alternative algebra
A non-associative algebra is alternative if
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1.
(left alternative laws) , and
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2.
(right alternative laws) ,
for any , where is the associator on .
Remarks
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Let be alternative and suppose . From the fact that , we can deduce that the associator is anti-commutative, when one of the three coordinates is held fixed. That is, for any ,
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(a)
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(b)
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(c)
Put more succinctly,
where , the symmetric group on three letters, and is the sign (http://planetmath.org/SignatureOfAPermutation) of .
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(a)
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An alternative algebra is a flexible algebra, provided that the algebra is not Boolean (http://planetmath.org/BooleanLattice) (characteristic (http://planetmath.org/Characteristic) ). To see this, replace in the first anti-commutative identities above with and the result follows.
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Artin’s Theorem: If a non-associative algebra is not Boolean, then is alternative iff every subalgebra of generated by two elements is associative. The proof is clear from the above discussion.
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A commutative alternative algebra is a Jordan algebra. This is true since shows that the Jordan identity is satisfied.
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Alternativity can be defined for a general ring : it is a non-associative ring such that for any , and . Equivalently, an alternative ring is an alternative algebra over .
Title | alternative algebra |
Canonical name | AlternativeAlgebra |
Date of creation | 2013-03-22 14:43:24 |
Last modified on | 2013-03-22 14:43:24 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 17D05 |
Related topic | Associator |
Related topic | FlexibleAlgebra |
Defines | Artin’s theorem on alternative algebras |
Defines | alternative ring |
Defines | left alternative law |
Defines | right alternative law |