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 power-associative (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 $H_3(𝕆)$, the algebra of $3\times 3$ Hermitian matrices over the octonions.

Title	Jordan algebra
Canonical name	JordanAlgebra
Date of creation	2013-03-22 14:52:15
Last modified on	2013-03-22 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