Lie Algebras and their representations

Aug-Dec 2012 (11:30-13:00 Mondays, Wednesdays)

Books:

  1. Introduction to Lie Algebras and Representation Theory, by J.E. Humphreys.
  2. Representation Theory: a first course, by W. Fulton and J. Harris.
  3. Lie Algebras of Finite and Affine Type, by R. W. Carter.
  4. Lie groups and Lie algebras, Bourbaki, chapters 4-6, 7-9.
First midterm exam on October 9 (Tuesday) afternoon.

Topics covered

Lecture 1 (Aug 13): Preliminaries - Lie algebras, subalgebras, homomorphism, representations, examples.
Suggested reading: Humphreys, sections 1,2.

Lecture 2 (Aug 17): The adjoint representation, analysis of representations of sl2.
Suggested reading: Fulton and Harris, lecture 11; Humphreys, section 7.

Lecture 3 (Aug 22): Representations of sl2 (contd).
Suggested reading: Fulton and Harris, lecture 11; Humphreys, section 7.
Assignment 1 (due Sep 3): Humphreys: Section 1: 1, 4, 6, 12;   Section 2: 2, 5, 6;   Section 7: 2, 4, 6, 7.
Lecture 4 (Aug 27): Direct sum, dual, tensor product; representations of sl3.
Suggested reading: Fulton and Harris, lecture 12.
Suggested viewing: Tensor products.
Key facts about tensor products.

Lecture 5 (Aug 29): Representations of sl3 (contd).
Suggested reading: Fulton and Harris, lecture 12.
Click here for sl3 paper (hexagonal ruled) (for more, see U. Goertz's webpage).
Assignment 2 (due Sep 12): Problems 1-6 from the tensor product worksheet.
Lecture 6 (Sep 3): Representations of sl3 (contd).
Suggested reading: Fulton and Harris, lecture 12.

Lecture 7 (Sep 5): Simple Lie algebras, statements of complete reducibility and abstract Jordan decomposition theorems; Cartan subalgebras, weight and root space decompositions.
Suggested reading: Fulton and Harris, lecture 14.

Lecture 8 (Sep 10): Analyzing representations of simple Lie algebras (contd).
Suggested reading: Fulton and Harris, lecture 14.
Assignment 3 (due Sep 21): is here.
Lecture 9 (Sep 12): The Weyl group, positive and negative roots, dominant integral weights index irreducible representations.
Suggested reading: Fulton and Harris, lecture 14.

Lecture 10 (Sep 17): The Killing form.
Suggested reading: Fulton and Harris, lecture 14.

Lecture 11 (Sep 21): Root systems.

Lecture 12 (Oct 1): Root systems (contd).

Lecture 13 (Oct 3): Root systems (contd).

Lecture 14 (Oct 8): Coxeter-Dynkin diagrams, Classification of root systems.
Suggested viewing:Classification of root systems.
Assignment 4 (due Oct 15): is here.
Lecture 15 (Oct 10): Construction of root systems.

Lectures 16, 17 (Oct 15, 17): Construction of Lie algebras associated to root systems - classical types.
Assignment 5 (due Oct 29): root systems.

Assignment 6 (due Nov 5): Weyl group.
Lecture 18 (Oct 22): Construction of finite dimensional simple Lie algebras by generators and relations (Serre's theorem).
Suggested reading: Carter, sections 7.3, 7.4, 7.5.

Lecture 19 (Oct 29): The Universal enveloping algebra, PBW theorem.
Suggested reading: Carter, sections 9.1, 9.2.

Lecture 20 (Nov 5): Verma modules.
Assignment 7 (due Nov 12): Universal enveloping algebras.