Lie Groups

1. Front Matter

   Pages i-xi

   PDF

2. Compact Groups

   1. Front Matter

      Pages 1-1

      PDF

   2. Haar Measure

      • Daniel Bump

      Pages 3-5

   3. Schur Orthogonality

      • Daniel Bump

      Pages 6-16

   4. Compact Operators

      • Daniel Bump

      Pages 17-20

   5. The Peter-Weyl Theorem

      • Daniel Bump

      Pages 21-26

3. Lie Group Fundamentals

   1. Front Matter

      Pages 27-27

      PDF

   2. Lie Subgroups of GL(n, ℂ)

      • Daniel Bump

      Pages 29-35

   3. Vector Fields

      • Daniel Bump

      Pages 36-40

   4. Left-Invariant Vector Fields

      • Daniel Bump

      Pages 41-45

   5. The Exponential Map

      • Daniel Bump

      Pages 46-49

   6. Tensors and Universal Properties

      • Daniel Bump

      Pages 50-53

   7. The Universal Enveloping Algebra

      • Daniel Bump

      Pages 54-57

   8. Extension of Scalars

      • Daniel Bump

      Pages 58-61

   9. Representations of sl(2, ℂ)

      • Daniel Bump

      Pages 62-68

   10. The Universal Cover

       • Daniel Bump

       Pages 69-78

   11. The Local Frobenius Theorem

       • Daniel Bump

       Pages 79-85

   12. Tori

       • Daniel Bump

       Pages 86-93

   13. Geodesics and Maximal Tori

       • Daniel Bump

       Pages 94-106

   14. Topological Proof of Cartan’s Theorem

       • Daniel Bump

       Pages 107-111

This book aims to be a course in Lie groups that can be covered in one year with a group of good graduate students. I have attempted to address a problem that anyone teaching this subject must have, which is that the amount of essential material is too much to cover. One approach to this problem is to emphasize the beautiful representation theory of compact groups, and indeed this book can be used for a course of this type if after Chapter 25 one skips ahead to Part III. But I did not want to omit important topics such as the Bruhat decomposition and the theory of symmetric spaces. For these subjects, compact groups are not sufficient. Part I covers standard general properties of representations of compact groups (including Lie groups and other compact groups, such as finite or p­ adic ones). These include Schur orthogonality, properties of matrix coefficients and the Peter-Weyl Theorem.

• Cohomology
• Fundamental group
• Matrix
• Matrix Theory
• Representation theory
• algebra
• homology

From the reviews:

"This book is a nice and rich introduction to the beautiful theory of Lie groups and its connection to many other areas of mathematics." (Karl-Hermann Neeb, Mathematical Reviews, 2005f)

"As Lie theory prerequisites can pose a great hurdle to number-theory students attracted to this program, Bump’s book will find an enthusiastic clientele even in an already crowded market. It will particularly delight readers who already know some of this material: the many short chapters generally begin with a map of the precise regress necessary to start wherever one ought. Summing Up: Highly recommended." (D.V. Feldman, CHOICE, Vol. 42 (8), April, 2005)

"This book is intended for a one-year graduate course on Lie groups and Lie algebras. The author proceeds beyond the representation theory of compact Lie groups … and provides a carefully chosen range of material to give the student the bigger picture." (L’Enseignement Mathematique, Vol. 50 (3-4), 2004)

"This book aims to be a course in Lie groups that can be covered in one year with a group of seasoned graduate students. … offers a wealth of complementary, partly quite recent material that is not found in any other textbook on Lie groups. … this book covers an unusually wide spectrum of topics … . the entire presentation is utmost thorough, comprehensive, lucid and absolutely user-friendly. … All together, this graduate text his a highly interesting, valuable and welcome addition … . (Werner Kleinert, Zentralblatt MATH, Vol. 1053, 2005)

"Reductive Lie groups and their representations form a very broad field. The aim of the book is to select essential topics for a year course for graduate students … . The book is nicely written and efficiently organized. … The presented book brings a beautiful selection of a number of further important additional topics, which are worth to include into a course. It is a very important addition to existingliterature on the subject." (EMS Newsletter, June, 2005)

"This book gives an introduction on the graduate level to the subject of Lie groups, Lie algebras and their representation theory. The presentation is well organized and clear … . this book is a very interesting and valuable addition to the list of already existing books on Lie groups." (J. Mahnkopf, Monatshefte für Mathematik, Vol. 147 (3), 2006)

• Department of Mathematics, Stanford University, Stanford, USA

  Daniel Bump

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