Mathematics and Physics in Heidelberg Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik Mathematisches Institut Karl-Theodor-Brücke Surprise Fakultät für Physik und Astronomie Fakultät für Physik und Astronomie Institut für Theoretische Physik Universität Heidelberg Universität Heidelberg Universität Heidelberg

Lecture course in the Summer Semester 2019

(Nach Absprache findet diese Veranstaltung in englischer Sprache statt!)

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: