Lecturer

Marc Burger

Coordinator

Alessio Savini

Goals of the course

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.

Brief description

The topics included in the course are:

1. Topological groups and Haar measure.
2. Definition of Lie groups.
3. Examples of local fields and examples of discrete subgroups.
4. Lie subgroups.
5. Lie algebras and relation with Lie groups: exponential map, adjoint representation.
6. Semisimplicity, nilpotency, solvability, compactness: Killing form, Lie's and Engel's theorems.
7. Definition of algebraic groups and relation with Lie groups.

Prerequisites

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.

Exercise sessions take place on Thursdays (every second week) in HG D 3.2 from 8.00 to 10.00. The first exercise session will be on the 27th of september.

The exercise sheet will be uploaded on this page every Tuesday before the exercise session. From that moment the student has to hand in the exercises in room J 68 within 9 days. The corrected exercises are left in the same room.

Exercise Sheet	Due by	Solutions
Exercise sheet 1	Thursday 4/10/2018	Solution 1
Exercise sheet 2	Thursday 18/10/2018	Solution 2
Exercise sheet 3 ( VERSION UPDATED)	Thursday 1/11/2018	Solution 3
Exercise sheet 4	Thursday 15/11/2018	Solution 4
Exercise sheet 5	Thursday 29/11/2018	Solution 5
Exercise sheet 6	Thursday 13/12/2018	Solution 6

The exam will consist of an oral examination of 30 minutes.

Here you can find some notes of the course: Introduction to Lie groups.

First part of the handwritten notes of the course: Introduction to Lie groups, notes by Marc Burger, first part. Second part of the handwritten notes of the course: Introduction to Lie groups, notes by Marc Burger, second part.

For more references:

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