Professor: Mark Haiman
Office: 855 Evans
Office hours: WF 1:30-3:00 or by appointment
Phone: (510) 642-4318

This course is a one-semester introduction to Lie groups and their Lie algebras. In addition, I will devote some time to algebraic groups and Hopf algebras, in preparation for Reshetikhin's covering quantum groups in 261B in the Spring.

Prerequisites: Background in algebra and topology equivalent to 202A and 250A. Although 214 (Differential Manifolds) is the official prerequisite, I will review in the lectures those bits of differential geometry that we will need.

Textbooks:

• Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd Edition
• Armand Borel, Linear Algebraic Groups, 2nd Enlarged Edition

Planned syllabus:

1. Definition and elementary properties of real and complex Lie groups
2. Closed subgroups of GL_n, classical Lie groups
3. Lie algebra of a Lie group, exponential map
4. Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem
5. Baker-Campbell-Hausdorff formula
6. Homomorphisms, covering groups, Chevalley's theorem on Lie subgroups, Lie algebra-Lie group correspondence
7. General structure theory of Lie algebras: solvable, nilpotent and semisimple Lie algebras, Lie and Engels' theorems, Cartan's criterion
8. Ext groups of Lie algebra representations, theorems of Weyl, Levi and Ado, proof of the Lie algebra-Lie group correspondence completed.
9. Representation theory of sl₂
10. Classification of complex semisimple Lie algebras
11. Finite dimensional representations of semisimple Lie groups and algebras
12. Compact Lie groups and semisimple complex Lie groups
13. Algebraic groups, Hopf algebras

Homework Assignments:

• Problems, Set 1 [Problem 7 changed 9/30]
• Problems, Set 2 [Problem 3 changed 9/30]
• Problems, Set 3 [Problem 1(a) changed 10/3, Problem 14 clarified 10/22]
• Problems, Set 4 [Problem 11 changed 11/21 and again 12/3]
• Problems, Set 5 [Problem 6 changed 12/3]
• Problems, Set 6

Grading: Based on homework assignments. No exams will be given.