Introduction to Lie Groups Spring 2024

Lecturer
Daniele Semola
Coordinator
Raphael Appenzeller

News

Content

Link to the Vorlesungsverzeichnis. 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.

Possible topics are:

Prerequisites are 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.

Some notes for the lectures are available here:

Notes of Lectures 1,2,3,
Topological groups.
Notes of Lectures 4,5,6,7,8,9,
Haar measure and homogeneous spaces.
Notes of Chapters 3.1, 3.2,
Lie groups and Lie algebras.
Notes of Chapter 3.3,
Lie algebra of a Lie group.
Notes of Chapter 3.4,
Exponential map.
Notes of Chapter 3.5,
Lie subgroup - subalgebra correspondence.
Notes of Chapter 3.6,
Adjoint representation.
Notes of Chapter 4.1,
Structure theory: Solvable Lie groups and algebras
Notes of Chapter 4.2,
Nilpotent Lie groups and algebras
Notes of Chapter 4.3,
The Killing form and Cartan's criterion for solvability
Notes of Chapter 4.4,
Semisimple Lie groups and algebras

The lecture recordings can be accessed here. There is also a diary.

Exercises

Exercise classes take place every second week on Thursday (replacing the lecture). The first exercise class takes place on Thursday 29. Feb.

Approximately every second week there will be an exercise sheet. Working on the exercise sheet is highly recommended. If you want feedback to your solutions, you can hand them in, either in the physical box in room HG J 68 or online (need to be in the ETH-network or use VPN).

Exercise sheet Hand in by Solutions
Exercise sheet 1 Monday, 11. 3. 2024, 12:00 Solutions 1
Exercise sheet 2 Monday, 25. 3. 2024, 12:00 Solutions 2
Exercise sheet 3 Monday, 15. 4. 2024, 12:00 Solutions 3
Exercise sheet 4 Monday, 29. 4. 2024, 12:00 Solutions 4
Exercise sheet 5 Monday, 20. 5. 2024, 12:00 Solutions 5
Exercise sheet 6 Monday, 27. 5. 2024, 12:00 Solutions 6
Exercise sheet 7 no hand in Solutions 7
Additional exercise sheet complementary to the actual content of the course. no hand in. (last update May 27)

Literature

The course will basically follow Alessandra Iozzi's notes From topological groups to Lie groups, which are still in revision.