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associator (Definition)

Let $ A$ be a non-associative algebra over a field. The associator of $ A$, denoted by $ [\ , , ]$, is a trilinear map from $ A\times A\times A$ to $ A$ given by:

$\displaystyle [\ a,b,c\ ]=(ab)c-a(bc).$

Just as the commutator measures how close an algebra is to being commutative, the associator measures how close it is to being associative. $ [\ , , ]=0$ identically iff $ A$ is associative.

Bibliography

1
R. D. Schafer, An Introduction on Nonassociative Algebras, Dover, New York (1995).



"associator" is owned by CWoo.
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See Also: alternative algebra, power-associative algebra, flexible algebra, commutator

Also defines:  anti-associative
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Cross-references: iff, associative, commutative, algebra, commutator, map, field, non-associative algebra
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This is version 7 of associator, born on 2004-10-10, modified 2006-10-02.
Object id is 6348, canonical name is Associator.
Accessed 2585 times total.

Classification:
AMS MSC17A01 (Nonassociative rings and algebras :: General nonassociative rings :: General theory)

Pending Errata and Addenda
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Discussion
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commutator by HkBst on 2004-10-16 12:30:15
commutator should point to
http://planetmath.org/encyclopedia/CommutatorLieAlgebra.html
[ reply | up ]
algebra or ring by HkBst on 2004-10-16 11:31:38
couldn't the associater be defined for rings? They seem to have all necessary operations.
[ reply | up ]

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