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Theory and Applications of Simple Lie-Algebras

Module PH2136

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 WS 2021/2 (current)

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
WS 2021/2SS 2018WS 2011/2

Basic Information

PH2136 is a semester module in German or English language at Master’s level which is offered irregular.

This Module is included in the following catalogues within the study programs in physics.

  • Specific catalogue of special courses for nuclear, particle, and astrophysics
  • Complementary catalogue of special courses for condensed matter physics
  • Complementary catalogue of special courses for Biophysics
  • Complementary catalogue of special courses for Applied and Engineering Physics

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

Total workloadContact hoursCredits (ECTS)
150 h 30 h 5 CP

Responsible coordinator of the module PH2136 is Norbert Kaiser.

Content, Learning Outcome and Preconditions


definition and basic notions for Lie algebras, low-dimensional examples, representations of sl(2,C), classical Lie algebra, root space decomposition of semi-simple Lie algebras, root systems and their complete classification through Dynkin diagrams, exceptional Lie algebras, representations and fundamental dominant weights, real and coplex Clifford algebras and their complete classification  

Learning Outcome

The student knows the basic notions of Lie algebras and the differences between real and complex Lie algebras.The The student knows the basic that semi-simple Lie algebras possess a root space decomposition and that the corresponding root systems can be completely classified.

The student knows that representations of semi-simple Lie algebras are determined by highest weights and that these a integral linear combinations of fundamental dominant weights.

The student knows the generalization of the Dirac algebra in form of the complete classification of the real and complex Clifford algebras.


linear algebra

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

VO 2 Theory and applications of simple Lie-algebras Kaiser, N. Fri, 16:00–18:00, PH HS3
and singular or moved dates

Learning and Teaching Methods

Starting from the axioms of a Lie algebra, we study first low-dimensional examples. The in physics most important Lie algebras of the orthogonal und unitary groups, and of the Lorentz group are treated in detail. A major part of the lecture concentrates on the classification of complex semi-simple Lie algebra by root systems and Dynkin diagrams. The final part deals with the description of the representations of semi-simple Lie algebras by weight diagrams ad hihest weight vectors.


Presentation on the black board, lecture notes will be handed out.


*Introduction to Lie algebras, K. Erdmann and M. J. Wildom, Springer *Lie algebras in particle physics, H. Georgi, *Lie-Gruppen und Lie-Algebren in der Physik, M. Boehm, Springer

Module Exam

Description of exams and course work

There will be an oral exam of 25 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using comprehension questions and sample calculations.

For example an assignment in the exam might be:

  • Which isomorphisms exist between low-dimensional Lie algebras?
  • How can one classify simple Lie algebras by Dynkin diagrams?

Exam Repetition

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

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