# Thread: Group Theory

1. I was wondering if anyone could tell me, is Group theory typically taught to undergraduate or postgraduate mathematics students?

Also is it considered to be hard?

Thanks in advance, sox.

2.

3. I don't now at what stage Group theory is taught but being new to the subject myself I found the following site very useful:

http://dogschool.tripod.com/index.html

From what I've seen so far, Group theory seems to be very intuitive and easy to grasp.

4. It depends on how deep the course material is. It could even be taught as part of a senior year high school algebra course, since it doesn't require any calculus.

5. I was taught group theory in an "abstract algebra" class in undergrad. Yes, it seemed difficult.

6. Originally Posted by sox
I was wondering if anyone could tell me, is Group theory typically taught to undergraduate or postgraduate mathematics students?

Also is it considered to be hard?

Thanks in advance, sox.
You would see basic group theory in either an undergraduate or graduate first class in algebra.

It is not hard. It is probably the easiest subject in algebra, at least at the introductory level. The complete classification of finite simple groups is hard, but is not usually part of the subject matter in most classes.

There are lots of good books that treat the subject. Either Rotman's or Hall's books on group theory are good. Also Artin's or Lang's books on algebra.

7. Thanks for all the replies.

I'm currently looking at things like Lie Algebras, SU(n), SO(n) and a load of other stuff like adjoints, roots and weights?? Then the class moves on to things like Dynkin diagrams.

It's totally beyond me at the moment.

8. And I am not surprised! As Rocket says, the basic concept of a group is simple to understand; the integers are a group with addition as the group operation, the additive inverse and additive null being well-defined.

One then quickly moves on to groups whose elements are not "everyday objects" like integers, but rather are, say, permutations (the so-called symmetric groups).

This takes a little getting used to, but falls into place given the willingness to think abstractly.

The Lie groups are a different matter altogether. It is my opinion, and that is all it is, that these guys are quite impossible to understand unless one has done quite a bit of manifold theory, since a Lie group is a manifold endowed with the structure of a group (which could easily be stated the other way round).

Understanding these dual properties (groups and manifolds) are essential to understanding the Lie algebras.

What is your level of education? Are you taking this as a formal class, or are you trying to self learn? If the latter, I suggest to get groups and manifolds under your belt first, assuming you have a good grounding in calculus and the theory of vector spaces.

Otherwise, start there. Don't try to run before you can walk!

9. Its a formal class as part of my particle physics and string theory postgrad course.

As for manifold theory, I've not studied that as a subject. Used the idea in GR but thats it.

The book I'm waiting on is by Georgi. It can't come soon enough.

Out of interest, you say for integers that addition is an operator; are all operators that simple or reduce to something that simple? That is things like add, multiply etc? Or can you have more complicated operators?

10. Originally Posted by sox
Its a formal class as part of my particle physics and string theory postgrad course.

As for manifold theory, I've not studied that as a subject. Used the idea in GR but thats it.
Well, I must say I do find this strange. I fail to see how can a graduate student studying string theory, and expected to do original research in this area in subsequent years - and not too far ahead at that - would not be expected to know about elementary group theory, the Lie groups and everything there there is to know about manifolds.

Of course, I may easily be wrong, never having been a graduate student in any area of physics.

Where do you do your graduate study in particle physics and string theory, as a matter of interest?

11. Originally Posted by Guitarist
Where do you do your graduate study in particle physics and string theory, as a matter of interest?
Centre for Particle Theory at Durham University, England.

http://www.cpt.dur.ac.uk/

And this is the Handbook for the course I'm doing. It has a description for all the classes in it.

http://www.maths.dur.ac.uk/~dma0vh/CPT/handbook.pdf

12. Yeah, looks like an interesting course. I envy you!

Originally Posted by sox
Out of interest, you say for integers that addition is an operator; are all operators that simple or reduce to something that simple? That is things like add, multiply etc? Or can you have more complicated operators?
Well first, I said that addition etc is an operation. The term operator, also known as a "linear transformation", is generally reserved for linear maps that send a vector space to a vector space which may or may not be the same one. There is a fundamental connection between vector spaces and groups, which you may or may not know about yet, but for now I would advise not muddling standard terms.

In group theory, operations are by definition any closed binary "action" of the form where the "times" denotes the Cartesian product on the underlying sets (actually in the standard lexicon, the term "binary" is taken to imply closure).

I am unsure what you would mean by a "complicated" operation. Do you think, for example, that matrix multiplication is a "complicated operation"?

If you want to start a thread on group theory in general and the Lie groups in particular, feel free. There are many here who can help you through the preliminaries

13. and many more who enjoy watching the learning process you'll go through, picking up on bits and pieces as you go.

14. Originally Posted by sox
Thanks for all the replies.

I'm currently looking at things like Lie Algebras, SU(n), SO(n) and a load of other stuff like adjoints, roots and weights?? Then the class moves on to things like Dynkin diagrams.

It's totally beyond me at the moment.
Lie groups and Lie algebras are another kettle of fish entirely. A Lie group is a group that is also a differential manifold and for which the group operation is smooth. The representation theory of Lie groups involves quite a bit of mathematics that you have yet to see. You will need a lot more background before you can undertake a serious study.

