Maths - Conjugation

In order to calculate how a point moves when being rotated in 3D we can use the 'sandwich product' as discussed on this page.

But where does this sandwich product come from? It seems arbitrary, why this function? Is this something that is specific to quaternions or something about rotations that needs this form?

It turns out that the 'sandwich product' is not only used for quaternions. It can be used to represent many other types of transform ... see the pages about geometric algebra, rotors, conformal space and so on.

But the 'sandwich product' is still more general than that, it turns up in group theory and set theory, in this form it is known as a 'conjugacy' (not to be confused with the conjugate of a complex number or quaternion, here a conjugacy is the whole sandwich product).

I think that if we really want to understand where this comes from then we need to delve into this branch of mathematics.

In set theory conjugacy is a way to determine whether two sets (with a multiplication operation - typically a group) are for practical purposes equivalent, that is, isomorphic.

Category Theoretic Approach

As discussed on the page here there are two types of algebra associated with transformations (such as rotation) and these algebras must interwork correctly together.

Here we want to do both of these types of algebra using quaternions. To do that a quaternion must be used to represent both a point and a transform, a way to do this is to choose some arbitrary point and then represent all other points as transforms to or from the fixed point (see representable functors on page here). I guess this is a bit like using vectors to represent points in space by using a vector from the origin.

Depending whether we choose a transform 'to' or 'from' will depend on whether a mapping between any two point varies covarient or contravariently with our 'representable' transforms as follows.

Let 'A', 'X' and 'Y' be elements of a class 'C'.

Assume that both 'A' and 'f' are fixed because:

covarient contravarient
covarient contra
  • If 'f' and 'h' are known then we can derive 'g' (by composition)
  • If 'f' and 'g' are known then we cannot derive 'h' (see concepts of section and retraction).

So given some fixed 'f' the 'g' depends on 'h' which I have drawn below as an arrow between arrows.

  • If 'f' and 'g' are known then we can derive 'h' (by composition)
  • If 'f' and 'h' are known then we cannot derive 'g' (see concepts of section and retraction).

So given some fixed 'f' the 'h' depends on 'g' which I have drawn below as an arrow between arrows.

covarient contravarient
This arrow goes in the same direction to f, hence it is covarient. This arrow goes in the opposite direction to f, hence it is contravarient.

So far in representables we have mapped a element of an object to an element of a homset, but we want to know if this map preserves structure, so here we will see what happens when we map a whole function 'f':

covarient contravarient
covarient contravarient

From the triangle on the right of the diagram we have:

hom(A,Y) = f • hom(A,X)

that is post compose with f

So f maps to:

hom(A,X) -> hom(A,Y)

which is:

hom(A,X) -> f • hom(A,X)

From the triangle on the right of the diagram we have:

hom(A,X) = hom(A,Y) • f

that is pre compose with f

So f maps to:

hom(A,X) -> hom(A,Y)

which is:

hom(A,Y) • f -> hom(A,Y)

So in one case we apply the transform by pre-multipling by 'f' and in the other case we apply the transform by post-multipling by 'f'. However we want the transform to be independent of the arbitrary point 'A', so we end up with a combination of both pre and post-multiplying.

Search For A More Intuitive Explanation

Here we will try to explain this in terms of quaternions:

quaternion conjugacy

We define some mapping φ from G to G'. We want G and G' to be equivalent to each other, this means that G and G' are isomorphic. This isomorphism means that:

φ(q1*q2) = φ(q1) * φ(q2)

where:

So what is this mapping φ? We can use:

φ(v) = q v q-1

so

φ(q1*q2) = q q1 q2 q-1
= q q1 q-1 q q2 q-1
= φ(q1)*φ(q2)

(since q-1 q =1)

So we can see that this sandwich product (conjugacy) has the right properties for an isomorphism (I have investigated what functions form an isomrphism on this page).

What would happen if instead of the sandwich product we tried a simple product:

φ(v) = q v

in this case we would have:

φ(q1*q2) = q q1 q2
but

φ(q1)*φ(q2) = q q1 q q2

which is different so a simple product does not form an isomorphism.

So what about matrix algebra? We can use the mutiplication of a matrix by a vector to rotate a point, why does this work?


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Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

This is the best book that I have found, so far, to introduce this topic:
flag flag flag flag flag flag Contemporary Abstract Algebra written by Joseph Gallian - This investigates algebra (Groups, Rings, Fields) using an abstract (set theory) approach. It is very readable for such an abstract topic but the inclusion of 1 page histories of the great mathematicians and other background information.

 

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