Topological groups and Haar measure. Definition of Lie groups, examples of local fields and examples of discrete subgroups; basic properties; Lie subgroups. Lie algebras and relation with Lie groups: exponential map, adjoint representation. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems. Definition of algebraic groups and relation with Lie groups.
The goal is to have a broad though foundational knowledge of the theory of Lie groups and their associated Lie algebras with an emphasis on the algebraic and topological aspects of it.
Topology and basic notions of measure theory. A basic understanding of the concepts of manifold, tangent space and vector field is useful, but could also be achieved throughout the semester.
The lecture will start on Wednesday 19th Feburary 2025 and takes place every week on
| exercise sheet | due by | solutions |
|---|---|---|
| Exercise sheet 1 | 06.03 | Solutions 1 | Exercise sheet 2 | 20.03 | 09.04 | 17.04 | 08.05 | 22.05 |
The general rule is that exercise sheet \(k\) will be released on Thursday of week \(2k−1\) and it will be due on Thursday of week \(2k+1\).
It is not mandatory to hand-in solutions and your solutions do not contribute in any way to your final grade. However, it is an opportunity to get some (irregular) feedback on your understanding of the material covered in class.
Please, upload your solution via the SAM upload tool. Read here on how to use the SAM upload tool.
In order to access the website you will need a NETHZ-account and you will have to be connected to the ETH-network. From outside the ETH network you can connect to the ETH network via VPN. Here are instructions on how to do that.
Make sure that your solution is one PDF file and that its file name is formatted in the following way:
solution_<number of exercise sheet>_<your last name>_<your first name>.pdf
For example: If your first name is Alice, your last name is Miller, and you want to hand-in your solution to exercise sheet number 2, then you will have to upload your solution as one PDF file with the file name solution_2_Miller_Alice.pdf.
Here are the lecture notes (27.02) up to the point of the current lecture.
| week | date | event | topic |
|---|---|---|---|
| 1 | 19.02 | Lecture (notes) | topological groups, locally compact topological groups |
| 20.02 | Lecture | examples of (non-) compact and (non-) locally compact topological groups; first properties of top. groups | |
| 2 | 26.02 | Lecture | local homomorphisms, Haar measure, Homogeneous spaces |
| 27.02 | Lecture | ||
| 3 | 05.03 | Lecture | Weil formula and lattices |
| 06.03 | Exercise Class | (notes) |
Ex. Class 06.03 Remark 1: \( O(1,1) \) has four connected compoenents. We only saw two of them in class. The other ones have determinant \(-1\).
Ex. Class 06.03 Remark 2: In general topological groups need not be first-countable. For example think of an uncountable product of topological groups with two elements. This will be a topological group which is NOT first countable. However, all metric spaces are even second-countable, so they are in particular first countable.
Ex. Class 06.03 Remark 3: If \( X \) is locally compact and has a base of cardinality \( K \), then \( C(X,X) \) has also a base of cardinality \( K \). In particular, if \( X \) is second-countable so is \( \operatorname{Homeo}(X) \).