- Lecturer
- Alessandra Iozzi
- Coordinator
- Lisa Ricci

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.

Exercise sessions take place on Thursdays every second week in ** HG E 1.2 ** from ** 10-12**. The first exercise session will be on the 30th of September.

The lecture will **start on Wednesday, 22nd of September 2021** and takes place every week on

Lectures will be in-person. According to the ETH Guidelines a Covid Certificate together with an ID card or Legi should be presented to enter the room for in-person teaching.

It is possible to follow the lecture live on Zoom. The recurring **Zoom meetings** for the lectures can be found here.
The lecture is recorded and the **recordings and lecture notes** will be uploaded on this page.

Every second week on Thursday there will be an **exercise session** (instead of the lecture). The first exercise session is on Thursday 30th September.

The exercise sheet will be uploaded on this page. The exercise sheet k will be uploaded on Friday of week 2k - 1 and it will be due on Thursday of week 2k + 1.

**Please, upload your solution via 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 `

**Example**:
` solution_2_Surname_Name.pdf`

.

exercise sheet | due by | upload link | solutions | comments |
---|---|---|---|---|

Exercise sheet 1 | 07/10/2021 | upload ex sheet 1 | Solution 1 | |

Exercise sheet 2 | 21/10/2021 | upload ex sheet 2 | Solution 2 | new version 28.10.21 (typo in ex 4b) |

Exercise sheet 3 | 4/11/2021 | upload ex sheet 3 | Solution 3 | |

Exercise sheet 4 | 18/11/2021 | upload ex sheet 4 | Solution 4 | new version 12.11.21 (ex6: added assumption simply connected on H) new version 25.11.21 (ex 6: remark about smooth covering) |

Exercise sheet 5 | 02/12/2021 | upload ex sheet 5 | Solution 5 | |

Exercise sheet 6 | 16/12/2021 | upload ex sheet 6 | Solution 6 |

This course has its own subforum in the D-MATH forum. Feel free to ask questions and discuss about the lecture and exercises there.

The lecture will follow these notes.

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