# associator

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Let $A$ be a non-associative algebra over a field.  The \emph{associator} of $A$, denoted by $[\ , , ]$, is a \PMlinkname{trilinear}{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.