Insights Representations Of The Special Unitary Group SU(2)

[fresh_42]

Part 1.

The second part of my journey to the manifold $SU(2)$ deals with some representations. We start with some bases and cite the classification theorem of representations of the three-dimensional, simple Lie algebra.

Continue reading ...

 

[lavinia]

These notes would be helpful for a student who is learning about Lie groups because they work through an important specific example - the example of $SU(2,C)$.

The student would have to master the Lie group technology in a different place.

I especially like the way the Hopf fibration is worked out.

The introductory section on spheres is not specific to $SU(2,C)$ so for me personally it was distracting. I also found the initial example of a local Lie group distracting.

The first paragraph of Part 1 says that it hopes to pique interest in Lie group mathematics. For this, some comment on why representations are important/ interesting - in mathematics - would have helped.

There is a lot of calculation here and some people might like an intuitive beacon to light the way along the journey.

Other thoughts:
- The notes do not require showing that the sphere is a manifold.

- For intuition, one might describe the actions of special orthogonal groups on spheres as rotations. To me this would be more intuitive than matrix multiplication. Even non-mathematicians can imagine a rotation and would immediately see that any rotation must have two fixed poles. The stabilizer then acts transitively on the tangent sphere at the poles. For $SO(3)$ the stabilizer also acts without fixed points and one sees that $SO(3)$ is the tangent circle bundle of the 2 sphere. Perhaps one could go from here to illustrate the difference between $SO(3)$ and $SU(2,C)$.
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[fresh_42]

Thank you for the detailed review, @lavinia.

You are absolutely right, that the initial example and the spheres feel distracting. It disturbed me, too. The reason is, that I originally wanted to focus on vector fields instead of the group. I began by noting, that there is this general vision of vectors attached to points on one hand and the abstract formulas on the other. I thought some examples with actual curves (flows, 1-parameter groups), groups and specific functions would be helpful, as they are often banned to exercises or get lost in the "bigger" theory. That's where those two paragraphs came from. As I looked closer into the example of SU(2) I got more and more involved with it instead of my original purpose vector fields.

So the actual distraction had been SU(2). To be honest, I wanted to understand connections better, esp. Ehresmann and Levi-Civita and hope to deal with it (on the example of SU(2) again) in a third part. So the two parts so far are more of a "what has happened before" part of the story. But the more I've read about SU(2), the more I found it interesting. I kept the distracting parts, as I recognized, that they are a good to quote or a copy & paste source for answers on PF. Up to now, I used the various notations of derivatives as well as the stereographic projection in an answer to a thread here. And as one-parameter groups are essential to the theory, I kept this part. And why not have a list of spheres of small dimensions, when one of them is meant to be the primary example of actual calculations? That's basically the reason for the felt (by you and by me) inhomogeneous structure and why the article is a bit of a collection of formulas.

So thanks, again, and I'll see if I can add a couple of explanations which you suggested.

 

[dextercioby]

It would have helped to describe in maximum 2 paragraphs the connection between a local Lie group and a global Lie group and from here the connection between the notions of globally isomorphic Lie groups and locally isomorphically Lie groups. Physicists usually gloss over these important definitions and theorems.
:)