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Introduction to Lie Algebras and Representation Theory

J.E. Humphreys
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Graduate Texts in Mathematics 9
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Hunacek
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I have considerable sentimental attachment to this book. Back in the fall semester of 1973 I was a first-year graduate student at the Courant Institute of New York University, where I had occasion to take two courses (graduate algebra and algebraic topology) with James Humphreys. In a conversation with him one day, he happened to mention Lie algebras, and made them sound quite interesting.

By the end of the 1974–75 academic year, neither Humphreys nor I were still at Courant; he took a position at the University of Massachusetts in Amherst and I, having realized that Courant was way too applied for my taste, got my Masters degree there and went to Rutgers University for a PhD. While there I met Robert Wilson, who specialized in Lie algebras, and approached him about being my advisor; he suggested that I spend a semester reading the book under review and lecturing to him once or twice a week on the contents. So it is that having first heard about Lie algebras from Humphreys the person, I first read about them from Humphreys the book.

The book, number 9 in the Springer Graduate Text in Mathematics series, was fairly new (it was published in 1972) when this happened. Forty years later, it is still in print, having gone through at least seven printings in the interim and also having been translated into Russian and Chinese. When it first came out it was, I think, the clearest and most accessible introduction to the subject; even now, it is difficult if not impossible to find a text that does a better job of teaching this material at the introductory graduate level. It is still routinely used as a reference for people interested in the subject, and as a text for graduate courses in this area; in fact, it may well be the market leader for such courses.

It deserves to be. Assuming only a good understanding of linear and abstract algebra, Humphreys starts with the definition of a Lie algebra and, about one hundred pages later, has gone through their basic properties and established the classification theorem for semisimple Lie algebras over an algebraically closed field of characteristic 0. (The first chapter, which introduces the basic notions, allows the underlying field to be arbitrary, but throughout most of the rest of the book it is assumed that the field has characteristic 0, though occasional remarks about characteristic p are made. The theory of Lie algebras in prime characteristic, which Humphreys omitted because he thought it more suitable for a second course, is nicely discussed in Jacobson’s Lie Algebras, which is now a Dover paperback; Jacobson also includes in his book a proof of the difficult theorem of Ado-Iwasawa, another topic that Humphreys thought was beyond the scope of a first course.)

In the remaining chapters of the book, Humphreys discusses the characteristic 0 representation theory of semisimple Lie algebras in terms of weight spaces, and Chevalley groups and algebras. These chapters are significantly more demanding than the rest of the book.

The text incorporated a number of ideas in the development of the subject that were, back then, fairly novel; for example, instead of working with Cartan subalgebras, Humphreys works with maximal toral subalgebras. For the Lie algebras under consideration, Cartan subalgebras and maximal toral subalgebras turn out to be precisely the same thing (a fact which Humphreys proves about halfway through the book), but initial use of the latter allows a smoother development of the theory. In addition, root systems, so useful in the classification theory, are developed axiomatically in the book.

I said earlier that this book was the most accessible account of Lie algebras then available, but I should qualify this by pointing out that “accessible” does not mean the same thing as “easy”. Reading the book was a demanding experience, and I was certainly glad that I had the benefit of an expert to talk to as I did so. The writing style is dense (there are only 162 pages of text), and many details are left to the reader to fill in; looking at the book now, I see a number of 38-year-old penciled-in paragraphs providing further explanation. (Since I generally abhor writing in books, the fact that I did so then provides some indication of how necessary I must have felt these explanatory notes were. Of course, all this was before the days of computers; now I’d probably just keep a running pdf file.) I don’t say this as a criticism, since I believe that at the graduate level, requiring this kind of intellectual commitment from a reader is not at all a bad idea.

For average undergraduates (at least today’s average undergraduates), I think this book is probably too hard, notwithstanding the statement in the preface that the first 90 pages or so “might well be read by a bright undergraduate”. For anybody interested in offering an undergraduate course on Lie algebras (perhaps, say, as a senior seminar) today, I would recommend Erdmann and Wildon’s Introduction to Lie Algebras (also published by Springer, in a different series), or (if you want a course more focused on representation theory than on the classification theory) Henderson’s Representations of Lie Algebras: An Introduction Through gln. There are also undergraduate-level textbooks that combine the purely algebraic theory of Lie algebras with Lie groups; my personal favorite here is Stillwell’s Naive Lie Theory.

A brief word about the exercises: there are quite a lot of them, with a wide range of difficulty. (I also see in my copy of the book some penciled-in solutions, including one where, in evident frustration, I began by writing “ignore the hint” and then wrote out a completely different solution.) There are no solutions provided in the book, which I think is a good thing; I learned a great deal by puzzling through many of the more difficult ones (and in some cases, when puzzling through them didn’t quite work, discussing them with my advisor).

I suppose that, looking back at a book with almost 40 years of hindsight, one is bound to think of things that might have been done better. The one most obvious example I can think of in reference to this book is motivation: this is, unapologetically, a book about the purely algebraic theory of Lie algebras, and therefore Lie groups receive little or no attention at all. As a consequence, Humphreys just plunges into the definition of a Lie algebra on page 1 with no explanation for why anybody would want to study these strange objects or how they arose historically. The Jacobi identity, after all, is not the most obvious thing in the world, but Humphreys simply says “this is called the Jacobi identity” and leaves it at that. The funny thing is, though, that this didn’t seem to bother me when I was first learning this material; I was so caught up in the beautiful theory surrounding Lie algebras that I didn’t really care about how these ideas came to be defined in the first place. As I got older, though, I found myself getting more and more interested in the reasons behind some definition or concept.

Since the publication of Humphreys’ book, it has become clear that a great deal of the flavor of the theory of Lie groups can be made accessible without recourse to differential manifolds by focusing on matrix groups, and so it should be possible to provide, in an introductory chapter, at least some intuitive and expository discussion of how Lie groups lead to Lie algebras; this is the way, for example, that Henderson’s book begins. Of course, this was not common practice when Humphreys’ book was first written: Roger Howe’s influential article “Very Basic Lie Theory” was not published in the American Mathematical Monthly until 1983, and Brian Hall’s beautiful book Lie Groups, Lie Algebras and Representations: An Elementary Introduction was not published until about twenty years after that. So one can understand why Humphreys chose not to make such a substantial detour in the exposition.

In view of the fact that most of the books reviewed in this column are either new or newly re-issued, one might wonder why this book, which is undoubtedly already quite familiar to anybody with an interest in Lie algebras, is being reviewed. Aside from the fact that it seems a pity that such an influential book was not the subject of a review, writing this gives me an opportunity to express my appreciation to Humphreys for producing a book that had such a strong influence on my mathematical development. All in all, this is an extraordinary book, one that has earned its reputation as a classic.

Mark Hunacek ( teaches mathematics at Iowa State University.

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