MATH 251, Lie Groups, Spring 2022:

Office hours:: MW2-3 and by appointment

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 a continuation of Lie Groups 251B, taught in Winter 2022. We will first concentrate on the representation theory of semisimple Lie agebras. This will include a proof of the character formula, decomposition of tensor products of representations and duality theorems (such as Schur-Weyl duality, Howe duality). Further topics may include real non-compact Lie groups, symmetric spaces and q-deformations of universal enveloping algebras, known as quantum groups. There will not be a fixed course book. The books and lecture notes 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:

Lecture notes for Math 251B Winter 2022

Borcherds' lecture notes

Ziller's lecture notes

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)

Some lecture notes for this course:

Lecture 2 March 30


Please ignore material below the line for now:

Homework 1

Homework 2

Homework 3

Solutions of some homework problems

Proof of Lie product formula

Homework 4

Lecture notes: complete reducibility

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