Representations of Compact Groups
Module MA5054
This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.
Module version of SS 2018
There are historic module descriptions of this module. A module description is valid until replaced by a newer one.
Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.
| available module versions | ||
|---|---|---|
| SS 2019 | SS 2018 | SS 2015 |
Basic Information
MA5054 is a semester module in English language at Master’s level which is offered irregular.
This module description is valid from SS 2018 to SS 2018.
| Total workload | Contact hours | Credits (ECTS) |
|---|---|---|
| 180 h | 45 h | 6 CP |
Content, Learning Outcome and Preconditions
Content
Subjects to be covered include: Lie groups and algebras, the exponential map, the 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.
Learning Outcome
After successful completion of this module, students will have a basic understanding of the representation theory of compact groups. Key learning goals include a thorough grasp of the relationship between a Lie group and its Lie algebra, and the ability to explain the resulting consequences for representation theory. Students know how to analyze Lie groups and their representations from algebraic, differential geometric and analytical viewpoints. This involves, in particular, the application of methods for constructing and decomposing representations.
Preconditions
MA1001 - Analysis 1
MA1002 - Analysis 2
MA1101 - Linear Algebra and Discrete Structures1
MA1002 - Linear Algebra and Discrete Structures 2
MA1002 - Analysis 2
MA1101 - Linear Algebra and Discrete Structures1
MA1002 - Linear Algebra and Discrete Structures 2
Courses, Learning and Teaching Methods and Literature
Courses and Schedule
| Type | SWS | Title | Lecturer(s) | Dates | Links |
|---|---|---|---|---|---|
| VO | 2 | Representations of Compact Groups | König, R. | ||
| UE | 1 | Representations of Compact Groups (Exercise Session) | König, R. |
Learning and Teaching Methods
Lectures, Exercises, Homework
This module consists of lectures together with accompanying practice sessions. The lectures will convey relevant concepts, mathematical results and proof methods along with concrete examples by means of board work and through discussion with the students. This is intended to motivate students to study the literature and approach problems independently. Practice sessions will be offered which complement the theoretical material developed in the lectures. Exercise sheets and solutions will be provided, which will help students to deepen their understanding, and independently assess their progress. The practice sessions will be guided initially, but will proceed throughout the term to more independent work by students individually as well as in small groups.
This module consists of lectures together with accompanying practice sessions. The lectures will convey relevant concepts, mathematical results and proof methods along with concrete examples by means of board work and through discussion with the students. This is intended to motivate students to study the literature and approach problems independently. Practice sessions will be offered which complement the theoretical material developed in the lectures. Exercise sheets and solutions will be provided, which will help students to deepen their understanding, and independently assess their progress. The practice sessions will be guided initially, but will proceed throughout the term to more independent work by students individually as well as in small groups.
Media
Board work, Exercise sheets
Literature
Primarily: Barry Simon, Representations of Finite and Compact Groups, AMS 1996.
Module Exam
Description of exams and course work
The module examination is based on an oral exam (20 minutes).
The exam will assess the ability to give precise definitions, prove selected theorems involving algebraic, differential geometric and analytical arguments, and apply results and techniques to analyze, classify and construct various algebraic objects appearing in the study of Lie groups and algebras. This includes, in particular, the problem of decomposing representations into irreducibles. With suitably chosen examples, the ability to transfer results and techniques to the analysis of related problems involving representation-theoretic concepts will be tested.
The exam will assess the ability to give precise definitions, prove selected theorems involving algebraic, differential geometric and analytical arguments, and apply results and techniques to analyze, classify and construct various algebraic objects appearing in the study of Lie groups and algebras. This includes, in particular, the problem of decomposing representations into irreducibles. With suitably chosen examples, the ability to transfer results and techniques to the analysis of related problems involving representation-theoretic concepts will be tested.
Exam Repetition
The exam may be repeated at the end of the semester.