How can we interpret the representations of SU(2) and $\mathfrak{su}(2)$ in physics?

I have studied a lot of mathematics including representation theory and differential geometry, so I understand SU(2) and $\mathfrak{su}(2)$ from a mathematical background, however I have not studied any physics. So when I look to understanding these (and other Lie groups) in the context of physics I quickly get confused about the standard model and so on. Nevertheless, there seems to be an interesting story between mathematics and physics here.

Could anyone please provide , from a mathematicians point of view, how do the representations SU(2) and $\mathfrak{su}(2)$ works in the world of physics? Similarly, any information regarding SU(3) and $\mathfrak{su}(3)$ would be interesting. Any useful sources would also be greatly appreciated.

Thankyou in advance!

  • $\begingroup$ Maybe start with seeing how U(1) works with electromagnetism then move on to non-Abelian for the weak force interactions. $\endgroup$ – user121049 May 1 '17 at 8:32
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    $\begingroup$ In quantum mechanics angular momentum (spin) $k\in\Bbb Z/2$ corresponds to (or is) the irreducible representation of $SU(2)$ of dimension $2k+1$. You will find representations of $SO(3)$ (and thus of $SU(2)$) also in classical physics under the name multipole expansion. $\endgroup$ – user8268 May 1 '17 at 12:43

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