Explaining SO(3) and U(2) Lie Group Relationships to Non-Experts

[belliott4488]

What's the correct way to state the relationship between these two Lie groups? One is the "covering group" of the other, right? Okay, then - what's that mean, to a non-expert?

I know the basics, i.e. SO(3) can be represented by rotation matrices in 3-space, and U(2) does the same in a complex 2-space, but how are the two connected?

What I'd really like to know is how to explain to non-physicists (like the engineers I work with) how it is that quaternions are used to represent body orientations in 3-space and why the angles pick up a factor of 1/2. I know it's connected to the business of a 2-pi rotation in complex 2-space picking up a factor of -1 so that you have to do a rotation by 4-pi to get back to your initial orientation ... but I don't really know what that means.

Any helpful pictures or explanations?

Thanks,
Bruce

 

[mathwonk]

you might consult michael artins algebra book.

the base point of the connection is probably the fact that the complex one dimensional projective space, is homeomorphic to a 2 sphere in 3 space.

 

[matt grime]

It means that there is a 2 to 1 map (homeomorphism) from one to the other, and SU(2) is simply connected.

We thus have the space of deck transformations (this is a group action on the preimages of a point, essentially) as C_2.

OK, that didn't help. But to visualize things, we have "The Soup Bowl Trick" to help us.

Imagine you are holding a bowl of soup in the palm of one hand in front of you. Your task it so spin it through 720 degress without spilling a drop. You can do this - start with it in your right hand, lift it up so you look like the statue of liberty, now rotate your hand clockwise - you'll now have to bend your elbow to do this and bring the bowl down and under your armpit. So, the bowl has spun twice round, in one loop...

 

[belliott4488]

Well, thanks, Matt, but I've never been able to get from that trick (also known as the Filipino candle dance, after a traditional folk dance where they do the same thing with candle on the palms of their hands - as well as on their heads) to understanding the space of two complex dimensions.

We live in the space of three real dimensions, don't we? So what do such tricks have to do with U(2)?

 

[matt grime]

It tells you what is happening with the rotations, i.e. it shows that the group SO(2) has fundamental group C_2. It isn't saying anything about SU(2) at all. That there is a double cover by SU(2) is a nice mathematical fact. I seem to remember you can use the unit quarternions in a very nice way to write down the map.

 

[mathwonk]

projective one space is by definition the set of complex lines throiugh the origin of complex 2 dimensional space.

an element of U(2) is a complex linear map of C^2 and hence carries lines through C^2 to lines through C^2, i.e. induces a map of the projective line to itself.

now the complex projective line is homeomorphic to a sphere in R^3, hence an element of U(2) induces a map of S^2 to itself. one then needs to check that an element of U(2) induces a length preserving map of the sphere to itself.

i.e. this is the map U(2)-->O(3).

 

[belliott4488]

Whoa, mathwonk - you said a mouthful! I think I'd like to reach the point where I could read your last post and actually understand it ... time to go get an algebra book and start working on it, maybe!

Thanks,
Bruce

 

[mathwonk]

try artins algebra for a less abstract, more explicit discussion

if the covering map is supposed to be 2:1, try to see why two elements of U (2) induce the same map of linrs through the origin. minus signs will presumably be involved.