18.745 - Introduction to Lie Algebras (Fall 2018)

Instructor: Professor Victor Kac

Email: kac [at] math . mit .edu

Office: Room 2-176

Office Hours: By Appointment

Grader: Gurbir Dhillon gdhillon@mit.edu

Lectures: TR 2:30-4:00 2-143

COURSE OVERVIEW

Lecture 1, 2: Introduction, basic definitions
Lecture 3-5: Engel and Lie theorems
Lecture 6-9: Cartan subalgebras and Chevally's conjugacy theorem
Lecture 10: Cartan's criterion
Lecture 11-14: Structure of semi-simple Lie algebras
Lecture 15-20: Classification and construction of simple Lie algebras
Lecture 21: Compact form and Weyl group
Lecture 22: Universal enveloping algebra, PBW theorem, Casimir element and cohomology of Lie algebras
Lecture 23: Weyl, Levi and Maltsev theorems
Lecture 24: Verma modules
Lecture 25, 26: Finite-dimensional representations of semisimple Lie algebras. Weyl character and dimension formulae

RECOMMENDED BOOKS

  • J.P. Serre, "Complex Semi Simple Lie Algebras"
  • J.E. Humphreys, "Introduction to Lie Algebras and Representation Theory"
  • A.L. Onishchik, E.B. Vinberg, "Lie Groups and Algebraic Groups"

COURSE REQUIREMENTS

  • Typing two lectures ( using TeX is preferred, MS Word is ok; contact typers of previous lectures for templates). Lecture notes should be handed in within a week of the date of the lecture.
  • Doing exercises given in the course of lecture; collected and graded weekly. The exercises from the previous two lectures are due every Tuesday (if holiday, then the next Tuesday).
  • No final exam