Course description
Symmetries play a crucial role in modern theoretical physics, from the Lorentz symmetry of any relativistic system to internal symmetries and gauge invariance. The structure of these symmetries is encoded in the Lie Algebras that generate the symmetry transformations. We have seen them in action in theories with global symmetries such as the O(N) vector model and in Yang-Mills theories. To treat integrable models, which have infinitely many symmetry generators, Lie Algebras are however not enough and it becomes necessary to study their extensions.
This course builds on the basic concepts of Lie algebras as treated in the course Advanced Concepts in Theoretical Physics and develops all the necessary knowledge to get to the Cartan classification of complex semi-simple Lie algebras. We will meet concepts such as the Cartan subalgebra, root diagrams and Dynkin diagrams. The generalization to affine Lie algebras is treated next. Advanced topics include superalgebras, quantum groups and Yangians. We also discuss the physical context in which these mathematical structures play a role.
