Representations of compact groups [MA5054]
Sommersemester 2019
Lecturer: 
Prof. Dr. Robert König 
Assistant: 
Frederik vom Ende 
Lecture: 
Fridays 10:15am – 11:45am, room 02.10.011 (room code 5610.02.011) 
Anmeldung 
Exercises: 
Fridays 9:15am – 10:00am, room 02.10.011 (room code 5610.02.011) 

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
 PeterWeyl 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
Exercises 
Solutions 
Topics 
To be discussed on 
Assignment 1 
Solution 1 
Topological and Lie groups, unitary groups 
May 3rd 
Assignment 2 
Solution 2 
Tangent space, exponential map, matrix Lie algebras 
May 10th 
Assignment 3 
Solution 3 
LieTrotter formula, representations of Lie algebras 
May 17th 
Assignment 4 
Solution 4 
Adjoint representation, connectedness 
May 24th 
Assignment 5 
Solution 5 
Wedge product, left uniform continuity 
May 31st 
Assignment 6 
Solution 6 
Haar measures, modular function and Heisenberg group 
June 7th 
Assignment 7 
Solution 7 
Dual and canonical representations 
June 14th 
Assignment 8 
Solution 8 
Irreducible representations ("irreps"), Schur's Lemma 
June 21st 
Assignment 9 
Solution 9 
Irreps and characters 
June 28th 
Assignment 10 
Solution 10 
Irreps of tori and of the dihedral group 
July 5th 
Assignment 11 
Solution 11 
PeterWeyl and Fourier Analysis 
July 12th 
Assignment 12 
Solution 12 
Limits of PeterWeyl, SU(n) revisited 
July 19th 
Assignment 13 
Solution 13 
Representations of SU(2), su(2), complexification 
July 26th 
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)
In addition, I will provide notes:
current version (updated on July 19) and
revised version (updated on August 16: proof of Theorem 4.2 restructured)