You are here

Representations of Lie Algebras: An Introduction Through \(\mathbf{gl}_n\)

Anthony Henderson
Cambridge University Press
Publication Date: 
Number of Pages: 
Australian Mathematical Society Lecture Series 22
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

For many years now, I have believed that the world needs more elementary (say, undergraduate-level) books on the purely algebraic theory of Lie algebras. The last decade or so has seen the publication of several books that attempt to make the theory of Lie groups accessible to undergraduates by focusing on matrix groups (thereby eliminating the need to talk about differentiable manifolds, while still retaining a lot of the flavor of the theory). Books along these lines include the excellent Naive Lie Theory by Stillwell and (at a somewhat higher, early graduate, level) Hall’s Lie Groups, Lie Algebras and Representations; another very good example is Tapp’s Matrix Groups for Undergraduates. There is also Pollatsek’s Lie Groups: A Problem-Oriented Introduction via Matrix Groups, but, as previously noted here, this is really a problem book rather than a textbook. Books like these do define Lie algebras and talk about them to some degree, but in all of them Lie algebras play a supporting role; Lie groups (of matrices) are the main objects of study.

On the other hand, reasonably elementary books that are devoted to the algebraic theory of Lie algebras as entities in their own right are somewhat harder to find. I first learned the subject in graduate school in a reading course at Rutgers University from Humphreys’ classic Introduction to Lie Algebras and Representation Theory, with occasional excursions to Jacobson’s Lie Algebras (then an Interscience book, now a Dover paperback) to learn the characteristic p theory of modular Lie algebras. The Humphreys text (one of the earlier entries in the Graduate Texts in Mathematics series, and still in print by Springer-Verlag after about 40 years) was considerably more accessible than the one by Jacobson, which I view as being completely beyond the reach of the vast majority of undergraduates. The preface to Humphreys says that the first four chapters might well be accessible to a bright undergraduate, but I think this is perhaps over-optimistic: although it is an excellent book, it is quite densely written, and the first seventy or eighty pages might well consume an entire semester. Then, about six years ago, Erdmann and Wildon’s Introduction to Lie Algebras was written, and (Humphreys’ preface notwithstanding) that book was, I thought, the only text that was genuinely within the reach of advanced undergraduates.

Until now, that is. Henderson’s book, though substantively quite different than the one by Erdmann and Wildon, is also one that could, with good likelihood of success, be used to teach Lie algebras to serious undergraduates, and it manages this by essentially using the same trick that the books of Stillwell/Hall/Tapp use in the theory of Lie groups: it focuses on matrices. By concentrating on one particular Lie algebra, namely the set gln of all n x n complex matrices under the commutator bracket operation [A, B] = AB – BA, Henderson is able to use calculations that are specific to matrices and therefore avoid having to engage in more general arguments.

The statement that this book is accessible to undergraduates should come with a caveat attached. Its origin is in a set of lectures delivered by the author to students in Australia, where mathematics undergraduates, as in Great Britain, are more advanced than here in the United States. Henderson’s students, for example, have already had some prior exposure to group representation theory. And, of course, anybody planning to study Lie algebras, even in the concrete context of matrix algebras, is going to want to have a good grounding in linear algebra: Irving Kaplansky once wrote, in a survey article on Lie algebras in Saaty’s Lectures in Modern Mathematics, that some Lie algebraic arguments are “linear algebra raised to the nth power” and that an “apprentice algebraist”, working through the theory of Lie algebras, will “find everything he knows [about linear algebra] used in a dazzling array of spectacular arguments”. Thus, the blurb on the back cover of this book stating that a reader should be familiar with linear algebra up to and including the Jordan Canonical Form is certainly not surprising, but does perhaps have the effect of narrowing the field somewhat. (Perhaps anticipating this, the Erdmann/Wildon text contains an appendix of about ten pages summarizing the more sophisticated aspects of linear algebra that are necessary; this text does not do this, but does spend some time in the first chapter discussing multilinear algebra.)

In addition, any person undertaking a study of Lie algebras will want to be fairly familiar with the rudiments of abstract algebra. Nevertheless, despite these inevitable prerequisites, this book is well written enough to still be accessible to well-prepared and well-motivated senior-level sophisticated undergraduates, perhaps in some kind of senior seminar. In terms of overall accessibility, I would put this book between Erdmann/Wildon and Humphreys, and closer to the former than the latter.

Another interesting and unusual feature of the text is that it rethinks the traditional approach to teaching the elementary theory of Lie algebras. Most courses along these lines, I suspect, have as their goal the classification of simple (and hence semisimple, since they are direct sums of simple) Lie algebras over an algebraically closed field of characteristic 0. The books by Humphreys and Erdmann/Wildon reflect this approach, although the former also contains a lot of subsequent material on representation theory as well. The problem with this agenda, though, is that if one starts from scratch with the definition of a Lie algebra and a summary of the basic properties and results, it is almost impossible to get through the classification theory without resorting to extensive hand-waving at the end of the semester. Henderson has set himself the goal of devising a semester course at the advanced undergraduate level in which the student actually sees, without omitted proofs, one “peak” (his term) of the theory. He has chosen as his desired “peak” the theory of representations of the Lie algebra gln, and has attempted to reach that peak in as direct a manner as possible.

Towards this end, Henderson has made a conscious decision to omit from the text a number of topics that are often covered in introductory books on this subject. Topics such as Cartan subalgebras, the root space decomposition of a simple Lie algebra and Dynkin diagrams, for example, which are essential tools in the classification theory of semisimple Lie algebras, are not developed here at all. Perhaps more surprisingly, a number of famous theorems (including, for example, Cartan’s criterion for semi-simplicity in terms of the Killing form, Engel’s theorem on nilpotence, and Lie’s theorem on solvable subalgebras of gln) are also not proved. These omissions are understandable given the author’s goals, but at the same time, they may make the book less attractive as a text for an introductory graduate course.

The book begins with an introductory and motivational chapter that, by means of explicit calculations and with a little multilinear algebra (developed as needed), shows that one problem (determining homomorphisms from the matrix group GLn to the matrix group GLm) is essentially equivalent to another (that of studying the mappings from the set of all n x n matrices to the set of all m x m matrices that preserve the commutator; or, in algebraic language, studying the homomorphisms from the Lie algebra gln to the Lie algebra glm). This material is not really used in the sequel (which in fact mostly avoids any reference to Lie groups), but does motivate the central question of this text, namely the classification of the integral representations of gln . A chapter like this seems like an excellent way to start the text, given the fact that the definition of a Lie algebra is not the most natural thing in the world. Neither Humphreys’ text, nor the one by Erdmann and Wildon, provide this much detail about the connections between Lie algebras and Lie groups.

Lie algebras themselves are formally defined in chapter 2, which also provides some basic examples and introductory definitions. (The book has a standing convention, mentioned prior to chapter 1, that all Lie algebras are over the field of complex numbers. Of course, a potential problem with such standing conventions — even when, as here, they are prominently announced at the beginning — is that “drop in” readers who are not reading the book from the beginning may be confused or misled.) Chapter 3 continues with more definitions (ideal, subalgebra, nilpotent, solvable, etc.) and examples; one nice feature here is a simple, calculational proof of the simplicity of the Lie algebra sln of n x n matrices with trace 0. (Contrast this, for example, with the approach taken by Humphreys: he proves the simplicity of sl2 directly but does not establish the simplicity of sln (and the other classical Lie algebras) until much later in the book, as a consequence of the root space decomposition. Erdmann and Wildon do mention the simplicity of sln fairly early on, but leave the proof as an exercise.)

Chapter 4 of the text introduces the idea of a representation of a Lie algebra L, done via the notion of an L-module, which is a vector space on which L operates in such a way a way as to induce a Lie algebra homomorphism from L into gln. The next three chapters then elaborate on this idea, with the ultimate goal of classifying integral representations of gln covered in chapter 7 (this is the “peak” referred to earlier); along the way we also see the classification of representations of sl2, a result which plays a significant role in the classification theory of (complex) semisimple Lie algebras.

The final chapter of the text, entitled “Guide to Further Reading”, is a relatively short exposition of some of the ideas of Lie algebra theory that lie just beyond the boundaries of the text, including the classification theory of semisimple Lie algebras and the representation theory of general Lie algebras. There are no proofs here, just an explication of the main ideas with some examples and bibliographic references.

Each chapter, save for the first and last, ends with a collection of eight or nine exercises, solutions to all of which are in the back of the book. An instructor using the book as a text may therefore need to come up with problems on his or her own to supplement the book’s exercises.

In summary, this is a well-written book that offers an interesting and novel perspective on teaching an introductory Lie algebras course. Of course, using this book as a text would require agreement with the author’s fairly unconventional approach to the material, but anybody who agrees with this approach, or is open to the possibility of being persuaded to agree with it, should certainly give this book a serious look. In fact, I think that anyone who is interested in Lie algebras should give this book a serious look, if for no other reason than the fact that any nicely written account of beautiful mathematics deserves to be noticed.

Mark Hunacek ( teaches mathematics at Iowa State University.

1. Motivation: representations of Lie groups
2. Definition of a Lie algebra
3. Basic structure of a Lie algebra
4. Modules over a Lie algebra
5. The theory of sl2-modules
6. General theory of modules
7. Integral gln-modules
8. Guide to further reading
Appendix: solutions to the exercises