Associator
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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
[edit]For a non-associative ring or algebra R, the associator is the multilinear map given by
Just as the commutator
measures the degree of non-commutativity, the associator measures the degree of non-associativity of R. For an associative ring or algebra the associator is identically zero.
The associator in any ring obeys the identity
The associator is alternating precisely when R is an alternative ring.
The associator is symmetric in its two rightmost arguments when 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
The nucleus is an associative subring of R.
Quasigroup theory
[edit]A quasigroup Q is a set with a binary operation such that for each a, b in Q, the equations and have unique solutions x, y in Q. In a quasigroup Q, the associator is the map defined by the equation
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
[edit]In higher-dimensional algebra, where there may be non-identity morphisms between algebraic expressions, an associator is an isomorphism
Category theory
[edit]In category theory, the associator expresses the associative properties of the internal product functor in monoidal categories.
See also
[edit]- Commutator
- Non-associative algebra
- Quasi-bialgebra – discusses the Drinfeld associator
References
[edit]- 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.