**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:**

- Definition and elementary properties of real and complex Lie groups
- Closed subgroups of
*GL*, classical Lie groups_{n} - Lie algebra of a Lie group, exponential map
- Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem
- Baker-Campbell-Hausdorff formula
- Homomorphisms, covering groups, Chevalley's theorem on Lie subgroups, Lie algebra-Lie group correspondence
- General structure theory of Lie algebras: solvable, nilpotent and semisimple Lie algebras, Lie and Engels' theorems, Cartan's criterion
- Ext groups of Lie algebra representations, theorems of Weyl, Levi and Ado, proof of the Lie algebra-Lie group correspondence completed.
- Representation theory of
*sl*_{2} - Classification of complex semisimple Lie algebras
- Finite dimensional representations of semisimple Lie groups and algebras
- Compact Lie groups and semisimple complex Lie groups
- 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.