Lie Groups: A Problem-Oriented Introduction via Matrix Groups

Lie groups play such a central role in mathematics and its applications that all mathematics students should be aware of them. From computer graphics to quantum theory, from differential equations to number theory, Lie groups and algebras are everywhere. Nevertheless, most mathematics undergraduates have never heard of them.

Of course, the reason for this is that even the definition of a Lie group (i.e., a differentiable manifold which is also a group and whose group operations are smooth functions) seems to require more background knowledge than most undergraduates have. In a famous article on “Very Basic Lie Theory” (American Mathematical Monthly, 1983), Roger Howe argued that the absence of the theory of Lie groups from most undergraduate programs was both scandalous and unnecessary. Since then, several books attempting to respond to Howe’s challenge have appeared. Harriet Pollatsek’s Lie Groups is her entry in this list.

Howe’s article, and many of the books that it inspired, still laid a heavy emphasis on the underlying topology and analysis, with the result that only advanced undergraduates (in fact, probably only rather strong advanced undergraduates) could really understand the material. Pollatsek is not satisfied with that approach. At Smith College, she has been teaching Lie Groups to sophomores, using a problem-driven approach that only assumes knowledge of linear algebra and multivariable calculus. From that course was born this book.

Lie Groups is fundamentally a problem book: each chapter presents a sequence of problems that students should solve. In each chapter, the main sections are followed by a summary section called “Putting the pieces together.” Finally, a section called “A broader view” tries to give a wider context to the material in that chapter. Everything important is done through problems. Thus, this is a book to be used, not a book to be read.

I must admit that I find teaching this material to sophomore mathematics majors to be a little bit too much of a reach. Instead, I used the book in a seminar course that followed our standard one-semester introduction to abstract algebra. In the class were six seniors, three juniors, and (yes!) a sophomore; all had taken algebra, either in the previous semester or a year before. Most, but not all, had taken other advanced courses as well, including Real Analysis. My students were assigned problems to solve and present in class, and we developed the theory in that way. Every once in a while, I would step in and give a broader view.

Did it work? Yes, it mostly did, though in the process I discovered that some aspects have to be tweaked a little bit. Because Pollatsek doesn’t want to assume any algebra background, there are lots of “check that SL(2,R) is a group” problems — more than I or my students wanted. Because she doesn’t want to assume much knowledge of topology, she does not have the chance to make much of connectedness, compactness, covering spaces, and (most significant in this context) simple connectedness. This seems like an opportunity lost.

Some choices I just don’t understand. Why, for example, is there a section on differential equations? It seems totally unrelated to the rest of the book. Why make such a meal of continuity versus differentiability in the definition of one-parameter subgroups? In this setting, one might as well just state that all one-parameter subgroups are differentiable and be done with it. Why not do more with the fact that SU(2) is a three-sphere? (My students were fascinated.) Why work only with the one-dimensional Lorenz group? Why not have more pictures? After all, it’s easy to draw pictures of the one-dimensional Lie groups.

In many of the problems, there is more handholding than I wanted my students to have; at some points, this included “hints” that led towards unnecessarily messy ways of doing things. Pollatsek clearly thinks that students at this level prefer computation to theory. She is probably right, but I would rather not indulge that preference.

Some issues had to do with the fact that my students knew some things that her students didn’t (what a group is, what a normal subgroup is) but did not have a sure grasp of some things that her students clearly do have well under control (the example that stands out in my mind is the Jacobian matrix of a differentiable map). The bibliography is also a little strange, too selective for my taste. This proved significant towards the end, when my students were working on term papers and all of them wanted the same books… which were not in our library.

There is no representation theory at all. The adjoint representation is described, but not put into any sort of context. With no representation theory, one cannot do much more than say “the physicists use this a lot,” without showing how or why. That seems like a pity.

Pollatsek’s book faces stiff competition. Perhaps the closest competitor is John Stillwell’s Naïve Lie Theory, a beautiful book aimed at upper-level undergraduates that is very much in the same spirit. Kristopher Tapp’s Matrix Groups for Undergraduates is also in a similar spirit. The main difference, of course, is that Stillwell and Tapp have written textbooks, while Pollatsek gives us a problem book. Using it requires buy-in: you need to want to teach in this way, and you will need to convince your students that the work is worth doing. If that applies to you, give this book a try.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He’ll be doing the Lie groups thing again soon.