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, 2019Instructor
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: