Lie groups beyond an introduction.

*(English)*Zbl 0862.22006
Progress in Mathematics (Boston, Mass.). 140. Boston: Birkhäuser. xv, 604 p. (1996).

Lie theory plays a role in a variety of mathematical disciplines and draws on many sources. Thus when writing a textbook on Lie groups the choice of prerequisites assumed to be known by the reader is a difficult task. So for example the proof of the functorial connections between Lie groups and Lie algebras (which are essential for the theory) requires a fair amount of preparation and space without giving to much insight into the techniques used in the study of the structure of Lie groups.

Comparing Knapp’s book with classical texts like S. Helgason’s [Differential geometry, Lie groups, and symmetric spaces. Pure and applied mathematics, Vol. 80, New York etc. (1978; Zbl 0451.53038)] or V. S. Varadarajan’s [Lie groups, Lie algebras, and their representations. Prentice-Hall series in modern analysis. Englewood Cliffs, NJ (1974; Zbl 0371.22001)] two differences meet the eye: The down-playing of differential geometry and the emphasis on matrix groups as the prime examples of the theory. The declared goal of the book under review is to provide the structure theoretic background needed in the representation theory of semisimple Lie groups. The approach chosen is a mixture of algebra and analysis with a minimum of differential geometry. In fact, the concepts from differential geometry and topology needed to set up the correspondence between Lie subalgebras and analytic subgroups or to lift algebra homomorphisms to group homomorphisms (like integrable vector fields or universal covering groups) are assumed to be known. In this way Knapp leads the reader with a basic knowledge of Lie groups (as it is provided in many courses on differential geometry) into substantial structure theory very quickly.

Most of the results treated by Knapp can also be found in Helgason or Varadarajan (or both). Exceptions are some more representation theory oriented topics like the Peter-Weyl theory, the Harish-Chandra isomorphism or parabolic subgroups, for which one had to refer to more specialized sources like G. Warner’s [Harmonic analysis on semisimple Lie groups. I. Grundlehren der math. Wiss., Vol. 188, Berlin etc. (1972; Zbl 0265.22020)] or N. R. Wallach’s [Real reductive groups. I. Pure and applied mathematics, Vol. 132, Boston etc. (1992; Zbl 0785.22001)] before. In some cases new proofs are given. A special feature of this book is the fairly detailed collection of historical notes (almost twenty pages) which is quite interesting in itself.

Knapp’s prize winning style of expository writing is well known and needs no further commenting on. I consider his book a welcome addition to the literature on an important branch of mathematics. A few more details on the contents of the book: Chapter I is an introduction to finite dimensional Lie algebras presenting standard results like Lie’s and Engel’s Theorems, the Cartan Criterion, and the representation theory of \(sl(2, \mathbb{C})\). The second chapter is devoted to the classification of complex semisimple Lie algebras introducing the machinery of root systems and Weyl groups on the way. Chapter III deals with the universal enveloping algebra with the Poincaré-Birkhoff-Witt Theorem as a main result.

In Chapter IV compact Lie groups and their representations are treated. This chapter is much more analytic in spirit. The main themes presented are the Peter-Weyl theory, centralizers of tori and the analytic Weyl group.

Chapter V which is again mostly algebraic is on the finite dimensional representation theory of Lie groups and Lie algebras. In addition to highest weight theory and the Weyl Character Formula the author also introduces the Harish-Chandra Isomorphism and parabolic subalgebras. Fundamental weights, however, only occur in the exercises and the Borel-Weil Theorem is mentioned only three hundred pages later in the notes on further topics.

As far as global structure theory goes the Chapters VI and VII are the heart of the book. Here one finds the Cartan, Bruhat, Iwasawa, and Harish-Chandra decompositions as well as a new approach to the classification of real semisimple Lie algebras using Vogan diagrams which describe the effect of suitable Cartan involutions on the Dynkin diagram. The last chapter is a short introduction to integration on Lie groups. Most notably the Weyl integration formula is proved.

There are three appendices, one on multilinear algebra, one on Lie’s Third Theorem and one collecting a lot of useful structural data for simple Lie algebras.

Comparing Knapp’s book with classical texts like S. Helgason’s [Differential geometry, Lie groups, and symmetric spaces. Pure and applied mathematics, Vol. 80, New York etc. (1978; Zbl 0451.53038)] or V. S. Varadarajan’s [Lie groups, Lie algebras, and their representations. Prentice-Hall series in modern analysis. Englewood Cliffs, NJ (1974; Zbl 0371.22001)] two differences meet the eye: The down-playing of differential geometry and the emphasis on matrix groups as the prime examples of the theory. The declared goal of the book under review is to provide the structure theoretic background needed in the representation theory of semisimple Lie groups. The approach chosen is a mixture of algebra and analysis with a minimum of differential geometry. In fact, the concepts from differential geometry and topology needed to set up the correspondence between Lie subalgebras and analytic subgroups or to lift algebra homomorphisms to group homomorphisms (like integrable vector fields or universal covering groups) are assumed to be known. In this way Knapp leads the reader with a basic knowledge of Lie groups (as it is provided in many courses on differential geometry) into substantial structure theory very quickly.

Most of the results treated by Knapp can also be found in Helgason or Varadarajan (or both). Exceptions are some more representation theory oriented topics like the Peter-Weyl theory, the Harish-Chandra isomorphism or parabolic subgroups, for which one had to refer to more specialized sources like G. Warner’s [Harmonic analysis on semisimple Lie groups. I. Grundlehren der math. Wiss., Vol. 188, Berlin etc. (1972; Zbl 0265.22020)] or N. R. Wallach’s [Real reductive groups. I. Pure and applied mathematics, Vol. 132, Boston etc. (1992; Zbl 0785.22001)] before. In some cases new proofs are given. A special feature of this book is the fairly detailed collection of historical notes (almost twenty pages) which is quite interesting in itself.

Knapp’s prize winning style of expository writing is well known and needs no further commenting on. I consider his book a welcome addition to the literature on an important branch of mathematics. A few more details on the contents of the book: Chapter I is an introduction to finite dimensional Lie algebras presenting standard results like Lie’s and Engel’s Theorems, the Cartan Criterion, and the representation theory of \(sl(2, \mathbb{C})\). The second chapter is devoted to the classification of complex semisimple Lie algebras introducing the machinery of root systems and Weyl groups on the way. Chapter III deals with the universal enveloping algebra with the Poincaré-Birkhoff-Witt Theorem as a main result.

In Chapter IV compact Lie groups and their representations are treated. This chapter is much more analytic in spirit. The main themes presented are the Peter-Weyl theory, centralizers of tori and the analytic Weyl group.

Chapter V which is again mostly algebraic is on the finite dimensional representation theory of Lie groups and Lie algebras. In addition to highest weight theory and the Weyl Character Formula the author also introduces the Harish-Chandra Isomorphism and parabolic subalgebras. Fundamental weights, however, only occur in the exercises and the Borel-Weil Theorem is mentioned only three hundred pages later in the notes on further topics.

As far as global structure theory goes the Chapters VI and VII are the heart of the book. Here one finds the Cartan, Bruhat, Iwasawa, and Harish-Chandra decompositions as well as a new approach to the classification of real semisimple Lie algebras using Vogan diagrams which describe the effect of suitable Cartan involutions on the Dynkin diagram. The last chapter is a short introduction to integration on Lie groups. Most notably the Weyl integration formula is proved.

There are three appendices, one on multilinear algebra, one on Lie’s Third Theorem and one collecting a lot of useful structural data for simple Lie algebras.

Reviewer: J.Hilgert (Clausthal)