What combination of generators can produce a particular SU(2) matrix?

[Einj]

Hello everyone,
I have a question that will probably turn out to be trivial. I have the following matrix:

$$
U=\text{diag}(e^{2i\alpha},e^{-i\alpha},e^{-i\alpha}).
$$

This seems to me as an SU(2) matrix in the adjoint representation since it's unitary and has determinant 1. Am I right?

If so, for a small value of $\alpha$ from what combination of the generators can I obtain it?Thanks!

 

[fresh_42]

SU(3). Which are your generators?

 

[Einj]

Oh you're right. Then my mistake was to try to find a combination of the J=1 generators of SU(2) while I should have looked for the SU(3) generators! Then I guess my question becomes even dumber: how do I distinguish an matrix belong to the adjoint of SU(2) from one belonging to the fundamental of SU(3)?

 

[fresh_42]

I don't know the automorphisms $su(2)$. They are in $GL(3,ℂ)$, that's right.
Have a look here. Just calculate it.

The fundamental group of $SU(2,ℂ)$ is $SO(3,ℝ)$ - as far as I can see - generated by the following rotations

$$\left[ {\begin{array}{*{20}{c}} {e^{it}}&{0}\\ {0}&{e^{-it}} \end{array}} \right] , \left[ {\begin{array}{*{20}{c}} {cos (t)}&{sin (t)}\\ {-sin (t)}&{cos (t)} \end{array}} \right] , \left[ {\begin{array}{*{20}{c}} {cos (t)}&{i * sin (t)}\\ {i *sin (t)}&{cos (t)} \end{array}} \right]$$

Here are further informations.