Representations of Compact Groups

What it is about

You will learn the representation theory of compact groups. The module conveys the relationships between Lie groups and their Lie algebras, central results such as the Peter–Weyl theorem, the structure of maximal tori, roots, weights and the Weyl character formula as well as representations of the classical groups. In the end you will be able to construct, decompose and analyze representations from algebraic, differential-geometric and analytic perspectives.

What you will be able to do

• Understand the relation between Lie groups and Lie algebras
• Apply the Peter–Weyl theorem
• Analyze maximal tori, roots and weights
• Utilize the Weyl group and Weyl character formula
• Construct representations of the classical groups and decompose them into irreducible components
• Combine methods from algebra, differential geometry and analysis

What the module consists of

• LectureConveying concepts, results and proof methods with examples
• ExerciseDeepening the lecture material, accompanying exercises and solutions, guided and increasingly independent work

Teaching method

• Lecture (blackboard, discussion)Introduction of terms, theorems and proof methods; motivation for self-study
• Exercise sheet with solutionsDeepening the material, self-check and consolidation through problems
• Guided exercise groupsInitial supervision, later support for independent and group-based problem solving