Solving Lie Group Exercise: Proving U Generates G

[Palindrom]

Hi.

I'm now studying Lie Groups, and have received the following exercise to solve. I have absolutely no idea where to begin, so please give me a direction.

Let U be any neighborhood of e. Prove that any element of G can be written as a finite product of elements from U (i.e., U generates G).

 

[Palindrom]

Nobody? I just need a small hint, as I have no intuition yet...

 

[marlon]

Nobody? I just need a small hint, as I have no intuition yet...

Let me give you a general overview of how the concept of generator is used in Lie algebra.
Look at these differential equations dx/dt = x and dy/dt = y.

I can write these like this [tex]\frac {dx^{i}}{dt} = X^{i}(x')[/tex] where the [tex]X^{i}[/tex] denotes a vector field equal to [tex]( x \frac{d}{dx},y \frac{d}{dy} )[/tex] and the x' denotes the variables, here : (x,y).
This field is an operator.

Now suppose we write the solutions of the above differential equaltions as [tex]x^{i} = p(t,y)[/tex], where t is a parameter and y is the solution for t = 0 (Cauchy existence theorem)

If you now expand this solution in a Taylor series with respect to parameter t (so you will get a power series in terms of t and the coefficients are derivatives of solution p with respect to t). You should be able to write down the solution in terms of an exponential that contains the generator X.

Keep in mind that X is defined as the first derivative of the solution p(t,y) with respect to t for t = 0. So you know what the first coefficient is of the t-term in the expansion.
If you are able to do that, you are well on your way.

In extension. If you look at transformations with generator G that leave the differential equation invariant, you will find a nice connection between G (generator of symmetry transformations) and X (generator of the solutions of the differential equation)

regards
marlon

 

[Palindrom]

May God help me.

Well first of all, thanks.

Second of all, the only thing we did in class was defining a Lie Group. I have no idea what it means to differentiate, let alone diff. equations... Isn't there an elementary way to approach this problem?

We're not going to talk about diff. equations until the middle of the semester, and this exercise is to be handed in in a week and a half, so I'm guessing he didn't mean something like that.

Did you say Taylor expansion? God.

 

[marlon]

May God help me.
Well first of all, thanks.
Second of all, the only thing we did in class was defining a Lie Group. I have no idea what it means to differentiate, let alone diff. equations... Isn't there an elementary way to approach this problem?
We're not going to talk about diff. equations until the middle of the semester, and this exercise is to be handed in in a week and a half, so I'm guessing he didn't mean something like that.
Did you say Taylor expansion? God.

Err, ok. Could you please give more info on what you have seen so far concerning Lie-Algebra. I mean within what context do you need to solve this problem ?

Have you seen the concept of a commutator ?

Please provide me with some more info if you can

marlon

 

[Palindrom]

I'd be happy to.

Here's what I've seen:

Def: A Lie group is a smooth manifold G, provided with m:GxG->G and s:G->G, s.t. (G,m) is a group, s(g)=g^(-1), and m,s are smooth.

And that's all...

 

[Palindrom]

To be exact, we also said that an equivalent def. would have been to require t(g,h)=gh^(-1) is smooth, and a few examples.

 

[mathwonk]

this statement seems false for lie groups with more than one component, such as GL(n). but what do i know? (not much) just my intuition.

if you assume conectedness, then the standard method is to show that the set of elements which can be written as such a sum is both open and closed.

this seems fairly easy, and openness is trivial.

for closedness I used that the nbhd contains a symmetric one, i.e. that you may assume: if y-x is in U then so is x-y.

 

[Palindrom]

The exercise is as I stated it. I'm absolutely not in the position to say you're wrong, but it didn't say "Prove or give a counter example", it said "Prove"...

And I'm still trying to understand why you're not losing generality with your assumption. :)

 

[matt grime]

I tend to go with mathwonk on this one: SO(n) in O(n).

counter examples seem to be constructible.

the closest result i can think of is cosets on compact lie groups.

 

[Palindrom]

I, too, agree now with mathwonk. I even understand why he's not losing generality.

I think you've given me enough hints for that exercise, and I should be able to solve it on my own now. If I get stuck with it, I'll bother you again.

Thanks a lot.

 

[Palindrom]

O.K., I'm still stuck. I can't seem to show closedness. Help?

Edit: Is it true that m and s are always open maps?
And, is this statement (the one in the head of the thread) also true in topological groups, or do I need the smoothness of m and s?

 

[Palindrom]

uh... mathwonk? Closedness?