Lie-Gruppen, Lie-Algebren und ihre Darstellungen/Lie groups, Lie algebras and their representations

The seminar assumes that participants are already familiar with the basic concepts of Lie groups and Lie algebras. After a small review of these basic concepts, we discuss basic structure theorems, e.g. of nilpotent and solvable Lie groups and algebras, and its relation to the Killing form (Cartan's second criterion). We then turn to semi-simple Lie algbras, its relations to compact Lie groups, the role of maximal tori, root systems and Weyl groups, and the classification of semi-simple Lie algebras. We want to emphasize the geometric understanding of these theorems, e.g. to related the Dynkin diagram to the classical groups. Then we discuss the representation theory of semi-simple Lie algebras, in particular how to construct representations of highest weight and how they classify the representations. We also discuss symmetric spaces. Emphasis is also put on the exceptional group G₂ which appears e.g. as the automorphism group of the octonions.