Group Theory Basics: Where Can I Learn More?

[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.

 

[BoulderHead]

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

 

[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...

 

[rutwig]

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.

 

[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]

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

 

[quantumdude]

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.

 

[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?

 

[rutwig]

Here a book that has been a quite interest source for many physicists/


http://www.cns.gatech.edu/GroupTheory/index.html

 

[quantumdude]

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]

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?

 

[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.

 

[marcus]

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.

 

[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.

 

[marcus]

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

 

[Kalimaa23]

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.

 

[rutwig]

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]

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 doesn't 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/

 

[marcus]

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]

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.

 

[KillaMarcilla]

Wait, wait, wait, is group theory just that mathematical dealy wherein you count numbers by grouping them in various ways, like if you want to prove there are more numbers between 0 and 1 than there are integers greater than zero?

 

[chroot]

Originally posted by KillaMarcilla
Wait, wait, wait, is group theory just that mathematical dealy wherein you count numbers by grouping them in various ways, like if you want to prove there are more numbers between 0 and 1 than there are integers greater than zero?

No. Group theory deals with sets of mathematical entities and operations upon those entities.

For example, take the set of real numbers and the addition operation. Together, the set of reals and the addition operation form a group. Groups have the following properties:

1) The result of applying the operator to any two elements of the group is itself an element of the group. (The sum of any two reals is itself a real.)

2) Every group has an identity element, such that the operation of any element with the identity returns that element. (The sum of any real with zero is left unchanged -- zero is the identity.)

3) Every element in a group has an inverse element. (The inverse of 1, for example, is -1.)

4) For any three elements in the group, (A + B) + C is the same as A + (B + C).

Marcus is talking about groups of 3x3 matrices. These groups are given names like SO(3) and so on to reflect the various characteristics that elements of each group share. The operation on these groups is that of matrix multiplication.

- Warren

 

[KLscilevothma]

Introductory book on group theory

Mathematical Groups (teach yourself) by Tony Barnard and Hugh Neill is a good book that introduces basic concepts of group. Topics include properties of group, notations, cyclic groups, isomorphism, etc. There are sufficient examples for beginners to understand and suitable for senior high school students or above.

 

[marcus]

Does everybody know matrix multiplication
and what a matrix transpose is?

(you get the transpose of a square matrix by flipping it over its main diagonal)

If you dont, please ask. If A is a square matrix the transpose is A^t.

If anyone has different notation from me they like better I am open to changing notation as long as I can type it easily.

We might have a small informal workshop on matrix groups
right here in the "Groups for Dummies" thread with no fanfare.
It might work, and no harm done if it didnt. But someone besides me has to do the lion's share of the explaining or I will get
too boring and monotonous.

There is a cool kind of matrix whose transpose is equal to its inverse (something you don't normally expect!)

A^t = A^-1


√1/2 -√1/2
√1/2 √1/2

 

[KillaMarcilla]

Yeah, that's what I meant, chroot

h0 h0, and here I thought Group Theory was some arcane mystery, unknowable to low-level undergraduates like myself

 

[Lonewolf]

What is the Lie algebra, and how does it relate to its respective Lie group?

 

[quantumdude]

Originally posted by KillaMarcilla
h0 h0, and here I thought Group Theory was some arcane mystery, unknowable to low-level undergraduates like myself

Actually, group theory is one of the few mathematical subjects that has no prerequisites. It is purely axiomatic and logical.

 

[quantumdude]

Originally posted by Lonewolf
What is the Lie algebra, and how does it relate to its respective Lie group?

A Lie algebra is a nonAbelian algebra whose elements a_i satisfy the following properties:

1. [a_i,a_i]=0 (a_i commutes with itself.)
2. [a_j+a_k,a_i]=[a_j,a_i]+[a_k,a_i] (Linearity of commutator.)
3. [a_i,[a_j,a_k]]+[a_j,[a_k,a_i]]+[a_k,[a_i,a_j]]=0 (Jacobi identity.)

The relation between the Lie algebra and the Lie group is that the elements of the algebra generate the group.

Example:

Consider the Lie algebra of the angular momentum operators in quantum mechanics:

[J_i,J_j]=i(hbar)ε_ijkJ_k

The elements J_i generate the Lie group of rotations D about a normal vector n through an angle φ as follows:

D(n;φ)=exp(-iJ^.nφ/(hbar))

Let me know if you want to go into more detail.

edit: typo

 

[marcus]

Originally posted by Lonewolf
What is the Lie algebra, and how does it relate to its respective Lie group?

a big question.
I hope others will help answer. gradually filling in the gaps
in the picture.

A Lie group is a group that is a smooth manifold
and each point x in a manifold has a tangent space T_x
and the tangent space at any point is a vector space

That is intuitive I guess----the space of tangent vectors at a point, in the 2D case a tangent plane.

But if the manifold is a group then there is a special element, the identity element of the group. Call it e.

So there is a special tangent space T_e of tangent vectors at the identity e.

that is what the Lie algebra is, as a set. But it has a lot of uses and more structure than you expect from just a vector space (with a zero and vector addition and all). There is a way of parlaying the group multiplication (which goes on down in the manifold) up into the tangent space---so that you get an operation up in the tangent space sort of like multiplying two vectors to get a third vector. This "multiplication-like" operation is called "bracket" and is written [A,B] where A,B are vectors in T_e the tangent space at the identity. It is not commutative, but then not all forms of multiplication are.

