Lie groups, Lie algebras, and representation theory are the main focus of this text. In order to keep the prerequisites to a minimum, the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples. The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation.

The text is divided into two parts. The first covers Lie groups and Lie algebras and the relationship between them, along with basic representation theory. The second part covers the theory of semisimple Lie groups and Lie algebras, beginning with a detailed analysis of the representations of SU(3). The author illustrates the general theory with numerous images pertaining to Lie algebras of rank two and rank three, including images of root systems, lattices of dominant integral weights, and weight diagrams. This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory. Brian Hall is an Associate Professor of Mathematics at the University of Notre Dame.

Contents

• Preface
• I: General Theory:
  • Matrix Lie Groups
  • Lie Algebras and the Exponential Mapping
  • The Bake-Campbel-Hausdorff Formula
  • Basic Representation Theory
• II: Semisimple Theory:
  • The Representations of SU(3)
  • Semisimple Lie Algebras
  • Representations of Complex Semisimple Lie Algebras
  • More on Roots and Weights

• Appendix A: A Quick Introduction to Groups
• Appendix B: Linear Algebra Review
• Appendix C: More on Lie Groups
• Appendix D: Clebsc-Gordan Theory for SU(2) and the Wigne-Eckart Theorem
• Appendix E: Computing Fundamental Groups of Matrix Lie Groups
• Bibliography