Lie Algebras and Groups


Lie algebras describe infinitesimal symmetries of physical systems. Therefore, they and their representation theory are extensively used in physics, most notably in quantum mechanics and particle physics. This course introduces semi-simple Lie algebras and the associated Lie groups for physicists. We discuss the essential tools, like the root and weight system, to efficiently work with them and their representations. As on explicit application of the mathematical framework, we discuss Grand Unified Theories (GUT). Moreover, we show how modern computer algebra tools like LieART can significantly help in all explicit computations throughout the course.

E6 root system A simple example is the visualisation of the root system of E6E_6 projected on the Coxeter plane, which you can see here. If you want to understand how it is created and connected to particle physics, you should take this course. Basic knowledge of core concepts in linear algebra, like vector spaces, eigenvalues and eigenvectors, is assumed. Some good books about the topic are:

  • Fuchs and Schweigert: Symmetries, Lie Algebras and Representations
  • Gilmore: Lie Groups, Lie Algebras, and Some of Their Applications
  • Fulton and Harris: Representation Theory
  • Georgi: Lie Algebras In Particle Physics: from Isospin To Unified Theories

The article Phys. Rep. 79 (1981) 1 by Slansky and the manual of the LieART package are good references, too.

Note: At multiple occasions, we will use Mathematica and it might be a good idea to set it up on your computer. Following the instructions on the main teaching website, you should be eventually able to run the notebook, which generates the projection of the E6E_6 root system above.

Exam: At the end of the semester, we will have a written exam. To better prepare for it, you will be provided a practice exam in the week before the last lecture. More details, will be announced here.

  1. Introduction and motivation
    Lecture03.10.2023 10:15, motivation SO(3)
  2. Some mathematical preliminaries
    Lecture10.10.2023 10:15
  3. Classical matrix groups
    Lecture17.10.2023 10:15
  4. Cartan subalgebra
    Lecture24.10.2023 10:15
  5. Root system
    Lecture31.10.2023 11:15, example sl(3,C)
  6. Simple root and Cartan matrix
    Lecture07.11.2023 11:15, E6 root system
  7. Classification and Dynkin diagrams
    Lecture14.11.2023 11:15
  8. Irreducible representations
    Lecture21.11.2023 11:15
  9. su\mathfrak{su}(NN) representations and Young tableaux
    Lecture28.11.2023 11:15
  10. Highest weight representations
    Lecture05.12.2023 11:15
  11. Characters and Weyl group
    Lecture12.12.2023 11:15
  12. Decomposition of tensor products and regular subalgebras
    Lecture19.12.2023 11:15
  13. Special subalgebras and branching rules
    Lecture09.01.2024 11:15
  14. Particle theory and the standard model
    Lecture16.01.2024 11:15
  15. The Georgi–Glashow model
    Lecture23.01.2024 11:15
Based on course 7, last update on September 22nd 2023, 07:00:50 | Build 277 on September 24th 2023, 14:45:50 | Times and dates shown in UTC+00:00 | Contact