• Source: 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 R, the associator is the multilinear map



[
⋅
,
⋅
,
⋅
]
:
R
×
R
×
R
→
R


{\displaystyle [\cdot ,\cdot ,\cdot ]:R\times R\times R\to R}

given by




[
x
,
y
,
z
]
=
(
x
y
)
z
−
x
(
y
z
)
.


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


Just as the commutator




[
x
,
y
]
=
x
y
−
y
x


{\displaystyle [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
=
[
w
x
,
y
,
z
]
−
[
w
,
x
y
,
z
]
+
[
w
,
x
,
y
z
]
.


{\displaystyle 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
]
=
{
0
}

.


{\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



⋅
:
Q
×
Q
→
Q


{\displaystyle \cdot :Q\times Q\to Q}

such that for each a, b in Q,
the equations



a
⋅
x
=
b


{\displaystyle a\cdot x=b}

and



y
⋅
a
=
b


{\displaystyle y\cdot a=b}

have unique solutions x, y in Q. In a quasigroup Q, the associator is the map



(
⋅
,
⋅
,
⋅
)
:
Q
×
Q
×
Q
→
Q


{\displaystyle (\cdot ,\cdot ,\cdot ):Q\times Q\times Q\to Q}

defined by the equation




(
a
⋅
b
)
⋅
c
=
(
a
⋅
(
b
⋅
c
)
)
⋅
(
a
,
b
,
c
)


{\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





a

x
,
y
,
z


:
(
x
y
)
z
↦
x
(
y
z
)
.


{\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.

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