associator

For a non-associative ring or algebra R, the associator is the multilinear map $[\backslash cdot,\backslash cdot,\backslash cdot]\; :\; R\; \backslash times\; R\; \backslash times\; R\; \backslash to\; R$ given by

: $[x,y,z]\; =\; (xy)z\; -\; x(yz).$

Just as the commutator

: $[x,\; y]\; =\; xy\; -\; yx$

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

: $w[x,y,z]\; +\; [w,x,y]z\; =\; [wx,y,z]\; -\; [w,xy,z]\; +\; [w,x,yz].$

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

: $[n,R,R]\; =\; [R,n,R]\; =\; [R,R,n]\; =\; \backslash \{0\backslash \}\; \backslash \; .$

The nucleus is an associative subring of R.

A quasigroup Q is a set with a binary operation $\backslash cdot\; :\; Q\; \backslash times\; Q\; \backslash to\; Q$ such that for each a, b in Q,

the equations $a\; \backslash cdot\; x\; =\; b$ and $y\; \backslash cdot\; a\; =\; b$ have unique solutions x, y in Q. In a quasigroup Q, the associator is the map $(\backslash cdot,\backslash cdot,\backslash cdot)\; :\; Q\; \backslash times\; Q\; \backslash times\; Q\; \backslash to\; Q$ defined by the equation

: $(a\backslash cdot\; b)\backslash cdot\; c\; =\; (a\backslash cdot\; (b\backslash cdot\; c))\backslash cdot\; (a,b,c)$

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

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

: $a\_\{x,y,z\}\; :\; (xy)z\; \backslash mapsto\; x(yz).$

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

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

• {{cite journal |title=Identities for the Associator in Alternative Algebras |first1=M. |last1=Bremner |first2=I. |last2=Hentzel |journal=Journal of Symbolic Computation |volume=33 |issue=3 |date=March 2002 |pages=255–273 |doi=10.1006/jsco.2001.0510 |citeseerx=10.1.1.85.1905 }}
• {{cite book |first=Richard D. |last=Schafer |title=An Introduction to Nonassociative Algebras |url=https://archive.org/details/introductiontono0000scha |url-access=registration |year=1995 |orig-date=1966 |publisher=Dover |isbn=0-486-68813-5 }}

Category:Non-associative algebra

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