I'm doing my essay on quaternions but I'm struggling to find material on them that isn't on Wikipedia.
Does any one know of any interesting theorems that I could look up. It would be a great help.

cheers.

I could loan you the excursions sheet we had to do on this, but that's all i can really suggest. I take it searching the library didn't work.

Have you checked the Wiki references at the bottom of the page?

Sadly, quaternions as a number system are rather disappointing:

Does any one know of any interesting theorems that I could look up

The sad truth is that there aren't any, at least in the complex number "holomorphic=analytic=conformal=awesome" type way (see 'quaternionic conformality' thread for my own shuddering disappointment on the matter). But a good line of extension lies in the world of lie groups: SU (2) the special unitary group in 2 dimensions (a sort of complex orthogonal matrix group)is isomorphic to the unit quaternions under multiplication and is a gateway to a world of awesome lie based maths.

Talk about what a lie group is, manifolds, group actions etc. Guaranteed first. Plus damn interesting.

Though if youre hooked on because it's a 'number system' then there is a theorem that states that and are the only rings containing the reals- it's a super hot theorem (can't remember the name though) but it's pretty hardcore. Anyone know the name?

Frobenius theorem perhaps ?

And yes, Quaternions are disappointing, nothing like the magnificence of complex numbers.

yeah, already got that one but thanks anyway. i'll focus more on group theory in that case, it's mainly what we're studying in algebra now anyway.

thanks for the help.

Got to agree with Xedi, there's good reasons why people don't bother with them much these days.

Well no, they're still really worth studying and bothering with, there's lots of interesting things going on there, and they really help in many situations.
Just that they aren't as beautiful as you would think they would be, seeing how complex numbers work great.

I'm classifying some finite simple groups for mine, and I would appreciate a point in the right direction as regards books. I'm not sure if Dave McCormick is a regular forum checker, but if anyone knows what books he's been using for this, I'd love to know.

"Got to agree with Xedi, there's good reasons why people don't bother with them much these days."

Thanks for you 'help'. I'm well aware that quaternions aren't at the cutting edge of research but that doesn't diminish their historical significance, or stop them being interesting. Maybe you could leave trolling to the forums that aren't designed specifically for academic support.

We did something very cool with Quarternions in Algebraic Topology today, I'll post it later if I have time, or Tom can.

Yeah, 'twas awesome, a sort of converse to the frobenius theorem. Won't spoil it now cos I want to do a bit of it at [Dg] tonight. But yeah will post the details later...

Trolling :-o The Frobenius theorem is a brilliant result, and they are historically important. They led to the development of a lot of the vector algebra today. It just seems that things get less interesting as you extend further beyond complex numbers. With the loss of associativity and commutitivity they become harder to work with and it seems that their main interest lies in physics. They do seem damn interesting (and are hence a fertile subject for 2nd year essay), it's just they seem a bit limited by their own nature.

edit: Errr, perhaps I should have posted the post in full first time. I just didn't expect many people would be online.

Stop it. If it helps, go to this site which gives good advice on forum etiquette:

beingniceonline

getting a bit off topic, please be nice, and start a new thread if you want to discuss something new.

I'll have to come clean: the above link was supposed to be a RickRoll, but it didn't work. I have yet to fully understand the subtleties of this trolling lark. I'll stop now, Steven's right.

Anyway- the converse to frobenius theorem...

The idea is that you prove frobenius using the fact that a normed division algebra containing immediately yields a normed vector space structure over (just forget about multiplication by anything non-real and the vector space axioms just fall out). Armed with this you construct an n-1 dimensional sphere (assuming your vector space is n dimensional) by way of and fluff out an inner product (usually :=re (v.u*) - where u* is the conjugate of u).

Using this we can costruct n-1 linearly independent vector fields on . We do so thus: choose a basis of the imaginary parts of (that is those orthogonal to ) and suppose then we have for each i. Each now defines a vector field since is orthogonal to x and hence tangent to the sphere (all quite easy to check), also (if you chose your original basis to be orthogonal this is much easier using inner product) we see that the are linearly independent.

