Associator

In abstract algebra, the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure. Associators are commonly studied as triple systems.

Ring theory

For a nonassociative ring or algebra ${\displaystyle R}$, the associator is the multilinear map ${\displaystyle [\cdot ,\cdot ,\cdot ]:R\times R\times R\to R}$ given by

${\displaystyle [x,y,z]=(xy)z-x(yz).}$

Just as the commutator

${\displaystyle [x,y]=xy-yx}$

measures the degree of noncommutativity, the associator measures the degree of nonassociativity of ${\displaystyle R}$. For an associative ring or algebra the associator is identically zero.

The associator in any ring obeys the identity

${\displaystyle w[x,y,z]+[w,x,y]z=[wx,y,z]-[w,xy,z]+[w,x,yz].}$

The associator is alternating precisely when ${\displaystyle R}$ is an alternative ring.

The associator is symmetric in its two rightmost arguments when ${\displaystyle R}$ is a pre-Lie algebra.

The nucleus is the set of elements that associate with all others: that is, the n in R such that

${\displaystyle [n,R,R]=[R,n,R]=[R,R,n]=\{0\}\ .}$

The nucleus is an associative subring of R.

Quasigroup theory

A quasigroup Q is a set with a binary operation ${\displaystyle \cdot :Q\times Q\to Q}$ such that for each a,b in Q, the equations ${\displaystyle a\cdot x=b}$ and ${\displaystyle y\cdot a=b}$ have unique solutions x,y in Q. In a quasigroup Q, the associator is the map ${\displaystyle (\cdot ,\cdot ,\cdot ):Q\times Q\times Q\to Q}$ defined by the equation

${\displaystyle (a\cdot b)\cdot c=(a\cdot (b\cdot c))\cdot (a,b,c)}$

for all a,b,c in Q. As with its ring theory analog, the quasigroup associator is a measure of nonassociativity of Q.

Higher-dimensional algebra

In higher-dimensional algebra, where there may be non-identity morphisms between algebraic expressions, an associator is an isomorphism

${\displaystyle a_{x,y,z}:(xy)z\mapsto x(yz).}$

Category theory

In category theory, the associator expresses the associative properties of the internal product functor in monoidal categories.

See also

• Commutator
• Non-associative algebra
• Quasi-bialgebra – discusses the Drinfeld associator

References

• Bremner, M.; Hentzel, I. (March 2002). "Identities for the Associator in Alternative Algebras". Journal of Symbolic Computation. 33 (3): 255–273. CiteSeerX 10.1.1.85.1905. doi:10.1006/jsco.2001.0510.
• Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5.