Associator
In abstract algebra, the term associator is used in different ways as a measure of the non-associativity of an algebraic structure. Associators are commonly studied as triple systems.
Ring theory
For a non-associative ring or algebra [math]\displaystyle{ R }[/math], the associator is the multilinear map [math]\displaystyle{ [\cdot,\cdot,\cdot] : R \times R \times R \to R }[/math] given by
- [math]\displaystyle{ [x,y,z] = (xy)z - x(yz). }[/math]
Just as the commutator
- [math]\displaystyle{ [x, y] = xy - yx }[/math]
measures the degree of non-commutativity, the associator measures the degree of non-associativity of [math]\displaystyle{ R }[/math]. For an associative ring or algebra the associator is identically zero.
The associator in any ring obeys the identity
- [math]\displaystyle{ w[x,y,z] + [w,x,y]z = [wx,y,z] - [w,xy,z] + [w,x,yz]. }[/math]
The associator is alternating precisely when [math]\displaystyle{ R }[/math] is an alternative ring.
The associator is symmetric in its two rightmost arguments when [math]\displaystyle{ R }[/math] 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
- [math]\displaystyle{ [n,R,R] = [R,n,R] = [R,R,n] = \{0\} \ . }[/math]
The nucleus is an associative subring of R.
Quasigroup theory
A quasigroup Q is a set with a binary operation [math]\displaystyle{ \cdot : Q \times Q \to Q }[/math] such that for each a, b in Q, the equations [math]\displaystyle{ a \cdot x = b }[/math] and [math]\displaystyle{ y \cdot a = b }[/math] have unique solutions x, y in Q. In a quasigroup Q, the associator is the map [math]\displaystyle{ (\cdot,\cdot,\cdot) : Q \times Q \times Q \to Q }[/math] defined by the equation
- [math]\displaystyle{ (a\cdot b)\cdot c = (a\cdot (b\cdot c))\cdot (a,b,c) }[/math]
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
- [math]\displaystyle{ a_{x,y,z} : (xy)z \mapsto x(yz). }[/math]
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. doi:10.1006/jsco.2001.0510.
- Schafer, Richard D. (1995) [1966]. An Introduction to Nonassociative Algebras. Dover. ISBN 0-486-68813-5. https://archive.org/details/introductiontono0000scha.
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