Groups and Lie Algebras

The aim of this module is to provide mathematical methods needed to describe symmetries in physics. For illustration, we will also discuss physical applications. The module starts with a hands-on discussion of finite groups and their representations. We introduce and define basic concepts (groups, representations, reducible/irreducible,…), provide proofs of elementary theorems (Schur’s lemma, Maschke’s theorem) and introduce techniques in representation theory (in particular, how to decompose a representation into irreducibles). In the second part, we discuss Lie groups and their representations, focussing on Matrix Lie groups, and state theorems such as the Peter-Weyl theorem and Schur-Weyl duality. We emphasize similarities between compact Lie groups and finite groups. The third part of the course discusses Lie algebras, focussing mainly on semi-simple complex Lie algebras. We discuss and explain their structure, partially include proofs, ending up with the classification theorem. Basics of the representation theory will also be covered.

Physics applications covered in the module include selection rules for molecules, crystal field splitting, as well as the symmetry of hadrons.

After successful completion of the module the students are able to:

1. State basic mathematical definitions: group, algebra, Lie algebra, representation.

2. Understand the natural structure preserving maps between the concepts mentioned in 1.

3. Decompose representations of finite groups into irreducible ones using the character table

4. Associate a Dynkin diagram to a complex semi-simple Lie algebra

5. Read off a complex semi-simple Lie algebra from a Dynkin diagram

6. Understand the meaning and importance of theorems on compact Lie groups (Peter-Weyl-theorem, Schur-Weyl duality)

7. Apply group theory to quantum mechanical systems

No preconditions in addition to the requirements for the Master’s program in Physics and Quantum Science and Technology

The module consists of lectures (4 SWS) and tutorial classes (2 SWS). The main teaching material will typically be presented on the blackboard. Weekly problem sets are offered to comprehend the lecture content better and improve their familiarity with them. The solutions to the problem sets are discussed in the weekly exercise classes

Blackboard

Standard textbooks on Group theory and representations, e.g.:

• Fulton and Harris: Representation theory, a first course, Springer 1991
• Fuchs and Schweigert: Symmetries, Lie algebras and representations. A graduate course for physicists. Cambridge monographs on mathematical physics 1997

There will be a written exam of 90-120 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using calculation problems and comprehension questions.

For example an assignment in the exam might be:

• Construct a root system from a given Dynkin diagram
• Decompose a given representation of a finite group into irreducibles using characters
• Decompose tensor products of representations GL(N,C) with the help of the symmetric group
• Compute the Killing form of a given Lie algebra

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.