15. Well my book came today so I'll see if it helps make any sense of it. I'll probably have to post a few questions though.

Thanks for all the contributions so far.

16. Excellent! Happy reading (don't forget the exercises though!).

But sure, ask away here. We are all (I hope) happy to help those who are trying help themselves (Yuk. Did I really say that?)

17. Ok I've started reading the book a bit, and there are two things that I'm not sure about.

1) Generators

I'm not entirely sure what they are. Are they constructs which enable us to move from the representation for the identity element of a group to some nearby representation?

So if we have for the representation for the identity element:

Then the representation for an element in the group near to the identity could be given by:

.

Where is a pre-factor, is the generator and is a continuous parameter that the group elements depend on.

Is this the right way of thinking of it?

2) Lie Algebras

After plodding through some derivation, the book gets to the following expression:

Where are the structure constants of the group and again are generators.

The book then says "The commutator in the algebra plays a role similair to the multiplication law for the group"

I don't understand how this relates to the group/representation multiplication law at all.

Any pointers would be greatly appreciated. I apologise in advance for the equations, I've simplified the constants alot.

To whom it may concern, all material above is copied or modified from: Lie Algebras in Particle Physics by Howard Georgi

18. Originally Posted by sox
Ok I've started reading the book a bit, and there are two things that I'm not sure about.

1) Generators

I'm not entirely sure what they are. Are they constructs which enable us to move from the representation for the identity element of a group to some nearby representation?

So if we have for the representation for the identity element:

Then the representation for an element in the group near to the identity could be given by:

.

Where is a pre-factor, is the generator and is a continuous parameter that the group elements depend on.

Is this the right way of thinking of it?

2) Lie Algebras

After plodding through some derivation, the book gets to the following expression:

Where are the structure constants of the group and again are generators.

The book then says "The commutator in the algebra plays a role similair to the multiplication law for the group"

I don't understand how this relates to the group/representation multiplication law at all.

Any pointers would be greatly appreciated. I apologise in advance for the equations, I've simplified the constants alot.

To whom it may concern, all material above is copied or modified from: Lie Algebras in Particle Physics by Howard Georgi
1. you need to understand elementary group theory before you even attempt to tackle Lie groups.

2. The Lie algebra for a Lie group is a non-associative algebra structure on the tangent space at the identity.

This probably makes little sense to you at this stage. That is because you need a LOT more background in mathematics before you will be ready for Lie groups. Based on what you have said in the past regarding your background, you are at least two years of intensive study from the requisite background.

19. Aye I think you're right.

Sadly I don't have 2 years though. I have about 7 weeks.

I'll take on board what you've said about tangent space though.

20. To Sox:

There's an introductory book on Lie Groups by J. Frank Adams called "Lectures on Lie Groups". I remember thinking of this book as a real gem when I was in grad school. It seems to be available on Amazon.

This was one of the only books on Lie Groups that I found both useful and easy to read.

I think that physicists are not necessarily the best people to learn mathematics from. They're good at doing math, but to teach this stuff to a beginner, you need to have a VERY deep mathematical intuition, which is something that I suspect many physicists lack.

21. If you want to understand the relationship between group operations and lie algebra operations though, think about the rotation group:

Think of doing a small rotation about the x-axis, a small rotation about the y-axis, the opposite rotation about x-axis, the opposite rotation about the y-axis. This combination of rotations:

Rotation_x * Rotation_y * Rotation_x^-1 * Rotation_y^-1 is approximately a rotation about the z-axis.

Infinitesimally this translates as [ L_x, L_y ] = L_z, where L_x, L_y etc are elements in the Lie algebra that generate the rotations about the x-axis, y-axis, etc. (I may have my signs wrong, but you get the idea ).

Incidentally, you may recognize a similar relationship between the angular momentum operators L_x, L_y, L_z in quantum mechanics.

22. Salsaonline,

I've just been looking at that book you reccomended on amazon and tried to look through the contents to see what the book contained.

The contents page wasnt very useful and the book itself only had a handful of pages that I could see, so I was wondering if you could tell me whats in the book?

Would any of the areas I mentioned on the first page of posts be included? I know there is a bit on roots and weights in the book.

I notice the book si quite cheap.

Scrap that, it's in the library! I'm off to get it! Thanks alot! Bye!

23. I've been studying some physics and it's used some of this group theory stuff.

I think I understand how the generators work now.

Does anyone understand how group theory works for Lorentz transformations?

That's the example I've got in mind which I think I understand, so was hoping I could cross-check my understanding with someone elses...

Cheers in advance for any help.

24. Originally Posted by sox
I've been studying some physics and it's used some of this group theory stuff.

I think I understand how the generators work now.

Does anyone understand how group theory works for Lorentz transformations?

That's the example I've got in mind which I think I understand, so was hoping I could cross-check my understanding with someone elses...

Cheers in advance for any help.
The Lorentz transformations form a group, the Lorentz group. It is a matrix group, so a Lie group.

Take a look at Naber's book, The Geometry of Minkowski Spacetime.

If you simply look at the group of transformations that preserve the Minkowski inner product, the analog of the usual orthogonal group, and add some conditions to preserve the direction of time, you can develop all of special relativity -- what you have are the orthochronous homogeneous Lorentz transformations.

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