Lecturer

Daniele Semola

Coordinator

Raphael Appenzeller

News

• There is a forum, where questions and related topics can be discussed.
• Information for doctoral students: please register under "401-3225-DRL Introduction to Lie Groups" in order for you to get the credits for following this course. If you started your phd after 2022, please contact the lecturer about getting credits (if you have not done so already).
• The lectures take place on Wednesdays, 8-10h, and Thursdays, 12-14h. The lectures take place in HG D1.1. The first lecture is on Wednesday, 21.02.2022.

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:

• Topological groups, homogeneous spaces, invariant measures: basic properties and examples
• Lie groups and their Lie algebras; correspondence between Lie subalgebras and Lie subgroups
• One parameter groups, the exponential map; the closed subgroup theorem; principal bundles and homogeneous spaces
• The adjoint representation, ideals and normal subgroups
• Solvable Lie groups and Lie algebras: Lie's theorem
• Nilpotent Lie groups and Lie algebras: Engel's theorem
• Semisimple Lie algebras: Dieudonne's theorem
• Levi decomposition
• Compact connected Lie groups, basic structure

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:

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).

Literature

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

• A. Knapp: "Lie groups beyond an Introduction" (Birkhaeuser)
• A. Sagle & R. Walde: "Introduction to Lie groups and Lie algebras" (Academic Press, '73)
• F. Warner: "Foundations of differentiable manifolds and Lie groups" (Springer)
• H. Samelson: "Notes on Lie algebras" (Springer, '90)
• S. Helgason: "Differential geometry, Lie groups and symmetric spaces" (Academic Press, '78)
• A. Knapp: "Lie groups, Lie algebras and cohomology" (Princeton University Press)