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
![{\displaystyle [\cdot ,\cdot ,\cdot ]:R\times R\times R\to R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c21c2636857129351846852fa03966fa47c83b99)
![{\displaystyle [x,y,z]=(xy)z-x(yz).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e44f6b788a502ce2833621d6871c1fa2a89993)
Just as the commutator
![{\displaystyle [x,y]=xy-yx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42b4220c8122ebd2a21c517ca80639581679cfa6)
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
![{\displaystyle w[x,y,z]+[w,x,y]z=[wx,y,z]-[w,xy,z]+[w,x,yz].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec38ef54e22fb31fb569021bea5ff79f12c756ee)
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
![{\displaystyle [n,R,R]=[R,n,R]=[R,R,n]=\{0\}\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb8a21a657e4199b3548278ec6100e3da78b9ee)
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.