Algebra Of Commuting And Associative Objects

Any set of objects together with a set of rules of allowable operations (such as multiplication or division) that behaves in an internally consistent ("closed") manner is called a Group.

Just as much that music can be conjured from the three operations of transposition, inversion and reflection that move us across a set of semitones comprising harmonic coherent overtones, a group is a closed set of operations (finger configurations) and objects (a fundamental and it`s harmonics) that keep the pianist on the keyboard and in tune.

Such a set must contain an identity element (e.g. zero or 1) and there must be a unique inverse for each object in the set. The set of two integers {+1,-1} is closed under multiplication, [x]. Being closed means that no operation takes an object from the set outside the set as we will arrive at +1,-1 with any operation.

Non-Commuting Algebras The group is said to abelian in the sense that it does not matter what order the numbers in the set are acted upon. The operation of multiplication is said to commute between integers as the order of multiplication has no impact on the final result.

We are familiar with sets of objects where the order of the operations does matter - think multiplication of matrices.The set of rotations described do depend on the order in which the transformations are applied. Get a book flip it`s cover then rotate it around its spine. Compare final position to what happens if you rotate it around its spine and then flip it! Different orientation in space. The operations of flipping and rotating do not commute.

The group of rotations is thus said to be Non-Abelian as its group operations are non- commutative. The group from which the quarks of QCD are realised is such a non-Abelian group.

Associative Algebras The other ordering property that one may ask of your set of objects with operation [.] is that it be associative in the sense that: g.(h.k)=(g.h).k so that the h may be associated to either number in the group. This is the case for set of Reals with the usual [x] operation. That this need not be the case for a set of objects with any operation is seen by considering the division operator [/] over the set of Real Integers{g=6, h=3,k=2}.

In this case the left hand side is g/(h/k)=6/(3/2)=4 where as the right-hand side is (g/h)/k=(6/3)/2=1.

The set is not associative under the / operation.

Now we note that the algebra of quaternians (described by complex-valued matrices) is non- Abelian (as it`s matrix operations do not commute). It is then natural to ask is there an algebra of object transformations that not only does not commute but is not associative as was the case for the division operator above. Such a Division Algebra does exist, being 8 dimensional it`s objects are called Octonians.

These complete the set {Reals, Complex, Quaternians, Octonians} and constitute the Clifford Algebra of hyper complex numbers. These underpin the symmetries realised by Dirac spinors.