Link to Springer web page for the book
Link to list of Corrections to the first printing
Color Pictures

How I came to write this book

The project that eventually became this book began in December of 1993, when I was preparing to teach a graduate course on Lie groups at the University of Virginia. The question confronting me was, "What book should I use?" A key pedagogical issue underlying this question was, "What to do about manifold theory?" Since a Lie group is a group that is also a manifold, manifold theory naturally plays a prominent role in most books on Lie groups. Unfortunately, however, it is not reasonable to assume that the students in a Lie groups course will be proficient in manifold theory. Certainly, I was not when I took my first Lie groups course and I knew that my students at Virginia would not be. I knew, then, that if I followed the approach in most books on Lie groups, I would have to spend considerable time at the beginning of the course covering manifold theory, including (ideally) the Frobenius theorem and integration of differential forms. My own first course on Lie groups was of this sort. We used the book of Frank Warner, "Foundations of Differentiable Manifolds and Lie Groups." I learned a lot about manifolds (for which I am glad!) and I learned some of the foundational parts of Lie group theory (such as the relationship between the group and the Lie algebra). But by then our one-semester course was almost over, so we weren't able to get much into representation theory or the theory of semisimple Lie groups.

Meanwhile, another pedagogical issue I faced as I prepared for my course at Virginia was how to make the theory of semisimple Lie groups digestible. In my second course on Lie groups, I finally met the machinery of semisimple Lie groups and Lie algebras: Cartan subalgebras, roots, weights, the Weyl group, the fundamental Weyl chamber, and all that. I found this overwhelming. I didn't really grasp what the roots really were or why I should care, and so the seemingly endless list of properties that the roots satisfy meant little to me. I was looking for a way to provide my students with some motivation for and perspective on this machinery before getting into all the gritty details. Since I could not find a book that addressed these two pedagogical issues in a way that seemed appropriate for my audience at Virginia, I did the only sensible thing: I wrote my own book.

Well, actually, what happened is that I began to write my own book. It took almost a decade from the time I began writing (December 1993) until my project became an honest-to-goodness published book (August 2003). I distributed some notes to my students during the spring semester of 1994, and in the years that followed, I revised and expanded the notes several times. Eventually, I posted a version of them on the web [http://www.arxiv.org/abs/math-ph/0005032]. Around the beginning of 2002, Springer-Verlag contacted me asking me if I was interested in having them publish the notes in book form. Given Springer's reputation and given that the timing was good for me, I agreed enthusiastically. By May of 2002, I had a signed contract with Springer and began working in earnest to turn the notes into a complete book. I sent the first complete draft of the book to Springer in December of 2002 and sent the final version to them in early summer of 2003. By August, I had a copy (with its nice yellow cover!) in my hands.

Features of the book

On the issue of manifold theory, my approach is to consider what I call "matrix Lie groups," that is, closed subgroups of GL(n,C). Most familiar examples of Lie groups, such as the special linear, orthogonal, and unitary groups, are of this sort. As the name suggests, every matrix Lie group is a Lie group, and while not every Lie group is a matrix Lie group, most of the interesting ones are. To define the Lie algebra of a matrix Lie group, I consider the exponential of an n x n matrix, which is defined by the usual power series (and which is described in many standard textbooks on differential equations). The Lie algebra of a matrix Lie group G is then defined as the set of matrices X such that exp(tX) lies in G for all real numbers t. The Lie algebra defined in this way is a real subspace of the space of all n x n matrices and is closed under the bracket (or commutator), [X,Y]:=XY-YX. This bracket operation makes the Lie algebra of a matrix Lie group into a Lie algebra in the abstract sense. In this way, one can study matrix Lie groups, their Lie algebras, and the exponential mapping from the Lie algebra to the Lie group, all without saying a word about manifolds.

Of course, having avoided manifold theory, I did not have at my disposal the Frobenius theorem when it came time to consider the "hard direction" of the correspondences between Lie groups and Lie algebras. An example of the hard direction is this: Under what conditions does a Lie algebra homomorphism give rise to a Lie group homomorphism? (Answer: If the domain group is simply connected.) This is to be compared to the "easy direction," namely, the easy result that every Lie group homomorphism gives rise to a Lie algebra homomorphism. Instead of using the Frobenius theorem to address the hard direction, I use the Baker--Campbell--Hausdorff formula.

I should point out that the recent book "Lie Groups: An Introduction Through Linear Groups," by Wulf Rossmann (Oxford Univ. Press, 2002), takes a similar approach using matrix (= linear) groups and the Baker--Campbell--Hausdorff formula. Thus there is considerable overlap between the first two chapters of Rossmann's book and the first three chapters of my book. Although there are some technical differences, I think our approaches (in these chapters) are essentially equivalent. Furthermore, I recently (6/04) had pointed out to me the book of J. Hilgert and K.-H. Neeb, "Lie-Gruppen und Lie-Algebren" (German) [Vieweg, 1991, ISBN 3-528-06432-3/pbk]. Based on my limited knowledge of German, it appears that Section I of that book is also quite similar to the first three chapters of my book.

Meanwhile, on the issue of the semisimple theory, my approach is to consider in detail the representation theory of SU(2) and SU(3) (or, at the level of complex Lie algebras, sl(2,C) and sl(3,C)) before going on to the general case. The example of SU(3) (more so than just the obligatory example of SU(2)) allows the reader to see roots, weights and Cartan subalgebras "in action" in a simple, concrete setting. This, I hope, leaves the reader emotionally prepared for the general case. In this respect, my book is similar to "Representation Theory: A First Course," by William Fulton and Joe Harris. That book is actually even more example-oriented than mine, in that after treating sl(3,C) it goes on to study the other classical Lie algebras, whereas my book turns at that point to the general case.

Even once I begin the general semisimple theory, I defer a detailed examination of root systems until the last chapter. I prefer to get to the "theorem of the highest weight" (the classification of the representations of a semisimple group, also known as "Cartan--Weyl theory"), as quickly as possible. Thus, in Chapters 6 and 7 I quote without proof those results on root systems needed to state the theorem of the highest weight, and then return to root systems in greater detail in Chapter 8, by which point the reader should have a greater appreciation for their importance. I include many pictures for the rank-two case in Chapter 5 (for the SU(3) case) and in Chapter 8 (for the remaining rank-two cases). I also include several color pictures for the rank-three case---these, I hope, earn the book some "coolness" points, whether or not the rest of the book succeeds. The models in the pictures were built using the (very cool) Zome system---see the Zome link at left for more information. Many more rank-three images, which could not be included in the book, can be seen by clicking here. The color pictures were generated using the vZome software, available from its creator, Scott Vorthmann, here.

In writing this book, I have tried to make the presentation accessible to physicists as well as mathematicians. While there are many useful books on Lie group methods in physics, I believe that graduate students in physics might benefit from a book that treats the mathematical constructions in a more systematic way. I have tried to give precise definitions and theorems---and, in most cases, complete proofs---while still maintaining a writing style and a level of concreteness that physicists will feel comfortable with. Time will tell whether I have succeeded.

I welcome comments and feedback of all sorts, at bhall@nd.edu. A pdf file at the top of this page lists all known typos in the first printing of the book. These have been corrected in the second printing,

which is available as of September, 2004. Please send me any additional corrections that you may

discover.