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

Syllabus: (Tentative syllabus, subject to change)

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. Homomorphisms; covering groups; Chevalley's theorem on Lie subgroups
5. Basic structure theory of Lie algebras: solvable, nilpotent and semisimple Lie algebras; Lie and Engels' theorems
6. Representation theory of sl₂
7. Classification of complex semisimple Lie algebras
8. Compact Lie groups and semisimple complex Lie groups
9. Universal enveloping algebra; Poincaré-Birkhoff-Witt theorem
10. Finite dimensional representations of semisimple Lie groups and algebras
11. Algebraic groups; Hopf algebras

Homework Assignments: To be posted later

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