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Jordan algebra
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(Definition)
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Let be a commutative ring with . An -algebra with multiplication not assumed to be associative is called a (commutative) Jordan algebra if
is commutative: , and
satisfies the Jordan identity:
,
for any .
The above can be restated as
, where is the commutator bracket, and
- for any
,
, where is the associator bracket.
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.
Remarks.
- If
is a Jordan algebra such that
, then is power-associative.
- If in addition
, then by replacing with in the Jordan identity and simplifying, is flexible.
- 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
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(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 . 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.
- An example of an exceptional Jordan algebra is
, the algebra of Hermitian matrices over the octonions.
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"Jordan algebra" is owned by CWoo.
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(view preamble)
| Other names: |
Jordan homomorphism, Jordan isomorphism |
| Also defines: |
Jordan identity, special Jordan algebra, exceptional Jordan algebra, Jordan algebra homomorphism, Jordan subalgebra, Jordan algebra isomorphism |
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Cross-references: octonions, Hermitian matrices, product, equation, right hand side, isomorphic, Lie algebras, invertible, addition, bijective, algebra, subset, associator, commutator bracket, commutative, associative, multiplication, commutative ring
There are 10 references to this entry.
This is version 7 of Jordan algebra, born on 2004-12-07, modified 2004-12-17.
Object id is 6547, canonical name is JordanAlgebra.
Accessed 8190 times total.
Classification:
| AMS MSC: | 17C05 (Nonassociative rings and algebras :: Jordan algebras :: Identities and free Jordan structures) |
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Pending Errata and Addenda
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