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.
The course is recommended to students (in F4) specializing either in mathematical physics or in mathematics, and also to interested PhD students. The course is given in English.