SO(2) is a continuous abelian group describing proper rotations in 2d space. Because all elements commute, each conjugacy class contains only one element. Thus, there are uncountably many classes that can be parameterized by the angle $0\le\phi<2\pi$. According to the group theory, the number of classes is equal to the number of irreducible representations (irreps). Thus, there should be uncountably many irreps. However, the irreps are known. They can be parametrized by integer (or half-integer) $J$, which has the meaning of the angular momentum in quantum mechanics. There are countably many of them. What are the remaining uncountably many representations? Am I missing something in these considerations?
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I think you are confusing ${\rm SO}\left(2\right)$ and ${\rm SU}\left(2\right)$ in the comment about quantum mechanics. – eranreches Jun 07 '19 at 07:51
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@eranreches No, SO(2) is a subgroup of SO(3). – yarchik Jun 07 '19 at 07:53
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The statement “the number of classes is equal to the number of irreducible representations” holds for finite groups, but not in general. And $SO(2,\mathbb R)$ is infinite.
José Carlos Santos
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Right, it is the Peter-Weyl theorem. What would be an intuitive explanation why this is the case? – yarchik Jun 07 '19 at 07:55
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I'm not sure I understand your question. The Peter-Weyl theorem, in the case of $SO(2,\mathbb R)$, simply asserts that the linear combinations of functions of the type$$\begin{bmatrix}\cos\theta&-\sin\theta\\sin\theta&\cos\theta\end{bmatrix}\mapsto e^{in\theta}$$($n\in\mathbb N$) are dense in the space of continuous functions from $SO(2,\mathbb R)$ into $\mathbb C$, with respect to the $\sup$ norm. – José Carlos Santos Jun 07 '19 at 08:04
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I wanted to intuitively understand why “the number of classes is equal to the number of irreducible representations” fails for infinite compact groups? – yarchik Jun 07 '19 at 08:11
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That is another question. Please post it as such. Concerning the question that you posted here, I believe that I provided an answer. – José Carlos Santos Jun 07 '19 at 08:13
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