Questions for compact lie group representations. I have two questions about representations of compact Lie group.


*

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

*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.
 A: 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.
