is commutative: , and
satisfies the Jordan identity: ,
for any .
The above can be restated as
If is a Jordan algebra, a subset is called a Jordan subalgebra if . Let and be two Jordan algebras. A Jordan algebra homomorphism, or simply Jordan homomorphism, from to is an algebra homomorphism that respects the above two laws. A Jordan algebra isomorphism is just a bijective Jordan algebra homomorphism.
If is a Jordan algebra such that , then is power-associative (http://planetmath.org/PowerAssociativeAlgebra).
If in addition , then by replacing with in the Jordan identity and simplifying, is flexible (http://planetmath.org/FlexibleAlgebra).
Given any associative algebra , we can define a Jordan algebra . To see this, let be an associative algebra with associative multiplication and suppose is invertible in . Define a new multiplication given by
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 . 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 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.
|Date of creation||2013-03-22 14:52:15|
|Last modified on||2013-03-22 14:52:15|
|Last modified by||CWoo (3771)|
|Defines||special Jordan algebra|
|Defines||exceptional Jordan algebra|
|Defines||Jordan algebra homomorphism|
|Defines||Jordan algebra isomorphism|