# Introduction to Lie Groups Autumn 2020

Lecturer
Prof. Dr. Alessandra Iozzi
Coordinator
Yannick Krifka (E-mail)

## Description

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.

## Goal

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.

## 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.

## Syllabus

A detailed syllabus will be made available here.

The exercise sheets will be released below. More challenging exercises are indicated by $$\dagger$$.

Exercise Sheet Solution Comment
Exercise Sheet 1 Solution 1 Update 18/09/2020: In Exercise 2 the maps $$X^Y$$ are supposed to be continuous. This is now explicit in the exercise and the notation $$X^Y$$ was replaced by $$C(Y,X)$$.
Exercise Sheet 2 Solution 2
Exercise Sheet 3 Solution 3
Exercise Sheet 4 Solution 4 A missing continuity assumption has been added to exercise 6.
Exercise Sheet 5 Solution 5
Exercise Sheet 6 Solution 6
Exercise Sheet 7 Solution 7

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 $$2k+1$$.

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.

Handed-in solutions that fail to comply with the above requirements will be ignored.

The lecture will start on Wednesday, 16th of September 2020.

TimeRoom
Wed 8-10 am HG D 1.2
Thu 08-10 am HG D 1.2

Every fourth time there will be an exercise class held by Yannick Krifka instead of a lecture. For a detailed schedule see below.

The lectures and exercise classes will be held in presence and, additionally, they will be live-streamed and recorded by ETH. The live-stream from room HG D 1.2 by ETH is available here and its recordings will be available here.

Update: Due to the new Corona regulations at ETH the lectures will be held via Zoom until further notice. You can find the Zoom-URLs in the table below.

DateEvent Zoom Meeting Zoom Recording Notes
Wed 16/09/20 Lecture Zoom Meeting Zoom Recording Notes
Thu 17/09/20 Lecture Zoom Meeting Zoom Recording 1st hour / Zoom Recording 2nd hour Notes / Separate Notes 2nd hour
Wed 23/09/20 Lecture Zoom Meeting Zoom Recording Notes
Thu 24/09/20 Exercise class Zoom Meeting Recording Notes
Wed 30/09/20 Lecture Notes
Thu 01/10/20 Lecture Notes
Wed 07/10/20 Lecture Notes
Thu 08/10/20 Exercise class Notes
Wed 14/10/20 Lecture Notes
Thu 15/10/20 Lecture Zoom Recording Notes
Wed 21/10/20 Lecture Notes
Thu 22/10/20 Exercise class Notes
Wed 28/10/20 Lecture Notes
Thu 29/10/20 Lecture Notes
Wed 04/11/20 Lecture Zoom Meeting Zoom Recording Notes
Thu 05/11/20 Exercise class Zoom Meeting Zoom Recording Notes
Wed 11/11/20 Lecture Zoom Recording Notes
Thu 12/11/20 Lecture Zoom Recording Notes
Wed 18/11/20 Lecture Zoom Meeting Zoom Recording Notes
Thu 19/11/20 Exercise class Zoom Meeting Zoom Recording Notes
Wed 25/11/20 Lecture Zoom Recording Notes
Thu 26/11/20 Lecture Zoom Recording Notes
Wed 02/12/20 Lecture Zoom Meeting Zoom Recording Notes
Thu 03/12/20 Exercise class Zoom Meeting Zoom Recording Notes
Wed 09/12/20 Lecture Zoom Meeting Zoom Recording Notes
Thu 10/12/20 Lecture Zoom Meeting Zoom Recording Notes
Wed 16/12/20 Lecture Zoom Meeting Zoom Recording Notes
Thu 17/12/20 Lecture Zoom Meeting Zoom Recording Notes

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 more or less these notes from a previous course. They are based on notes by Alessandra Iozzi, and they were type-set and modified by Stephan Tornier.