Associator identity

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Statement

Suppose R is a (possibly) non-associative ring with multiplication * and a:R^3 \to R is the associator:

a(x,y,z) := ((x * y) * z) - (x * (y * z))

Then, it satisfies the following identity for all w,x,y,z \in R:

\! w * a(x,y,z) + a(w,x,y) * z = a(w*x,y,z) - a(w,x*y,z) + a(w,x,y*z)

Related facts

The associator identity is the analogue for associativity of the Jacobi identity for commutativity.

The associator identity is also closely related to the associativity pentagon, which is a pentagon where the vertices are the different ways of associating a product of four letters and the edges correspond to ways of moving from one parenthesization to the next by applying associativity.

Associativitypentagon.png

Here, the legend on an arrow equals the head of the arrow minus the tail of the arrow. We see that the associativity pentagon directly gives us the associator identity.