A$ satisfies the Jordan identity: $({a}^{2}b)a={a}^{2}(ba)$,
for any $a,b\in A$.
The above can be restated as

1.
$[A,A]=0$, where $[,]$ is the commutator bracket, and

2.
for any $a\in A$, $[{a}^{2},A,a]=0$, where $[,,]$ is the associator^{} bracket.
If $A$ is a Jordan algebra, a subset $B\subseteq A$ is called a Jordan subalgebra if $BB\subseteq B$. Let $A$ and $B$ be two Jordan algebras. A Jordan algebra homomorphism, or simply Jordan homomorphism, from $A$ to $B$ is an algebra homomorphism that respects the above two laws. A Jordan algebra isomorphism is just a bijective Jordan algebra homomorphism.
Remarks.

•
If $A$ is a Jordan algebra such that $\mathrm{char}(A)\ne 2$, then $A$ is powerassociative (http://planetmath.org/PowerAssociativeAlgebra).

•
If in addition $2=1+1\ne \mathrm{char}(A)$, then by replacing $a$ with $a+1$ in the Jordan identity and simplifying, $A$ is flexible (http://planetmath.org/FlexibleAlgebra).

•
Given any associative algebra $A$, we can define a Jordan algebra ${A}^{+}$. To see this, let $A$ be an associative algebra with associative multiplication $\cdot $ and suppose $2=1+1$ is invertible^{} in $R$. Define a new multiplication given by
$$ab=\frac{1}{2}(a\cdot b+b\cdot a).$$ (1) It is readily checked that this new multiplication satisifies both the commutative law and the Jordan identity. Thus $A$ 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 $A$ 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 $3\times 3$ Hermitian matrices^{} over the octonions.
Title  Jordan algebra 
Canonical name  JordanAlgebra 
Date of creation  20130322 14:52:15 
Last modified on  20130322 14:52:15 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 17C05 
Synonym  Jordan homomorphism 
Synonym  Jordan isomorphism 
Defines  Jordan identity 
Defines  special Jordan algebra 
Defines  exceptional Jordan algebra 
Defines  Jordan algebra homomorphism 
Defines  Jordan subalgebra 
Defines  Jordan algebra isomorphism 