The groups O(3), SO(3) and SU(2)

[Rory9]

Homework Statement

How can irreducible representations of O(3) and SO(3) be determined from the irreducible representations of SU(2)?

The Attempt at a Solution

I believe there is a two-one homomorphic mapping from SU(2) to SO(3); is that enough for some shared representations? If I had an idea of *why* irreducible reps. can determined for O(3) and SO(3) from SU(2), I might have a better notion of *how* to go about proving it.

Cheers!

 

[George Jones]

Mathematically, what is a representation of a group G?

 

[Rory9]

Mathematically, what is a representation of a group G?

Typically a matrix, I believe, for which $$\Gamma(T_{1}T_{2}) = \Gamma(T_{1})\Gamma(T_{2})$$ holds, where $$T_{1}, T_{2}$$ belong to $$G$$

 

[dextercioby]

There are 2 isomorphisms you need to use:

$$\mbox{SO(3)}\simeq\frac{\mbox{SU(2)}}{\mathbb{Z}_{2}}$$

and

$$\mbox{O(3)} = \mbox{SO(3)} \times \{-1_{3\times 3}, 1_{3\times 3} \}$$

There are at least 50 books or so discussing the connection b/w SO(3) and SU(2), however there are many less computing all representations of O(3) starting from the ones of SO(3) deducted from the ones of SU(2).

 

[Rory9]

There are 2 isomorphisms you need to use:

$$\mbox{SO(3)}\simeq\frac{\mbox{SU(2)}}{\mathbb{Z}_{2}}$$

and

$$\mbox{O(3)} = \mbox{SO(3)} \times \{-1_{3\times 3}, 1_{3\times 3} \}$$

There are at least 50 books or so discussing the connection b/w SO(3) and SU(2), however there are many less computing all representations of O(3) starting from the ones of SO(3) deducted from the ones of SU(2).

Thank you very much for your answer. I understand the second statement, but what exactly are you doing in the first - simply slicing off the complex aspect by mathematical fiat?

Cheers :)