Understanding Lie Groups: A Simple Definition

[Matterwave]

Hi, so I didn't see exactly where group theory stuff goes...but since Lie groups are also manifolds, then I guess I can ask this here? If there's a better section, please move it.

I just have a simple question regarding the definition of a Lie group. My book defines it as a group which is also a smooth manifold (e.g. its elements can be smoothly and continuously parameterized), and the group operation induces a smooth map of the manifold into itself.

I understand the first part of the definition, but I'm having trouble understanding the second half. What does it mean for a group operation to "induce" a map, and what exactly does it mean to have a smooth map of a manifold into itself? I mean I can picture a map which maps a manifold into itself, like a map from R->R (a simple function f(x) should do the trick right), but why impose that restriction? My book says something about making the group structure compatible with the structure of a manifold, but I didn't quite understand that either. Some help would be nice, thanks. =]

 

[lavinia]

Hi, so I didn't see exactly where group theory stuff goes...but since Lie groups are also manifolds, then I guess I can ask this here? If there's a better section, please move it.

I just have a simple question regarding the definition of a Lie group. My book defines it as a group which is also a smooth manifold (e.g. its elements can be smoothly and continuously parameterized), and the group operation induces a smooth map of the manifold into itself.

I understand the first part of the definition, but I'm having trouble understanding the second half. What does it mean for a group operation to "induce" a map, and what exactly does it mean to have a smooth map of a manifold into itself? I mean I can picture a map which maps a manifold into itself, like a map from R->R (a simple function f(x) should do the trick right), but why impose that restriction? My book says something about making the group structure compatible with the structure of a manifold, but I didn't quite understand that either. Some help would be nice, thanks. =]

- A smooth map has partial derivatives of all orders. A simple function, f(x), may not even be continuous.

- There are topological groups that are not manifolds and whose group multiplication is only continuous.

- In a Lie group, the smoothness of the group multiplication implies rich mathematical structure.

- Lie groups appear naturally in many areas of mathematics and physics - e.g. differential geometry, quantum mechanics.

 

[Monocles]

Here is an intuitive explanation of what it means where I will try to appear to visual intuition. Let G denote the Lie group of unit complex numbers under multiplication, and pick two elements g and h from it, and let U be a small neighborhood of h and consider the image g(U). We note that gh is an element in g(U). Pick another element h' in U, and imagine slowly varying gh to gh'. This will trace out a smooth path in g(U) - it is precisely this smoothness that is meant by a smooth map induced by the group operation onto itself. Technically, this means it is infinitely differentiable.

Now the actual map induced by the group operation on G is defined by picking any g in G and then looking at the image g(G), i.e., replace all elements h with gh. What this would correspond to on our example is simply rotations of the unit complex circle. Do you see why this is true?

 

[HallsofIvy]

the group operation induces a smooth map of the manifold into itself.

For any element a in the group, the function f_a G-> G defined by f_a(x)= a*x (where * is the group operation) is a continuous function.

From that one can show that the map from GxG to G given by (a, b)-> a*b (where * is the group operation) is a continuous function and the map from G to G given by a-> a^{-1} is continuous.

 

[Matterwave]

Ok, I think I got it. Just to make sure I got it right, can you guys verify that my example below is correct? I work well with concrete examples.

Suppose I take the group of all rotations about the origin in 2-D. I can continuously parameterize this group by the angle theta through which I rotate, and therefore this group is mappable to an open set in R, and is therefore a manifold.
Next, if I pick out an element from the group, I will call it "a" - a rotation around angle pi/2, then for "b" a rotation around angle theta, a*b=rotation around angle theta+pi/2. If I vary my parameter theta smoothly (varying "b" continuously), then I obviously get the total rotation a*b which varies smoothly. Therefore, I conclude that the group of all rotations in a 2-D plane about the origin is a Lie Group.

This is correct?

 

[micromass]

Ok, I think I got it. Just to make sure I got it right, can you guys verify that my example below is correct? I work well with concrete examples.

Suppose I take the group of all rotations about the origin in 2-D. I can continuously parameterize this group by the angle theta through which I rotate, and therefore this group is mappable to an open set in R, and is therefore a manifold.
Next, if I pick out an element from the group, I will call it "a" - a rotation around angle pi/2, then for "b" a rotation around angle theta, a*b=rotation around angle theta+pi/2. If I vary my parameter theta smoothly (varying "b" continuously), then I obviously get the total rotation a*b which varies smoothly. Therefore, I conclude that the group of all rotations in a 2-D plane about the origin is a Lie Group.

This is correct?

This is indeed correct. Note that the elements of your manifold are \phi_\theta(x)=e^{ix}. In fact, we have a diffeomorphism that does

Rotations\rightarrow \text{Sphere in}~\mathbb{C}:~\phi_\theta\rightarrow e^{i\theta}.

The group operation on the rotations then correspond to the multiplication in the complex numbers. Since this multiplication is smooth, it follows that your group operation is also smooth...

 

[micromass]

Also, if I'm not mistakesn you'll have to show that taking the inverse is smooth...

 

[Matterwave]

baby steps! haha, ok thanks =D