associator

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

[\ 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.

References

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

Title	associator
Canonical name	Associator
Date of creation	2013-03-22 14:43:21
Last modified on	2013-03-22 14:43:21
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 17A01
Related topic	AlternativeAlgebra
Related topic	PowerAssociativeAlgebra
Related topic	FlexibleAlgebra
Related topic	Commutator
Defines	anti-associative