Representations of compact groups [MA5054]

Sommersemester 2015

Lecturer:	Prof. Dr. Robert König
Lecture:	Tuesday 2:15pm – 3:45pm, room 02.04.011 (room code M9/M10 5604.02.011)	Anmeldung
Exercises:	Monday 8:30am - 10:00am, room 03.10.011 (room code M5 5610.03.011)	Anmeldung

News

• The (oral) exam will take place on Wednesday, August 5. Location: Seminarraum (M5) (5610.03.011)
• Exam time slots: Doodle poll results.
• There will be no lecture on July 7. Instead, we'll have a lecture on Monday June 8, 8:30am - 10:00am (same room as above).
• The next exercise class will be on Monday June 15 (assignment 5).
• Exercise classes will take place once every two weeks (location/time: scheduled for Monday, 8:30am-10:00am based on the doodle poll).

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.

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 handwritten notes: (everything so far in one file )

• Overview
• Definition of Lie groups, examples
• One-parameter groups and the exponential map
• Lie group - Lie algebra correspondence, properties of exponential map
• Forms and Integration, the Weyl integration formula (part I)
• Representations: basic definitions
• Schur's Lemma, orthogonality of characters (corrected 06/09)
• The Peter-Weyl Theorem
• Lie algebra representations and SU(2)
• Weights
• Appendix 1: Root, root elements, fundamental translations
• Appendix 2: Maximal tori, Weyl group
• Appendix 3: Heighest weights and irreps

Exercises

• Assignment 7 -- Solutions
• Assignment 6 -- Solutions
• Assignment 5 -- Solutions
• Assignment 4 -- Solutions
• Assignment 3 -- Solutions
• Assignment 2 -- Solutions
• Assignment 1 -- Solutions

Other comments

You may be interested in the course Analysis on Groups, taught concurrently by Dr. Dominik Juestel. It will have some overlap with this course. See here: http://www-m7.ma.tum.de/bin/view/Analysis/AoG15#WikiVorlesung