Lie Algebras

Zhe-Xian Wan, I. N. Sneddon, M. Stark

• Publisher: Pergamon
• Year: 2014
• ISBN:
  9781483187303
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

Lie Algebras is based on lectures given by the author at the Institute of Mathematics, Academia Sinica. This book discusses the fundamentals of the Lie algebras theory formulated by S. Lie. The author explains that Lie algebras are algebraic structures employed when one studies Lie groups. The book also explains Engel's theorem, nilpotent linear Lie algebras, as well as the existence of Cartan subalgebras and their conjugacy. The text also addresses the Cartan decompositions and root systems of semi-simple Lie algebras and the dependence of structure of semi-simple Lie algebras on root systems. The text explains in details the fundamental systems of roots of semi simple Lie algebras and Weyl groups including the properties of the latter. The book addresses the group of automorphisms and the derivation algebra of a Lie algebra and Schur's lemma. The book then shows the characters of irreducible representations of semi simple Lie algebras. This book can be useful for students in advance algebra or who have a background in linear algebra.

Table of contents

• Front Cover
• Lie Algebras
• Copyright Page
• Table of Contents
• PREFACE
• CHAPTER 1. BASIC CONCEPTS
• CHAPTER 2. NILPOTENT AND SOLVABLE LIE ALGEBRAS
• CHAPTER 3. CARTAN SUBALGEBRAS
• CHAPTER 4. CARTAN'S CRITERION
• CHAPTER 5. CARTAN DECOMPOSITIONS AND ROOT SYSTEMS OF SEMISIMPLE LIE ALGEBRAS
• CHAPTER 6. FUNDAMENTAL SYSTEMS OF ROOTS OF SEMISIMPLE LIE ALGEBRAS AND WEYL GROUPS
• CHAPTER 7. CLASSIFICATION OF SIMPLE LIE ALGEBRAS
• CHAPTER 8. AUTOMORPHISMS OF SEMISIMPLE LIE ALGEBRAS
• CHAPTER 9. REPRESENTATIONS OF LIE ALGEBRAS
• CHAPTER 10. REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS
• CHAPTER 11. REPRESENTATIONS OF THE CLASSICAL LIE ALGEBRAS
• CHAPTER 12. SPIN REPRESENTATIONS AND THE EXCEPTIONAL LIE ALGEBRAS
• CHAPTER 13. POINCARÉ-BIRKHOFF-WITT THEOREM AND ITS APPLICATIONS TO REPRESENTATION THEORY OF SEMISIMPLE LIE ALGEBRAS
• CHAPTER 14. CHARACTERS OF IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS
• CHAPTER 15. REAL FORMS OF COMPLEX SEMISIMPLE LIE ALGEBRAS
• INDEX
• OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS