TITLE: Representation Theory of Lie Groups: Reference Request
QUESTION [12 upvotes]: I am looking for a reference that describes the correspondence between the (finite-dimensional) representations of (real) Lie groups and the representations of their Lie  algebras. More precisely, does anyone know of a text that contains proofs of statements below (modulo some assumptions):

If G is a compact, simply-connected Lie group, then its (finite dimensional) representations are in correspondence with those of the complexification of its Lie algebra;
If H is a (compact) Lie group with the same Lie algebra as G, then it is a quotient of G by a subgroup of the centre; the (finite dimensional) representations of H are precisely those for G that factor through the quotient.

I understand the representation theory of (finite-dimensional, complex, semisimple) Lie algebras, and have a (working) knowledge of differential geometry and algebraic topology; references that only consider matrix Lie groups are not preferred, though it would be nice if any particularly high-powered differential geometry\topology is kept to a minimum.

REPLY [4 votes]: As mentioned above, my book, "Lie groups, Lie algebras, and representations," discusses this question in detail. See http://www.amazon.com/Lie-Groups-Algebras-Representations-Introduction/dp/3319134663/ref=sr_1_3?s=books&ie=UTF8&qid=1457839028&sr=1-3&keywords=brian+hall. The book is now in its second edition; in this version, the relevant part is Chapter 5. I do things from the matrix group perspective. In addition, I use the Baker-Campbell-Hausdorff formula rather than appealing to the Frobenius theorem. I think the BCH formula shows most clearly "why the result is really true." As already pointed out, compactness is not needed in the general statement.
One can, however, look at the question specifically from the perspective of the compact case. The classification of the representations of the group and of the Lie algebra both take the form of a "theorem of the highest weight". (These are discussed in Part II of my book for the Lie algebra case and Part III for the group case.) In the simply connected case, we expect that the classifications will match up. To verify this directly (without appealing to the results of Chapter 5), we have to show that the possible highest weights in the group result are the same as in the Lie algebra result. I show this in Chapter 13 of the book; see Corollary 13.20. Since the argument is a bit involved, it is probably simplest just to look at the material in Chapter 5.