Group Theory For Dummies

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climbhi

[SOLVED] Group Theory For Dummies

I've become interested in learning about Group Theory. I don't know too much but I see it spring up all over the place and would just like to know what it is about and some of the basics. Could some one please point me in the direction of a good resource that wouldn't be too far over my head? Thanks.
 
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BoulderHead

Oops, I thought you said 'group therapy for dummies'. I guess I can't help after all.
 
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climbhi

Originally posted by BoulderHead
Oops, I thought you said 'group therapy for dummies'. I guess I can't help after all.
Well perhaps I need some of that too...
 
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Originally posted by climbhi
I've become interested in learning about Group Theory. I don't know too much but I see it spring up all over the place and would just like to know what it is about and some of the basics. Could some one please point me in the direction of a good resource that wouldn't be too far over my head? Thanks.
Good (introductory) references are:

M. A. Armstrong. Groups and symmetry. Springer Verlag 1988.

J. Rotman. An introduction to the theory of groups, Sprnger Verlag 1995.

J. D. Dixon, Problems in Group Theory. New York: Dover, 1973.

R. Mirman. Group Theory: An Intuitive Approach. World Scientific, 1995.
 
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damgo

Hmm, I used Dummit&Foote "Abstract Algebra" as an introductory textbook, and found it to be excellent. Though the emphasis is on mathematics rather than the physics applications (Lie groups, representations, etc). The nice thing about pure group theory is it requires basically zero prerequisites.
 

chroot

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I'll also recommend the Schaum's Outline of Group Theory. It doesn't specifically cover some of the more physically interesting topics such as groups of 3x3 matrices, but it gives you all of the tools necessary to understand just about any group-theoretical system.

- Warren
 

Tom Mattson

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Originally posted by chroot
I'll also recommend the Schaum's Outline of Group Theory.
Yep, and the one called Abstract Algebra, too. All in all, a 30 dollar committment.
 
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climbhi

Thanks for the replies, does anyone know of a good online (read free) source? I'm kind of interested in group theory to see how it relates to QM and what not, but also just for pure math. Would the Schaubs outlines work well for both if there is no good free source available?
 

Tom Mattson

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Originally posted by climbhi
Would the Schaubs outlines work well for both if there is no good free source available?
There are many free sources available, but I recommend the Schaum's outlines anyway, because they are loaded with solved examples and exercises with answers.
 

marcus

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Originally posted by climbhi
I've become interested in learning about Group Theory. I don't know too much but I see it spring up all over the place and would just like to know what it is about and some of the basics. Could some one please point me in the direction of a good resource that wouldn't be too far over my head? Thanks.
I'm wondering if it would be possible to have an entrylevel workshop here at PF on groups.

I mean a collective teach-each-other tutorial-----no one person doing all the teaching but trading around.

I see Tom and Rutwig and Chroot have posted online resources
and also hardcopy books to buy.

The big question is------is there enough interest?
A secondary question is-----could we stand to type all the subscripts, superscripts, matrices, and greek letters? PF is a great medium for non-hierarchical learning. But the sheer typing of symbols and inability to draw pictures imposes some limits on what one can handle here.

So I am skeptical that a group theory tutorial or workshop would get anywhere.

But just to see how it might go----here is my proposal

Focus on the simplest most classical groups central to basic physics--dimensions 2, 3, 4.

Focus on things like SO(3) the special orthogonal group. ["special" just means det = 1 in this case, think of rotations]

And SU(2) the special unitary group----because of its relation to SO(3) and the pauli spinmatrices. among other things.

And SL(2,C) because of its relation to the Lorentz group.

It seems to me that the goal should be not to snow anybody or discourage anybody----not to show off or try to pull rank on people (as non-PF people sometimes do when discussing math)----but simply to go over the group theory that is most basic and do it in an entrylevel way.

This might not be possible---it might simply not work.

Also it might be tiresome to try to type in matrices---even like the three pauli spinmatrices which are about as simple as 2x2 matrices can get would be sort of tedious to type into PF-style posts.

Anyway I am broaching the idea. Reactions? Better ideas of how to do it?
 
I like the sound of your idea, Marcus. I'd be interested once my exams are done. I like especially the sound of learning its applications to Physics. We get taught Group Theory, but only in the sense of pure maths.
 

marcus

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Originally posted by Lonewolf
I like the sound of your idea, Marcus. I'd be interested once my exams are done. I like especially the sound of learning its applications to Physics. We get taught Group Theory, but only in the sense of pure maths.
If anyone knows good notes on the web that correspond to what Lonewolf is talking about (basic classical group theory with an eye to applications in physics) please post a link.

Lonewolf, this thread may possibly remain dormant until you are thru exams. Depends on how interested the others are. When you or anybody returns I will probably get a notice by email. but to be sure, send me a PM.
 
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The book by Cvitanovic is one of the links. There are many others, but it should be specified whether one is interested on discrete, continuous (non differentiable) or Lie groups, or even generalizations like Kac-Moody groups, supergroups, etc. Each of the topics is a world in itself.
 

marcus

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Originally posted by rutwig
The book by Cvitanovic is one of the links. There are many others, but it should be specified whether one is interested on discrete, continuous (non differentiable) or Lie groups, or even generalizations like Kac-Moody groups, supergroups, etc. Each of the topics is a world in itself.
Rutwig I do not know if you have any interest in LQG or follow it at all but, if you do, then you probably have noted that
a recent result of Olaf Dreyer seems to force a change in the group from SU(2) to SO(3)
Lubos Motl has what seems to be a clearsighted outsiders
perspective on this (not being especially an advocate of LQG)

Have you any comment on this----perhaps the change seems insignificant given that one is a 2-fold cover of the other---or does it have some interesting ramifications?

