### Sommersemester 2020

Dozent: Dr. Leonardo Patimo## Wednesday 12-14 in SR 318 , Ernst-Zermelo-Straße 1

## Organization

Due to the current restrictions, the course will take place online.The plan is to have recorded video replacing the lectures and Live-Stream sessions for the exercises classes.

The videos will be available on ILIAS. (Please write me an e-mail if you need a Password to enter)

The Live-Stream exercise classes will take place on BigBlueButton room Kasparov SR318 (that you can access from here) on Wednesday at 12.15.

This session will also serve as an opportunity for questions about the course material. If you are interested in the course, please sign up on HISinOne!

## Content

Lie theory is a subject lying at the intersection of algebra and geometry: a Lie group is a smooth manifold with a group structure such that the group operations are smooth. Lie groups arise in a natural way as symmetries of geometric objects: prominent examples of Lie groups are the general linear group or the orthogonal group. In addition, also the tangent space of a Lie group is equipped in a natural way with a particular algebraic structure, known as Lie algebra.In this lecture course, we will introduce the notion of Lie groups and Lie algebras and discuss the correspondence between them. The focus of the course will be on compact Lie groups, an important class of Lie groups for which the theory is very rich and well-developed. We will then study and classify representations of compact Lie groups, that is smooth linear actions on vector spaces. As a concrete final goal, we will classify compact Lie groups in terms of more elementary data: root systems.

## Literature

- W. Soergel: Lecture notes Mannigfaltigkeiten und Liegruppen, available at http://home.mathematik.uni-freiburg.de/soergel/Skripten/XXML.pdf.
- M. Sepanski: Compact Lie Groups, Springer, 2007.

## Notes and Exercises

N. | Date | Notes | Exercises | Due Date | Solutions |
---|---|---|---|---|---|

1 | 11.05 | Notes 01 | Exercise Sheet 1 | 19.05 | Solutions 1 |

2 | 18.05 | Notes 02 | Exercise Sheet 2 | 26.05 | Solutions 2 |

3 | 25.05 | Notes 03 | Exercise Sheet 3 | 09.06 | Solutions 3 |

4 | 01.06 | Notes 04 | Exercise Sheet 4 | 16.06 | Solutions 4 |

5 | 08.06 | Notes 05 | |||

6 | 15.06 | Notes 06 | Exercise Sheet 5 | 30.06 | Solutions 5 |

7 | 22.06 | Notes 07 | |||

8 | 29.06 | Notes 08 | Exercise Sheet 6 | 14.07 | Solutions 6 |

9 | 06.07 | Notes 09 | |||

10 | 13.07 | Notes 10 | Exercise Sheet 7 | 28.07 | Solutions 7 |

11 | 20.07 | Notes 11 | |||

12 | 31.07 | Notes 12 |