MATH 251, Lie Groups, Fall 2019:

Office hours:: tbd

Office: APM 5256, tel. 534-2734


Prerequisites: A solid understanding and familiarity with basic concepts in algebra (groups, homomorphisms etc) and analysis (convergence, norms) as well as a good understanding of linear algebra. Ask me if in doubt.

Material: This is an introduction into the theory of Lie groups and Lie algebras. They play an important role in numerous areas in physics and mathematics, including geometry, algebra and combinatorics. We will follow the approach in the book by Hall (see below) by working with matrix groups, i.e. closed subgroups of the group Gl(n) of all invertible n by n matrices. We will first study the connection between Lie groups and Lie algebras. After that we plan to study representations of Lie groups, primarily in the context of Lie algebras. In particular, we want to prove Weyl's character formula, which, for Lie type A, will give us symmetric functions. There will not be a fixed course book. But the books by Hall, and later by Humphreys and by Fulton & Harris listed below should cover most of the material of the course. For material not covered in these books, we plan to make other resources available.

Some books/lecture notes related to the course:

Lie Groups, Lie Algebras, And Representations : An Elementary Introduction, Brian C. Hall (electronic copy available from our library)

Introduction to Lie Algebras and Representation Theory, James E. Humphreys, Springer (electronic copy available from our library)

Representation Theory. A First Course, Graduate Texts in Mathematics 129, Joe Harris and William Fulton, Springer (electronic copy available from our library)

Homework 1

Homework 2

Homework 3

Solutions of some homework problems

Proof of Lie product formula

Homework 4

Lecture notes: complete reducibility


Please ignore material below the line for now:

Below are some notes for certain topics of the course

Exponential map for matrix Lie groups

Matrix Lie groups are Lie groups

basic properties of tori

Here are some problems and remarks concerning this course:

Problems related to exponential map