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# Representations of Compact Groups

## Module MA5054

This Module is offered by TUM Department of Mathematics.

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 2019SS 2018SS 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.

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

### Courses, Learning and Teaching Methods and Literature

#### 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.

#### 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.

#### Exam Repetition

The exam may be repeated at the end of the semester.

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