Graduate Texts in Mathematics, 98. New York etc.: Springer-Verlag. X, 313 p., 24 figs. DM 128.00 (1985).
The representation theory of compact Lie groups is to a great extent the creation of Hermann Weyl and Elie Cartan. In outline it is simple, beautiful, and powerful; in detail it constitutes a fundamental tool in topology, invariant theory, theoretical physics, and representation theory itself. One can read accounts of the subject written by the partisans of each of these. At best, such accounts can provide a wealth of detailed and useful information; but it is difficult for the simplicity of the basic ideas to survive the demands of applications.
The present book comes from topologists. Assuming only linear algebra and advanced calculus (including Stoke’s theorem and Sard’s theorem), the authors develop the basic structure and representation theory of compact Lie groups: maximal tori, the Weyl group, root systems, the Cartan-Weyl highest weight theory, the Peter-Weyl theorem, Tannaka-Krein duality, and the Weyl character formula. They treat disconnected groups and representations over the reals and quaternions; these are not often considered in elementary texts, and they are important in many applications (notably topological ones).
The development is in general excellent. The proof of the Peter-Weyl theorem, for example, uses only some easy functional analysis; many books use information about the spectrum of the Laplacian. Students with a good background in analysis and a dislike for algebra will be pleased by the authors’ taste in many places.
There are some small problems. Occasionally proofs are a little too sketchy for a first textbook: . is connected because performing elementary row and column operations which do not involve multiplication by a negative scalar does not change components.” It is a little strange to see Lie algebras almost completely excluded from the text; but it is hard to find places where they would simplify matters much. (The classification of the representations of SU(2) without them is impressive.)
These are very small flaws, however, in an excellent book. I have already recommended it to beginning graduate students, and will continue to do so.