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MATH 650
Lie Groups
(Fall 2003)
Instructor:
Reyer Sjamaar
Meeting
Time & Room
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
Last modified:
August 13, 2003
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