### News

Notes on the construction of Haar measure and the Peter-Weyl theorem for compact topological groups from a previous version of the course.

By mutual agreement this course is taught
in English. The notes (in German) from Winter 15/16 are
** here**.

**Information** regarding the Exam.

** Time and place:**
Monday & Wednesday, 11-1, SR A

### Instructor

Prof. J. Walcher, walcher@uni-heidelberg.de

### Description

The course corresponds roughly to the module MB10 from the Mathematics Bachelor program. In comparison to the current module description, a stronger emphasis is placed on the representation theory (of finite and compact topological groups over the real and complex numbers) that is most important for applications in physics.

** Prerequisites:**
Linear algebra and elements of topology, basic notions of differential geometry
and Hilbert spaces is advantageous for full benefit in the middle section of the
course.

** References:**
Within the enormous amount of literature on the subject, a modern classic is:

W. Fulton and J. Harris, Representation Theory: A first course, Springer GTM 129

I also like:

B. Simon, Representations of Finite and Compact Groups, AMS Graduate Studies in Mathematics, Vol. 10

### Exercises

**Direction:**
Lukas Hahn, Sebastian Nill

**Routine:**
The weekly problem sets become available on Tuesday, 11 am. Solutions can be submitted
until the following Tuesday, 11:00 in the box in front of the deanery (semester-long two-person teams
are admissible), and are being discussed in the tutorials on Wednesday and Friday.

**Time and place:**

Wednesday, 16-18 in SR 8 with Sebastian Nill

Friday, 14-16 in SR 9 with Lukas Hahn

First meetings: April 24 and 26

**Registration**
in the Müsli

### Course Plan

Subject to change!

Week of | Content |
---|---|

April 15 | Introduction, Schur's Lemma, Tensor operations on representations |

April 24 | Finite-dimensional representations, characters, representation theory of finite groups |

April 29 | The irreducible representations of the symmetric group |

May 6 | Character table of the symmetric group |

May 13 | Haar measure for compact topological groups |

May 20 | Peter-Weyl theorem; Lie groups and Lie algebras |

May 27 | The classical groups; exponential map, adjoint representation, regularity |

June 3 | Baker-Campbell-Hausdorff formula |

June 10 | Simple connectedness |

June 17 | Representation theory of \(\mathfrak{sl}(2,{\mathbb C})\), beginnings of structure theory |

June 24 | Theorems of Engel and Lie, Cartan criterion |

July 1 | Semisimplicity vs. reductiveness, invariant volume form, reductiveness vs. compactness |

July 8 | root space decomposition of \(\mathfrak{sl}(n,{\mathbb C})\), Complete reducibility of representations |

July 15 | Cartan subalgebras, root spaces; classification of simple Lie algebras over \({\mathbb C}\) |

### Exam

**Regulations:**

The *regular exam* will be written on **Thursday, July 25, from 9am--11am**, place TBA.

Admission with 50% of possible homework points and a valid photo ID.
Doors open at 8:45. No registration necessary.

**Hardship:**