This course starts with a general introduction to Lie algebras with several examples from classical matrix Lie algebras. Next we discuss the classification in nilpotent, solvable, and semisimple Lie algebras. The main part of the course consists of a detailed study of semisimple Lie algebras and their representations. These algebras appear in several applications in atomic, nuclear, and particle physics. Besides, they have a central role in many branches of pure mathematics, in harmonic analysis, differential geometry, algebraic geometry, integrable systems and (symmetries of) differential equations.
We also discuss infinite-dimensional generalizations, including affine Kac-Moody algebras which play an important role in quantum field theory and string theory. Finally, we study quantum groups as deformations of semisimple Lie algebras. These are an important tool in the theory of quantum integrable systems and they also lead to interesting examples in noncommutative geometry.
After completed course, the PhD student should be able to:
- understand the relation between Lie algebras and Lie groups appearing as symmetries in physics models.
- be familiar with the structure theory of semisimple Lie algebras in terms of root diagrams and be able to derive the basic properties of Lie algebras from the root structure.
- understand the classification of representations of simple Lie algebras and be able to construct some of the standard representations, especially those which one meets in nuclear and particle physics.
- use associative algebra methods (universal enveloping algebra) for constructing representations of Lie algebras.
- understand the basic structure theory of infinite-dimensional affine algebras appearing in quantum field theory, and be able to construct some of their representations.
- understand the structure of quantum groups (Hopf algebras) which are deformations of classical Lie algebras and are relevant in the theory of integrable systems in physics.
