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.