# Representation Theory of Lie Groups: Reference Request

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.

• Why don't those who downvote leave suggestions of relevant texts? – Yemon Choi May 7 '15 at 9:52
• I think this is a perfectly reasonable reference request, seeking precise detail rather than "a book about X". Would anyone care to explain what is wrong with the question? – Yemon Choi May 7 '15 at 9:57
• @Yemon: re: your first question: presumably if you downvote a question you think it shouldn't be here, and so you wouldn't want to leave such a suggestion because you don't want to encourage similar questions in the future. Re: your second question, presumably people feel like these facts are too elementary to warrant asking about on MO. (For the record, I didn't downvote or vote to close.) – Qiaochu Yuan May 8 '15 at 17:50

Neither statement requires compactness as a hypothesis. The key result to both, and the only place where any work is needed, is the following:

If $G$ is a connected Lie group and $H$ is a Lie group, then the differentiation map

$$\text{Hom}(G, H) \to \text{Hom}(\mathfrak{g}, \mathfrak{h})$$

is injective (straightforward). If $G$ is simply connected, it is bijective (takes work).

Both statements you want are exercises assuming this result.

For the connection between representations of a simply connected Lie group and its Lie algebra, take $H = \text{GL}_n(\mathbb{C})$ and use the fact that complexification is left adjoint to the forgetful functor from complex to real Lie algebras.

For the connection between Lie groups having the same Lie algebra, if $G$ is a simply connected Lie group and $H$ is a connected Lie group with the same Lie algebra, then there is an isomorphism $\mathfrak{g} \sim \mathfrak{h}$, and by the above it lifts to a map $G \to H$ which is a local diffeomorphism and hence (with a bit of work) a covering map. It's an exercise to show that a covering map between connected topological groups is a quotient by a discrete subgroup of the center. Since $G \to H$ is surjective, a representation of $H$ is determined by the corresponding representation of $G$, and the only condition to check is that it factors.

The key result should be in any book that discusses the relationship between Lie groups and Lie algebras; to give two random examples, it is proven in Section 8.3 of Fulton and Harris, and it is Theorem 3.27 in Warner's Foundations of Differentiable Manifolds and Lie Groups.

• Fulton and Harris's treatment of this point is rather sketchy. – Brian Hall Mar 13 '16 at 3:14

A good introduction is given in the book Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction which is an extended version of http://arxiv.org/abs/math-ph/0005032 Some other books are mentioned in answers to this MO question: Learning about Lie groups

Depending on your own background, you might find the textbook by Brocker and tom Dieck useful: reference here. This gives a focused account of both the structure of compact Lie groups and their irreducible (necessarily finite dimensional) representations, relative to the standard highest weight theory for a complex semisimple Lie algebra.

There are of course many kinds of textbooks treating Lie groups, Lie algebras, and representation theory; but most of these deal also with non-compact groups and unitary representations, etc. Besides Hall's book, there are standard texts by Helgason, Knapp, Procesi, and others. Unfortunately, most of the books mentioned are fairly long and take a while to get into.

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.

I like the book by Claudio Procesi Lie Groups: An Approach through Invariants and Representations

Try also with "Compact Lie groups" by Sepanski. The answers to your questions are in Chapter 4.