And by a result in topology the number of linearly independent vector fields on is only n-1 for 1,3 and 7- these correspond to , and respectively - being the octonions. Thus these are the only possibles.

Another way of looking at this is as a converse to the hairy ball theorem- that states there is no continuous non vanishing vector field on for even n. Using as our vector space we have an inner product as defined above (but with a dot product in place of multiplication and conjugation componentwise) and hence a norm and a 2n-1 dimensional sphere.

We now have for , which does the job nicely as above...

Now for any we may apply the above ideas (but with i,j and k in place of the i to give 3 guaranteed linearly independent vector fields.

Now for any , the octonions give us seven...

By this method we can completely classify linearly independent vector fields on for n<15- after which point, things get complicated (clifford algebras apparently)... Which is awesome.

Get yourself a vector field today!

Any questions?

I'll have to come clean: the above link was supposed to be a RickRoll, but it didn't work. I have yet to fully understand the subtleties of this trolling lark. I'll stop now, Steven's right.

Better yet, use a site which re-directs links. On topic though, I like the sketch proof, it's pleasingly... short.

i'd quite like to practise my presentation at a discussion group sometime. if we can get 3 or 4 people to do it on the same night that would pretty much fill up the time.

i was hoping for next thursday at the earliest, since i haven't actually written it yet. anyone else interested?

Good idea, and I would definitely do the same were it not for the fact that my essay is on the same topic as one of Dave McCormick's discussion groups. You reckon you can find another 3 or 4 people?

I also have a similar problem with the Dave Cormickness. Also the I-haven't-chosen-a-title-yet-ness.

I'll turn up too, though I've done nothing at all yet. When is the real presentation supposed to be?

Mine is in week 9 but it is up to your tutor when it is.

well i know someone else who wants to do one too, i might be able to persuade another person as well. including cosmin that should be enough entertainment for one evening.

Cool. If I don't have anything by then I'll just practice my excursions presentation. :D

i have a couple of questions with regards to frobenius theorem.
firstly, what do you mean by linearly independant vector fields, and which result from topology allows you to make the claim required to finish the first proof, i'd like to reference it in my essay even if i don't prove it there.

thanks

A vector field may be considered as a function N from a manifold to its tangent space at that point- i.e. taking a point and sending it to a point on the hyperpane tangent to the manifold at that point. For a vector v to be in such a tangent plane say is clearly the same as saying <(v-x),n>=0 (where n is the normal to the manifold), but for this reduces to <(v-x),x>=0 (since x is normal to at x- differentiate and try for yourself).

Now linear independence of vector fields is just the statement that for all x is linearly independent. Notice particularly that if for some x then we have linear dependence vacuously (thus the vanishing vetor fields for n even which the hairy ball theorem shows are the only ones- do not count as L.I.)

To prove the result we need a lemma saying "if we can do linear independence- we can do orthonormality"- which is much like one from linear algebra, but a little tougher since we must keep continuity satisfied.

From here we realise that we need a function taking orthonormal bases to orthonormal bases- this is just the action of the clifford group, the set of invertible elements of the clifford algebra under multiplication (see clifford algebras on wikipedia or come to me for a chat for details). We see where this group can act and to what degree and get an elementary formula for the number of linearly independent vector fields on . [Can't find this A.T.M. but anyone who's been taking algebraic topology should have it (post now or forever hold your peace)]

This is then equal n only for n= 1,3,7. But if we had a normed division algebra of any other dimension m, then we would get the number of L.I. vector fields on as above. 7 we deal with separately- I don't know how...

But yeah apparently (after some research) not a topological result. But one in the field of topology. Apparently this is best dealt with in a paper by Eckmann, which I thoroughly intend to find and read soon...

Hope that helps...

I'm pretty sure I'd have to reference something like this so what kind of book would I find a proof of this nature, or in fact could you just post the name of a book so I can quote it that would be great.