M392C: Lie Groups and Algebras, Fall 2021


This page will contain notes, homework assignments and other information about my Lie theory course, to be taught in fall 2021. At this point (April), all that is here is a course description:

Material: We will cover the basic theory of Lie Groups and Lie algebras, from the Lie correspondence through the classification theorem (over the complex numbers) and highest weight modules. If there is time, then I would also like to talk about Kazhdan's property (T), and/or the structure theory of real Lie groups and algebras. This structure theory tells you how to think about real Lie groups, but we will not be able to cover their classification. This theory is a prerequisite to understanding infinite-dimensional representations of Lie groups, but we will not be able to cover any of that, either.

Prerequisites: I will assume you are comfortable with the material from our first year graduate courses in topology and algebra. From topology, you absolutely need fluency with the fundamental group, covering spaces, and the language of differentiable manifolds. We will use deRham coholomogy a little bit, but you could get by with just a high-level understanding of it. From algebra you need fluency with group theory and multilinear algebra (including bilinear and Hermitian forms, and tensor products). No Galois theory will be needed, only a tiny bit of commutative algebra, and nothing with a ground field other than the real or complex numbers. From analysis we need only a little: enough to understand statements about Haar measure, and maybe what a Banach space is.

Textbook: notes prepared by me.