## Representations of compact groups [MA5054]

### Sommersemester 2021

**Lecturer**: |
Prof. Dr. Robert König |

**Assistants**: |
Farzin Salek, Zahra Khanian |

**Lecture**: |
Thursdays 4:15pm – 5:45pm (zoom) |

**Exercises**: |
Fridays 4:00pm - 5:30pm (zoom) |

### News

Registration for the (oral) exam should open on September 13. Please sign up for it in TUMOnline by Friday, September 24. (If you have trouble doing so, please let me know.) Only registered students will receive information by email to schedule the exam.

### Contents

This course will serve as an introduction to the theory of Lie groups and their representations, a topic of central importance in physics. Subjects to be covered include:

- Lie groups and algebras, the exponential map
- Peter-Weyl theorem
- maximal tori, roots and weights
- the Weyl group, the Weyl character formula and representations of the classical groups.

In addition, some applications to physics may be discussed.

### Exercises

For the lecture on July 1: Please watch the two

prerecorded videos.

### Literature

There are plenty of excellent textbooks on these topics. The lectures will follow the second half of

- B. Simon, Representation of finite and compact groups, AMS (1996)

to a large extent. Other recommended literature (more may be provided during the course):

- A. Knapp, Lie groups beyond an introduction, Birkhaeuser (1996)
- R. Goodman and N. Wallach, Representations and Invariants of the Classical Groups, Cambridge University Press (1998)
- T. Broeckner and T. Dieck, Representations of Compact Lie Groups, Springer (1985)

### Some notes

Continuously updated (partial)

lecture notes/summary.

Files |
Lecture |

Review topological/smooth manifolds |
Lecture 1 |

Motivation (Laplace-Operator), Tangent space, differential maps |
Lecture 2 |

Vectorfields, integral curves, derivations, left-invariance |
Lecture 3 |

One-parameter groups, Lie algebra, Ad-map and differential |
Lecture 4 |

Connected component of the identity, Proof of Cartan's subgroup theorem |
Lecture 5 |

Proof of Cartan's subgroup theorem, quotient groups, differential forms, pushforward |
Lecture 6 |

Left-invariant forms, Haar measure, representations |
Lecture 7 |

Weyl's unitary trick, equivalence of representations, invariant subspaces and (ir)reps |
Lecture 8 |

Complete reducibility, Schur's lemma, intertwiners |
Lecture 9 |

Orthogonality relations, characters, Kuenneth formula |
Lecture 10 |

Lie algebra/Lie group representations, representations of sl(2,C), complexification |
Lecture 11 |

The Fritz-Peter/Hermann Weyl-Theorem |
Lecture 12 |