Fall 2013 -- Lie Groups and Representation Theory (Math 599/Math 742)

The goal of this course is to develop the more standard, as well as the more interesting, parts of the theory of Lie groups and Lie algebras, with an emphasis on representation theory. For a more detail description, see the Syllabus
Lectures are on Monday and Friday, 11:35--12:55 in Burnside 1234
Our alternate meeting time, for discussion sessions and make-up classes, is on Wednesday, 11:35--12:25, In Burnside 920

Final Exam: Take-home. From December 5, 12pm until December 6, midnight (11:59pm).

Final Project: As an alternative to the Final Exam. Possible topics are given in homework 2 and 3. You are welcome to suggest your own topic, but please let me know before November 15 what you want to do. The write-up should be between 10 and 15 pages in length, and it would be best if you could also tell the class about your findings.
The report is due on December 16, 10 am

Lecture Notes

Numbering is logical. So far, we've been covering roughly 1 1/2 "lectures" per week.

Lecture 1 (Introduction)
Lecture 2 (Finite Groups 1; Schur's Lemma)
Lecture 3 (Finite Groups 2; Characters, Main Theorem)
Lecture 4 (Symmetric Group 1; Irreducible Representations)
Lecture 5 (Symmetric Group 2; Character Table)
Lecture 6 (Compact Topological Groups 1; Haar measure)
Lecture 7 (Compact Topological Groups 2; Peter-Weyl theorem)
Lecture 8 (Lie Groups and Lie Algebras 1: Definitions and Examples)
Lecture 9 (Lie Groups and Lie Algebras 2; Exponential Map and BCH)
Lecture 10 (Structure Theory; Engel, Lie, Cartan-Killing)
Lecture 11 (Complements to structure and representation theory)
Lecture 12 (Complex semi-simple Lie algebras 1; Tori, weights and roots)
Lecture 13 (Complex semi-simple Lie algebras 2; Dynkin diagrams, classification)

Homework Problems

Registered students are asked to work on the complete problem set at home, and contribute their solutions during the discussion session.

Homework 1 (McKay Correspondence) discussed September 18, 25 Solutions of problems 1(vi), 2 and 3 for binary octahedral group (Raghad)
Homework 2 discussed October 19 Solutions (Selim)
Homework 3 discussed November 13 Solution of problem 3 (BCH formula) (Eric) Solutions of all problems but #3 (Selim)