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

There will be no lecture on May 13 (holiday). The exercises on Friday the 14th will take place as usual.

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
• Peter-Weyl 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	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

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)

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