I will edit this to add a link to Lubos Motl's paper, though
I would not be surprised if you had already noticed it.

http://www.arxiv.org/abs/gr-qc/0212096
 
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A good place for online textbooks is

http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html

I printed out "Abstract Algebra, the basic graduate year" by Prof. Robert Ash, and it looks pretty good so far.

A workshop would be pretty nice, since I had already planned to study some algebra this summer anyway.
I got an introduction this semester, and although most of my fellow physics students hated the abstactness of it all, it grew on me. Seems like a fun game to play.
 
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Originally posted by marcus
perhaps the change seems insignificant given that one is a 2-fold cover of the other---or does it have some interesting ramifications?
Interesting ramifications should be searched for experimentally, but it is not at all insignificant that the adequate group is not the simply connected universal cover, but some projection of it. With respect to the covering, this would indicate that the system makes no distinction of the covering elements (as happens at the tangent space level), and probably this has some significant consequences.
 

marcus

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Originally posted by Lonewolf
I like the sound of your idea, Marcus. I'd be interested once my exams are done. I like especially the sound of learning its applications to Physics. We get taught Group Theory, but only in the sense of pure maths.
Lonewolf, I got your PM that exams are over. I am here but
have been preoccupied with an LQG thread in "theoretical"
forum. The thread is about SO(3) and its Lie algebra
so(3). Good stuff to know. Marsden's introductory treatment is good.

Look at Marsden's Chapter 9 "An introduction to Lie groups" if you want.

for some people who have just posted here, they are waaaay
beyond that entrylevel introduction by Marsden. But if you and I want to start talking it has to be somewhere and the beginning is apt to be a good place.
Besides, Jerry Marsden is a CalTech professor and his approach
connects up to the physics-needs of CalTech students. It doesnt look at all "pure" to me, so you might like it.

Do you find anything in Chapter 9 interesting or whatever?

I will go fetch the link and edit it in here. Really nice of Marsden to put it online.

http://www.cds.caltech.edu/~marsden/bib_src/ms/Book/ [Broken]
 
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marcus

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Originally posted by rutwig
Interesting ramifications should be searched for experimentally, but it is not at all insignificant that the adequate group is not the simply connected universal cover, but some projection of it. With respect to the covering, this would indicate that the system makes no distinction of the covering elements (as happens at the tangent space level), and probably this has some significant consequences.
rutwig, thanks
hope to hear further---any thoughts you have about this
very interesting switch to SO(3) or news, if you receive any,
about them finding some way, cunning as they are, to switch groups yet again.
 

marcus

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openers for a workshop on groups

On the off chance that we might have a collective learning effort in classical Lie groups here----which might begin at least by being based on Marsden's chapter 9----I have pasted in this extract dealing with the group of rotations. It is a summary of rotation facts made a day or two ago for a thread in "theoretical" forum.
Maybe it is not the perfect thing for this thread but it is a start.

for the moment I am thinking of this very concretely---not at all abstractly---as 3x3 rotation matrices. Anyone else is welcome to take the lead here, but because nothing is happening as yet I will paste in this extract (essentially part of what is covered by Marsden)

Here are some basic facts about SO(3)
**************************************
SO(3) is a compact Lie group of dimension 3.

Its Lie algebra so(3) is the space of real skew-symmetric 3x3 matrices
with bracket [A,B] = AB - BA.

The Lie algebra so(3) can be identified with R3
the 3-tuples of real numbers by a vectorspace isomorphism
called the"hat map"

v = (v1,v2,v3) goes to v-hat, which is a skew-symmetric matrix
meaning its transpose its its NEGATIVE, and you just stash the three numbers into such a matrix like:

+0 -v3 +v2
+v3 +0 -v1
-v2 +v1 +0

v-hat is a matrix and apply it to any vector w and
you get vxw.

Everybody in freshman year got to play with v x w
the cross product of real 3D vectors
and R3 with ordinary vector addition and cross product v x w is kind of the ancestral Lie algebra from whence all the others came.

And the hat-map is a Lie algebra isomorphism

EULER'S THEOREM

Every element A in SO(3) not equal to the identity is a rotation
thru an angle φ about an axis w.

SO SO(3) IS JUST THE WAYS YOU CAN TURN A BALL---it is the group of rotations

THE EIGENVALUE LEMMA is that if A is in SO(3) one of its
eigenvalues has to be equal to 1.
The proof is just to look at the characteristic polynomial which is of degree three and consider cases.

Proof of Euler is just to look at the eigenvector with eigenvalue one----pssst! it is the axis of the rotation. Marsden takes three sentences to prove it.

A CANONICAL MATRIX FORM to write elements of SO(3) in
is

+1 +000 +000
+0 +cosφ -sinφ
+0 +sinφ cosφ

For typography I have to write 0 as +000
to leave space for the cosine and sine under it
maybe someone knows how to write handsomer matrices?

EXPONENTIAL MAP
Let t be a number and w be a vector in R3
Let |w| be the norm of w (sqrt sum of squares)
Let w^ be w-hat, the hat-map image of w in so(3), the Lie algebra. Then:

exp(tw^) is a rotation about axis w by angle t|w|


It is just a recipe to cook up a matrix giving any amount of rotation around any axis you want.
 

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