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:

• 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
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

• 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
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