# Lie Algebras

Course Aim

Students wanting to learn graduate-level algebra, and especially representation theory. A solid grasp of linear algebra will be assumed, as well as an ability to understand and construct quite sophisticated mathematical proofs.

Course Description

Learn the fundamental objects in algebra, especially representation theory, with hands-on experience in computing representations and constructing sophisticated proofs for some powerful (and quite beautiful!) results. Practice focuses on the basic structures of simple Lie algebras over the complex numbers, as well as the theory of highest weight representation including Verma modules and enveloping algebras, concluding with Weyl's character formula for finite-dimensional simple modules. Additional topics include root systems, Cartan subalgebras, Cartan/triangular decomposition, Dynkin diagrams, and the Killing form.

This is an alternating years course.

Course Contents

1) Definition and key examples of Lie algebras
2) Structure theory of Lie algebras
3) Root systems, Dynkin diagrams, and Weyl groups
4) Classification of finite-dimensional (semi-)simple Lie algebras
5) Highest weight modules, simple modules, and Verma modules
6) Weyl's character formula for finite-dimensional simple modules

Assessment

Homework: 100%. There will be roughly 6 sets of homework problems during the term.

Prerequisites or Prior Knowledge

A solid grasp of undergraduate linear algebra, as well as experience following long proofs and constructing your own proofs. Students must be very comfortable with proofs in order to understand the material in this course and complete the homework questions adequately. If you are unsure, please discuss this further with your academic mentor. Some prior knowledge of the representation theory of finite groups will also be helpful when grappling with analogous results for Lie algebras, but it is not completely necessary.

Textbooks

Introduction to Lie Algebras and Representation Theory, by James Humphreys

Reference Books

Representation Theory: A first course, by William Fulton and Joe Harris
Introduction to Lie algebras, by Karin Erdmann and Mark Wildon

ノート

Alternate years course: AY2024