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Representations of compact groups [MA5054]

Sommersemester 2021

Lecturer: Prof. Dr. Robert König
Assistants: Farzin Salek, Zahra Khanian
Lecture: Thursdays 4:15pm – 5:45pm (zoom)
Exercises: Fridays 4:00pm - 5:30pm (zoom)

News

Registration for the (oral) exam should open on September 13. Please sign up for it in TUMOnline by Friday, September 24. (If you have trouble doing so, please let me know.) Only registered students will receive information by email to schedule the exam.

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 Manifolds and groups April 23
Assignment 2 Solution 2 Tangent vectors and push-forwards April 30
Assignment 3 Solution 3 Vector fields May 7
Assignment 4 Solution 4 Exponential map and Lie algebra May 14
Assignment 5 Solution 5 Connected abelian lie groups May 21
Assignment 6 Solution 6 Exponential map and Lie algebra May 28
Assignment 7 Solution 7 Forms and Haar measure June 4
Assignment 8 Solution 8 Wedge product & left uniform continuity June 11
Assignment 9 Solution 9 The modular Function & representation postponed to June 25
Assignment 10 Solution 10 (Ir)reducible representations June 25
Assignment 11 Solution 11 Matrix elements of irreps July 2
Assignment 12 Solution 12 Representations of SU(2) July 9
Assignment 13 Solution 13 Peter-Weyl theorem July 16

For the lecture on July 1: Please watch the two prerecorded videos.

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):

Some notes

Continuously updated (partial) lecture notes/summary.

Files Lecture
Review topological/smooth manifolds Lecture 1
Motivation (Laplace-Operator), Tangent space, differential maps Lecture 2
Vectorfields, integral curves, derivations, left-invariance Lecture 3
One-parameter groups, Lie algebra, Ad-map and differential Lecture 4
Connected component of the identity, Proof of Cartan's subgroup theorem Lecture 5
Proof of Cartan's subgroup theorem, quotient groups, differential forms, pushforward Lecture 6
Left-invariant forms, Haar measure, representations Lecture 7
Weyl's unitary trick, equivalence of representations, invariant subspaces and (ir)reps Lecture 8
Complete reducibility, Schur's lemma, intertwiners Lecture 9
Orthogonality relations, characters, Kuenneth formula Lecture 10
Lie algebra/Lie group representations, representations of sl(2,C), complexification Lecture 11
The Fritz-Peter/Hermann Weyl-Theorem Lecture 12