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Lie Groups and Lie Algebras
(dt. Lie-Gruppen und Lie-Algebren)

Level, degree of commitment Advanced module, compulsory elective module
Forms of teaching and learning,
workload		 Lecture (4 SWS), recitation class (2 SWS),
270 hours (90 h attendance, 180 h private study)
Credit points,
formal requirements		 9 CP
Course requirement(s): Successful completion of at least 50 percent of the points from the weekly exercises.
Examination type: Written or oral examination (individual examination)
Language,
Grading		 German,
The grading is done with 0 to 15 points according to the examination regulations for the degree program B.Sc. Mathematics.
Duration,
frequency		 One semester,
Regularly alternating with the other advanced modules
Person in charge of the module's outline Prof. Dr. Ilka Agricola, Prof. Dr. István Heckenberger, Prof. Dr. Pablo Ramacher

Contents

  • Basic concepts about Lie groups and Lie algebras: Relationship between Lie groups and Lie algebras, exponential function, rough classification of Lie algebras, fundamental theorems (Engel, Lie, Cartan...).
  • Structural theory of simple Lie algebras: Cartan subalgebras, roots
  • Representation theory: fundamentals of finite-dimensional theory, highest weights.

Qualification Goals

The students

  • have become familiar with the algebraization of a fundamental concept of symmetry,
  • understand the interaction of geometric and algebraic methods,
  • have practiced mathematical ways of working (developing mathematical intuition and its formal justification, training the ability to abstract, reasoning),
  • have improved their oral communication skills in exercises by practicing free speech in front of an audience and in discussion.

Prerequisites

None. The competences taught in the following modules are recommended: either Foundations of Mathematics and Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis. Basic knowledge of algebra is advantageous, but not mandatory.


Applicability

Module imported from B.Sc. Mathematics.

It can be attended at FB12 in study program(s)

  • B.Sc. Mathematics
  • M.Sc. Computer Science
  • M.Sc. Mathematics
  • LAaG Mathematics
  • BA Minor Mathematics

When studying BA Minor Mathematics, this module can be attended in the study area Compulsory Elective Modules.

The module is assigned to variant A of the BSc minor program in Mathematics.


Recommended Reading

  • Fulton-Harris, Introduction to representation theory, Springer
  • Bröcker- tom Dieck, Representations of Compact Lie Groups, Springer
  • Goodman-Wallach, Representations and invariants of the classical groups, Cambridge University Press
  • Humphreys, J., Introduction to Lie Algebras and Representation Theory, Springer, 1972



Please note:

This page describes a module according to the latest valid module guide in Winter semester 2023/24. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:

  • Winter 2016/17 (no corresponding element)
  • Summer 2018 (no corresponding element)
  • Winter 2018/19 (no corresponding element)
  • Winter 2019/20 (no corresponding element)
  • Winter 2020/21 (no corresponding element)
  • Summer 2021 (no corresponding element)
  • Winter 2021/22 (no corresponding element)
  • Winter 2022/23
  • Winter 2023/24

The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.

The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.