Lie groups and representation theory

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

First lecture on April 15, 2019

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: