PlanetPhysics/Non Commutative Structure

Let $(C,\circ )$ be a structure consisting of a class , $C$, together with a binary [[../Cod/|operation]] $\circ$ defined for pairs of [[../TrivialGroupoid/|objects]] in $C$ (or elements of $C$ when the latter is a small class, i.e., a set). The structure-- and the operation $\circ$-- are said to be [[../AbelianCategory3/|noncommutative]] if

$\begin{array}{c}a\circ b\ne b\circ a\end{array}$

for either at least some or all of the $a,b$ pairs in $C$ for which the operation is defined.

A structure that is noncommutative is also called sometimes a \htmladdnormallink{non-Abelian {http://planetphysics.us/encyclopedia/AbelianCategory3.html} structure}, although the latter term is, in general, more often used to specify non-Abelian theories. A binary operation that is not commutative is said to be [[../AbelianCategory3/|non-commutative]] (or noncommutative ). Thus, a noncommutative structure can be alternatively defined as any structure whose binary operation is not commutative (that is, in the commutative case one has

$\begin{array}{c}a\circ b=b\circ a\end{array}$

for all $a,b$ pairs in $C$, and also that the operation $\circ$ is defined for all pairs in $C$).

An example of a commutative structure is the [[../CosmologicalConstant2/|field]] of real numbers-- with two commutative operations in this case-- which are the addition and multiplication over the reals.

A [[../AbelianCategory3/|commutative group]] is also called Abelian , whereas a [[../Cod/|category]] with structure that has [[../Commutativity/|commutative diagrams]] is not necessarily Abelian --unless it does satisfy the Ab1 to Ab6 axioms that define an [[../AbelianCategory2/|abelian category]] (or equivalently, if it has the properties specified in Mitchell's alternative definition of an Abelian category .)

An example of a non-commutative operation is the multiplication over $n\times n$ [[../Matrix/|matrices]]. Another example of a noncommutative algebra is a general [[../CCliffordAlgebra/|Clifford algebra]], which is of fundamental importance in the [[../CoIntersections/|algebraic]] theory of [[../QuantumSpinNetworkFunctor2/|observable]] [[../QuantumOperatorAlgebra5/|quantum operators]] and also in [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]].