The Lie algebra is kind of an "infinitesimal" version of the group and the group (at least neighborhood of the identity and actually more) can be regenerated from the algebra.

If you lose the group you can grow (at least a piece of) it back just from the tangent space at the identity---the algebra.

Maybe someone else will supply a rigorous definition or some more intuitive insight.

What I would like to do is stop here and look at one simple example of a Lie group and its Lie algebra. Then let someone else take over, if they want.

 

[marcus]

I just happened to notice that Greg Tom and chroot are browsing the math forum and any of them could give a rigorous def
of a Lie group (I have not given the definition so far) and its Lie algebra.

and that would be a step in the right direction (of collectivizing and getting several persons approaches)

 

[quantumdude]

Originally posted by marcus
I just happened to notice that Greg Tom and chroot are browsing the math forum and any of them could give a rigorous def
of a Lie group (I have not given the definition so far) and its Lie algebra.

Actually, I can't. They did not talk about this stuff in my Abstract Algebra course. I only know about it through my QM courses, which is why I talk in terms of examples. We need people such as Hurkyl, Lethe, SelfAdjoint, etc... to provide the rigorous generalities.

and that would be a step in the right direction (of collectivizing and getting several persons approaches)

I posted mine a few seconds before yours (see above).

 

[marcus]

it rarely a mistake to look at examples before studying
abstract definitions and my favorite example of a Lie group/algebra
is rotations/skew-symmetric matrices.

Everybody has had linear algebra so probably know
A^t the transpose of a (real) matrix and
A^-1 the matrix inverse
and may also know that an orthogonal matrix
(one that doesn't change the length of vectors
obviously a very valuable interesting kind and it
also does not change their inner product when you apply
it to two vectors)
this very nice kind of matrix is described by
A^t = A^-1

Now those things form a group because if A and B dont
change lengths or inner products then AB will not either
and you can also check the A^t = A^-1
condition for AB.

But they arent a vector space because if you add two A+B
is usually not that kind of matrix any more

Everybody knows the determinant and that detA^t = detA
and that detA^-1 = 1/detA
So if you look at the A^t = A^-1 condition in that light you will see that there is no possible thing that det A can be except +1 or -1. The matrices with det = +1 form a subgroup.

These are very nice simple useful Lie groups and the question is, what is the Lie algebra. What does the tangent space at the
identity matrix look like?

So you Lonewolf ask "what is the Lie algebra" and I am temporarily turning this question into a very concrete one: "what is the Lie algebra of this particular group of matrices, the orthogonal ones, or the subgroup of them which are simple rotations. We can try to answer that in either 2D or 3D.
Are there any questions so far?

Anybody who wishes is invited to take over explaining and discussing at this point.

 

[marcus]

Originally posted by Tom
...know about it through my QM courses, which is why I talk in terms of examples.


I posted mine a few seconds before yours (see above).

Glad to see you here
We may need examples far more than rigor
Would invite and encourage examples

 

[Hurkyl]

We need people such as Hurkyl, Lethe, SelfAdjoint, etc... to provide the rigorous generalities.

Eep!


Paraphrased from my abstract algebra text:

A Lie Algebra is simply a vector space A over a field F equipped with a bilinear operator [,] on A that satisfies [x, x] = 0 and the jacobi identity:

[[x, y], z] + [[y, z], x] + [[z, x], y] = 0


(If F does not have characteristic 2, [x, x] = 0 is equivalent to [x, y] = -[y, x])


I would like to point out that [x, y] is not defined by:

[x, y] = xy - yx

(or various similar definitions); it is merely a bilinear form that satisfies the Jacobi identity and [x, x] = 0.


However, for any associative algebra A, one may define the lie algebra A^- by defining the lie bracket as the commutator.


An example where [,] is not a commutator is (if I've done my arithmetic correctly) the real vector space R³ where [x, y] = x * y, where * is the vector cross product.


edit: fixed an omission

 

[chroot]

Quoting from John Baez' "Gauge Fields, Knots, and Gravity,"

"Lie algebras are a very powerful tool for studying Lie groups. Recall that a Lie group is a manifold that is also a group, such that the group operations are smooth. It turns out that the group structure is almost completely determined by its behavior near the identity. This, in turn, can be described in terms of an operation on the tangent space of the Lie group, called the 'Lie bracket.'

"To be more precise, suppose that G is a Lie group. We define the Lie algebra of G, often written g, to be the tangent space of the identity element of G. This is a vector space with the same dimension of G. A good way to think of Lie algebra elements is as tangent vectors to path in G that start at the identity. An example of this is the physicists' notion of an 'infinitesimal rotation.' If we let [gamma] be the path in SO(3) such that [gamma](t) corresponds to a rotation by the angle t (counterclockwise) about the z axis:

[gamma](t) =

  cos t   -sin t   0
  sin t   cos t    0
   0        0      1

"Then the tangent vector to [gamma] as it passes through the identity can be calculated by differentiating the components of [gamma](t) and setting t = 0:

[gamma]'(0) =

0  -1   0
1   0   0
0   0   0

This is an element of so(3), the Lie algebra of SO(3). Any such matrix, which is the tangent vector to a path through the identity of SO(3), is a member of so(3).

- Warren