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Matrix Groups: An Introduction to Lie Group Theory

Andrew Baker
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
David Roberts
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The first reference cited in Matrix Groups is a 1983 Monthly article by Roger Howe, Very Basic Lie Theory. In his article, Howe discusses a "pedagogy gap." On the one hand, Lie theory touches "a tremendous spectrum of mathematical areas," from number theory to physics. On the other hand, "it has barely penetrated the undergraduate curriculum." Howe identifies the source of the pedagogy gap as the fact that substantial prerequisites are required for the foundations of Lie theory. More optimistically, he writes,

While a complete discussion of Lie theory does require fairly elaborate preparation, a large portion of its essence is accessible on a much simpler level, appropriate to advanced undergraduate instruction.

Howe goes on to present the theory at what he views as the appropriate simpler level. A centerpiece of his presentation is the theorem that a closed subgroup of the matrix group GLn(R) automatically has the structure of a differentiable manifold and is thus a Lie group.

Baker writes in his preface that he was influenced by Howe's article, and the influence is easily seen. Following Howe, he aims to close the pedagogy gap by presenting just the essence of Lie theory. In Baker's words, "This work provides a first taste of the theory of Lie groups...providing an appetiser for a more substantial further course." The book is divided into three parts:

I. Chapters 1-6. Basic Ideas and Examples
II. Chapters 7-9. Matrix Groups as Lie Groups
III. Chapters 10-12. Compact Connected Lie Groups and their Classification

Baker suggests that a 30-lecture course to advanced undergraduates might center around Chapters 1, 3, and 7. The main result in Chapter 7 is the same as the one highlighted by Howe, that matrix groups are Lie groups. The book is designed so that instructors can "quarry" the remaining chapters in different ways to round out their courses. Each chapter ends with a collection of exercises, most of which explicitly involve matrices but are not routine. In keeping with the format of books in the Springer Undergraduate Mathematics Series, there are hints or solutions to most of these exercises.

My feeling is that Lie theory is indisputably central to mathematics and indeed should be taught more to undergraduates. However for this to happen successfully, I think one should understand Howe's quote displayed above much more radically. It is with some hesitation that I find myself in disagreement with Howe and Baker, both very distinguished mathematicians. But to my mind, they both pitch their treatments at an unnecessarily advanced level. A large portion of the essence of Lie theory, more than just a taste, is essentially algebraic in nature. Undergraduates can be introduced to this portion of Lie theory without the heavy doses of topology and analysis present in both Howe's paper and Baker's book.

From my viewpoint, the excess of topology in Matrix Groups begins in Chapter 1, which defines operator norms, product topologies, topological groups, compact spaces, and Hausdorff spaces, among other topological notions. All this, despite the fact that one can say a lot about subgroups of GLn(R) without any topology at all. After all, there is a very rich theory of matrix groups over arbitrary fields. On the analytic side, consider again Chapter 7's theorem that matrix groups are Lie groups. Certainly this theorem has a first-class pedigree, as it captures von Neumann's contribution to Hilbert's fifth problem. But as a practical matter, the proof involves careful consideration of quite a large number of inequalities. While many undergraduates may be able to follow this proof line-by-line, only quite exceptional undergraduates would really appreciate it. To me, this is prime material to be simply quoted, just as Baker simply quotes the important result that a compact Lie group is a matrix group.

What in the theory of Lie groups can be taught without providing a background in topology and assuming analytic sophistication? On the side of core theory, consider the very last exercise in Matrix Groups. It concerns the group of unitary n-by-n matrices Un and its subgroup T of diagonal unitary matrices. Among other questions at the same level, it asks what the normalizer N of T in Un is. The answer is that N is the subgroup of unitary monomial matrices, i.e. unitary matrices with exactly one nonzero entry in each row and column. In other words, N is the semidirect product of T and the group Sn of permutation matrices. This sort of thing is both accessible to undergraduates and truly central to Lie theory. But, perhaps because algebra is not given its due in this book, this exercise is flagged with two warning signs as especially challenging.

On the side of applications both within mathematics and beyond, a "first taste" of Lie theory should provide a representative sample. The fact that Lie theory is the most fecund source of finite simple groups could be mentioned. Two-dimensional hyperbolic geometry is governed by PSL2(R) and provides a highly visual application. Representations of the compact group SO3 on spaces of spherical harmonics could represent the major topics of general representation theory and special functions. The connection of these representations with the s-, p-, d- and f-orbitals of chemistry could be sketched as an obviously important application outside of mathematics. There would be many other possible topics. The point is that a first taste of Lie theory should communicate not just what Lie theory is, but also why Lie theory is important.

Howe's Very Basic Lie Theory opened with a discussion of the compact-open topology on spaces of functions. Howe later felt his discussion was incomplete and published a correction. The irony should not be lost — this material is not "very basic" and not the route to teach Lie theory to undergraduates! Baker's book stays within more reasonable bounds. However the pedagogical gap will not be closed until some author realizes that the great idea of providing only a "first taste" of Lie theory can liberate a text from technical details and yet still permit it to communicate a large portion of the essence of the theory.

David Roberts is an assistant professor of mathematics at the University of Minnesota, Morris.

Part I. Basic Ideas and Examples: Real and Complex Matrix Groups. Exponentials, Differential Equations and One Parameter Subgroups. Tangent Spaces and Lie Algebras. Algebras, Quaternions, and Symplectic Groups. Clifford Algebras and Spinor Groups. Lorentz Groups.- Part II. Matrix Groups as Lie Groups: Lie groups. Homogeneous Spaces. Connectivity of Matrix Groups.- Part III. Compact Connected Lie Groups and their Classification: Maximal Tori in Compact Connected Lie Groups. Semi-simple Factorization. Roots systems, Weyl Groups and Dynkin Diagrams.- Bibliography.- Index.