## Lie Algebras and Groups

USOSLie 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.

A simple example is the visualisation of the root system of $E_6$ 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 $E_6$ 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.

- Introduction and motivationLecture03.10.2023 10:15, motivation SO(3)
- Some mathematical preliminariesLecture10.10.2023 10:15
- Classical matrix groupsLecture17.10.2023 10:15
- Cartan subalgebraLecture24.10.2023 10:15
- Root systemLecture31.10.2023 11:15, example sl(3,C)
- Simple root and Cartan matrixLecture07.11.2023 11:15, E6 root system
- Classification and Dynkin diagramsLecture14.11.2023 11:15
- Irreducible representationsLecture21.11.2023 11:15
- $\mathfrak{su}$($N$) representations and Young tableauxLecture28.11.2023 11:15
- Highest weight representationsLecture05.12.2023 11:15
- Characters and Weyl groupLecture12.12.2023 11:15
- Decomposition of tensor products and regular subalgebrasLecture19.12.2023 11:15
- Special subalgebras and branching rulesLecture09.01.2024 11:15
- Particle theory and the standard modelLecture16.01.2024 11:15
- The Georgi–Glashow modelLecture23.01.2024 11:15