# 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

Title associator Associator 2013-03-22 14:43:21 2013-03-22 14:43:21 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 17A01 AlternativeAlgebra PowerAssociativeAlgebra FlexibleAlgebra Commutator anti-associative