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

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