Cornell Math - MATH 650, Fall 2003
MATH 650
Lie Groups
(Fall 2003)
Instructor: Reyer Sjamaar
Prerequisites: undergraduate analysis, manifold theory (as taught e.g. in MATH 652), some measure theory and general topology. A nodding acquaintance with covering spaces and fundamental groups would be helpful.
Description: The theory of Lie groups codifies the concept of "continuous symmetries". Lie groups are ubiquitous in differential geometry, mathematical physics, harmonic analysis, ordinary and partial differential equations, and many branches of algebra. We will introduce the exponential map and Dynkin's formula, study the relationship between Lie groups and Lie algebras, and prove the three fundamental theorems of Lie. We will derive the basic facts concerning group actions on manifolds and in particular on vector spaces ("representation theory"). We will then focus on compact Lie groups, obtain the Peter-Weyl theorem (a non-abelian generalization of Fourier analysis), and delve into the Cartan-Weyl theory of highest weights, which gives a picture of all irreducible representations of compact Lie groups. We end with the famous Weyl character formula.
Book: There will not be a required textbook. I will use chapters from the following books:
- Broecker and Tom Dieck, Representations of Compact Lie Groups
- Kolk and Duistermaat, Lie Groups
- Bourbaki, Groupes et algebres de Lie, Ch IX