Erdmann K., Wildon M. Introduction to Lie Algebras

Springer-Verlag London Limited 2006, 254 c
Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right. This book provides an elementary introduction to Lie algebras based on a lecture course given to fourth-year undergraduates. The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality. Numerous worked examples and exercises are provided to test understanding, along with more demanding problems, several of which have solutions. Introduction to Lie Algebras covers the core material required for almost all other work in Lie theory and provides a self-study guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics.Ideals and Homomorphisms
Low-Dimensional Lie Algebras
Solvable Lie Algebras and a Rough Classification
Subalgebras of gl(V)
Engel’s Theorem and Lie’s Theorem
Some Representation Theory
Representations of sl(2, C)
Cartan’s Criteria
The Root Space Decomposition
Root Systems
The Classical Lie Algebras
The Classification of Root Systems
Simple Lie Algebras
Further Directions
Appendix A: Linear Algebra
Appendix B: Weyl’s Theorem
Appendix C: Cartan Subalgebras
Appendix D: Weyl Groups
Appendix E: Answers to Selected Exercises