This book brought back fond memories. Back around 1975, when I was a graduate student, my thesis advisor and some of his friends on the faculty organized an informal seminar for the purpose of going through the (then) recently published book Linear Algebraic Groups by Humphreys. I was invited to join them, and for a semester we all met once or twice a week, taking turns to lecture on the text. I had a great time, for several reasons: the mathematics was very interesting, it was a wonderful opportunity to hobnob with the faculty in their native habitat, and I had some personal connection with Humphreys, having had him as a teacher several years back.
Humphreys’ book was not the only text in existence on the subject of algebraic groups (see, for example, Borel’s Linear Algebraic Groups), but was generally considered the most accessible, although it was by no means a particularly easy read. It was written for people with at least a year of graduate school under their belts, and the writing, though clear, was on the terse side. I found the material sufficiently interesting, however, that even though I never did any real work with algebraic groups, I did keep an eye out on other accounts of the subject that have appeared over the years, both textbooks (e.g., Springer’s Linear Algebraic Groups and Tauvel and Yu’s Lie Algebras and Algebraic Groups) and textbook-quality lecture notes (such as the set of notes by Milne, generously made freely available by him, along with notes on a number of other topics). Notwithstanding the undeniable quality of these more recent works, I always felt that Humphreys’ book remained, with one exception, the most accessible text on the subject.
The one exception is the book under review, first published in 2003 in hardcover and now reprinted (apparently without change) in the paperback version that is the subject of this review. The exposition here seems a bit more leisurely than that in Humphreys’ text, and the author does not strive for the kind of generality that is found in Humphreys. Indeed, the author of this book states in the preface that the books by Borel, Humphreys and Springer “contain much more material than we can cover here.”
Roughly speaking, an algebraic group is the algebraic analog of a Lie group, with algebraic geometry playing the role of differential geometry. Just as a Lie group is simultaneously a differentiable manifold and a group (with the group operations of multiplication and inversion being compatible with the manifold structure), an algebraic group is simultaneously a variety and a group, with the group operations being morphisms. A linear algebraic group is an algebraic group of matrices. Examples include the four infinite families of classical groups (special linear, symplectic, and special orthogonal in even and odd dimension).
So, just as the general definition of a Lie group requires some background in differentiable geometry to be comprehensible, the definition of an algebraic group requires some background in algebraic geometry. The beginning sections of chapter 1 develop part of the algebraic geometry that will be needed, though not in greater abstraction than is necessary for subsequent developments. In particular, in chapter 1 the author works exclusively with algebraic sets; general affine varieties are not introduced until about a hundred pages later, and fancy stuff like sheaves are not discussed at all. Projective varieties are not discussed much either. (This should be contrasted, for example, with the account given in Humphreys’ text, where general affine and projective varieties are introduced right away.) After spending about half of chapter 1 on algebraic geometry (including some discussion of Groebner bases), Geck defines a linear algebraic group, provides examples and proves some basic facts. The tangent space of an algebraic group is defined, and, for linear algebraic groups, it is shown that this tangent space is a Lie algebra. Subsequent sections in this chapter discuss the notion of a BN-pair or Tits system, and it is shown that the classical groups all have BN-pairs.
Chapter 2 generalizes the notion of an algebraic set to an abstract affine variety, and also provides a definition of a general affine algebraic group. It is proved that a general algebraic group is isomorphic to a linear algebraic group. Issues concerning abstract affine varieties and their morphisms (including the question of when a bijective morphism is an isomorphism) are discussed. In addition, the question of an algebraic group acting on a variety is examined (along with a first glimpse at invariant theory), and, in a subsequent section, is illustrated by a detailed look at the specific example of the special linear group acting on the variety of unipotent matrices.
Chapter 3 culminates in a discussion of the structure of connected solvable groups, using Borel subgroups (also developed in this chapter) as a tool. While Humphreys discusses this material in the context of an arbitrary algebraically closed ground field, Geck works with the algebraic closure of Fp, the finite field with p elements. This results in several simplifications in the presentation of the material.
In the fourth, final and most sophisticated chapter of the book, the author applies what has come before to study finite groups of Lie type, including representation theory and, in the final section of the chapter, a detailed study of the Suzuki groups. The discussion of representation theory assumes some prior background in the theory of group representations, say at the level of Part I of Serre’s famous book.
(Although I had heard the phrase “finite group of Lie type” before, I was never really clear as to what exactly the precise definition of the term was. I don’t feel bad about this, though, because even people with vastly more knowledge than I have about the subject matter have acknowledged a similar problem. Humphreys himself, for example, in a blog post at Math Overflow, states that it may be “hopeless to reach a consensus on what the definition should be”.) A glance at this discussion should help make clear why this review cannot even attempt to give a definition here.
Although the exposition is probably as reader-friendly as the nature of the material allows, the fact remains that the subject matter of this text is fairly difficult and sophisticated. Thus, although the advertising blurb on the back cover of the text announces that this book is suitable for “advanced undergraduates” (and the author indicates that the first two chapters have been used in undergraduate courses taught by him in Europe), I cannot really see any realistic likelihood of this book being successfully used as an undergraduate text at an average American university. (How often does one encounter, at such universities, any undergraduate courses in either algebraic geometry or algebraic groups?)
The prerequisites for this book include a solid background in abstract and linear algebra, including tensor products of vector spaces. And, although the term “Lie algebra” is defined from scratch, the definition is preceded by the phrase “Recall that…”, so presumably the author is also assuming that the reader is not totally unfamiliar with these objects. In addition to these specific prerequisites, there is assumed a general level of mathematical maturity that is simply not possessed by the average American undergraduate mathematics major.
For students at the graduate level, however, I cannot think of a more accessible introduction to this subject. The book is obviously written with the student reader in mind. Each chapter begins with an introduction spelling out what is going to be covered, and ends with a section that both surveys the literature on the subject matter of the chapter, and also gives a number of exercises, quite a few of which are accompanied by hints. In addition to these motivational discussions, the proofs in the book are generally quite detailed.
In conclusion: this book would be an excellent choice for an early-graduate course in algebraic geometry and algebraic groups, or (including the last chapter) for a more advanced graduate course on topics in finite groups.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.