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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:

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 Lie-Trotter 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 Peter-Weyl and Fourier Analysis July 12th
Assignment 12 Solution 12 Limits of Peter-Weyl, 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

to a large extent. Other recommended literature (more may be provided during the course):

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)