XVI

Preface

helped greatly if they have had a basic introduction to Lie theory. Though

the background material in Chapter 14 is reasonably self-contained, it is

intended as a survey, and so some of the more technical proofs have been

left out. For example, the basic theorems relating Lie algebras and Lie

groups are stated without proof, as is the existence of compact real forms

for complex semisimple groups and the classification of finite dimensional

representations of semisimple Lie algebras.

Each chapter ends with an exercise set. Many exercises involve filling in

details of proofs in the text or proving results that are needed elsewhere in

the text, while others supplement the text by exploring examples or addi-

tional material. Cross-references in the text to exercises indicate both the

chapter and the exercise number; that is, Exercise 2.5 refers to Exercise 5

of Chapter 2.

There are many individuals who contributed to the completion of this

text. Edward Dunne, Editor for the AMS book program, noticed an early

version of the course notes on my website and suggested that I consider turn-

ing them into a textbook. Without this suggestion and Ed's further advice

and encouragement, the text would not exist. Several of my colleagues pro-

vided valuable ideas and suggestions. I received encouragement and much

useful advice on issues in several complex variables from Hugo Rossi. Aaron

Bertram, Herb Clemens, Dragan Milicic, Paul Roberts, and Angelo Vistoli

gave me valuable advice on algebraic geometry and commutative algebra,

making up, in part, for my lack of expertise in these areas. Henryk Hecht,

Dragan Milicic, and Peter Trombi provided help on Lie theory and group

representations. Without Dragan's help and advice, the chapters on Lie

theory, algebraic groups, and the Borel-Weil-Bott theorem would not exist.

The proof of the Borel-Weil-Bott theorem presented in Chapter 16 is due

to Dragan, and he was the one who insisted that I approach structure the-

orems for semisimple Lie groups from the point of view of algebraic groups.

The students who took the course the three times it was offered while the

notes were being developed caught many errors and offered many useful

suggestions. One of these students, Laura Smithies, after leaving Utah with

a Ph.D. and taking a position at Kent State, volunteered to proofread the

entire manuscript. I gratefully accepted this offer, and the result was nu-

merous corrections and improvements. My sincere thanks goes out to all

of these individuals and to my wife,

, who showed great patience and

understanding while this seemingly endless project was underway.

Joseph L. Taylor