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I have two questions about representations of compact Lie group.

  1. If all irreducible representations of a compact Lie group are one-dimensinal, then G is abelian.

  2. An infinite compact Lie group has an infinitely number of non-equivalent irreducible representations.

I am very un-familiar with this topic, I know the finite group case, but didn't know how to start these two questions.

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  • $\begingroup$ What tools do you have to work with? For 1., consider the adjoint representation when $G$ is non-abelian... $\endgroup$ – Jason DeVito Dec 3 '14 at 19:01
  • $\begingroup$ I don't know adjoint representation... I know representative functions, matrix coefficients, orthogonal relations, Peter and Weyl theorem... I don't know whether they are helpful or not. $\endgroup$ – CYC Dec 3 '14 at 19:13
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If you know enough about representations of general compact groups to understand the Peter-Weyl theorem then you're more or less there. Knowing about representations of finite groups is good too; after all, compact groups are particularly nice because you can, more or less, get away with pretending that they're finite.

The trick here to make things simpler is to know a bit of Lie theory. Disclaimer: I'm most definitely not a Lie theorist, so these solutions are probably by no means optimal.

One way to see your first question is to know (or learn!) that every compact Lie group admits a faithful representation. Then identify $G$ with its image in this representation, which is a finite sum of $\mathrm{Aut}(V_i)$, for $\mathrm{Aut}(V_i)$ one-dimensional vector spaces. But then $G$ is just isomorphic to a subgroup of $(\mathbb{C}^\times)^n$ for some $n$ and you're done.

For the second, I believe we can actually say something stronger: a compact Lie group $G$ is finite if and only if it has finitely many irreducible representations (which are necessarily finite-dimensional, as $G$ is compact). You can see this using the Peter-Weyl theorem: the space $L^2(G)$ is isomorphic to the direct sum of the irreducible representations of $G$, with multiplicity. But then $L^2(G)$ is finite-dimensional if and only if $G$ is finite, and we're done.

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  • $\begingroup$ Why is $L^2(G)$ isomorphic to direct sum of the irreducible representation of G? and if it is true, why the last statement is true? $\endgroup$ – CYC Dec 3 '14 at 20:24
  • $\begingroup$ $L^2$ under the regular action decomposing as a direct sum of irreducibles with multiplicity is just the Peter-Weyl theorem, which you said you were familiar with. Try thinking about why the second statement is true -- can you think of a canonical way to attach to each element of $G$ a distinct function in $L^2(G)$? Are those functions linearly independent? If you can do that, you're done. $\endgroup$ – PL. Dec 3 '14 at 20:37
  • $\begingroup$ Please give me some details about the second statement, my brain is in a mess right now, I think it could help me understand this topic more. $\endgroup$ – CYC Dec 3 '14 at 20:55

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