Lie Groups and Representations
T-TH 10:40-12:00, Lockett 111
||T-TH 1:30 --2:30 PM and by request
||578-1608 and 225-337-2206 (cell)
|Text: || Lecture notes, see more information at the end.
Lie groups and their representations are central in several areas of mathematics. This
includes algebra, analysis, and parts of geometry and physics. In this course we will
mainly aim at some basic facts about Lie groups and geometric actions of Lie groups.
The course will approximately be structured as follows:
- Simple facts about manifolds.
- Lie groups.
- Basic representation theory.
- Representations of compact groups.
- Examples of linear groups.
- The Lie algebra of Linear groups and the exponential map.
- Homogeneous spaces.
- Homogeneous vector bundles.
- Continuous implies smooth/analytic.
- Square-integrable sections and induced representations.
- Representations of compact Lie groups. Decomposition
of induced representations. Fourier series on compact
- If there is time then we will also discuss the very important topic
of highest weight representations.
We will mainly deal with linear Lie groups to avoid time-consuming discussion about
the exponential map. But we will also give some comment on the general case.
There will be graded home works every second or third week. There will be one mid-term
take home test due beginning of class October 15. There will be a take home
final, due 1:30 pm on December 10. The final can be replaced by a presentation
We will not be using a single book, but our own lecture notes. I will try
to post as much of my own notes as possible. But here are some (in no particular
and good books related to what we will be doing:
- Frank W. Warner: Foundations of Differentiable Manifolds and Lie Groups. Springer.
- T. Brocker and T. Dieck: Representations of Compact Lie Groups. Springer
- M. R. Sepanski: Compact Lie Groups. AMS.
- W. Rossmann: Lie Groups: An Introduction through
Linear Groups. Oxford.
- A. Arvanitogeorgos: An Introduction to Lie Groups
and the Geometry of Homogeneous Spaces.
- H. Hall: Lie Groups, Lie Algebras, and
Representations: An Elementary Introduction. AMS
Some great advanced book:
- J.J. Duistermaat and J. A. C. Kolk: Lie
- S. Helgason: Differential Geometry, Lie Groups,
and Symmetric Spaces. AMS.
- V.S. Varadarajan: Lie Groups, Lie Algebras, and Their Representations.