Lie theory is essential to many parts of mathematics and theoretical physics, and lie algebras have become a central subject in their own right. This is the first and only book to provide a basic algebraic introduction that has been designed specifically for undergraduates, and that covers all the core material required for all other work in Lie theory. Everything is kept as simple as possible so the only mathematics required is linear algebra, and there are plenty of worked examples and exercises making it ideal for self-study. This will be an essential text for lecturers and undergraduates in mathematics and theoretical physics.

Klappentext

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.

Inhalt

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.