Introduction to Lie Algebras

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.

Based on a lecture course given to fourth-year undergraduates, this book provides an elementary introduction to Lie algebras. It starts with basic concepts. A section on low-dimensional Lie algebras provides readers with experience of some useful examples. This is followed by a discussion of solvable Lie algebras and a strategy towards a classification of finite-dimensional complex Lie algebras. The next chapters cover Engel's theorem, Lie's theorem and Cartan's criteria and introduce some representation theory. The root-space decomposition of a semisimple Lie algebra is discussed, and the classical Lie algebras studied in detail. The authors also classify root systems, and give an outline of Serre's construction of complex semisimple Lie algebras. An overview of further directions then concludes the book and shows the high degree to which Lie algebras influence present-day mathematics.

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.

Contenu

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.