Lectures: Monday 15:00-16:30, Thursday 15:00-16:30,
The lecture gives an introduction to the structure theory and representation theory of Lie algebras. The main focus here lies on the classification of finite-dimensional complex semisimple Lie algebras and their representations. The aim of this lecture is to provide the basic theory and knowledge on Lie algebras and representation theory, as it is necessary for further directions of Differential Geometry, Number Theory and many other areas. After introducing basic notions of Lie algebra theory we discuss the theorems of Engel and Lie, the Jordan-Chevalley decomposition, the Cartan criteria, Weyl's theorem, the theorems of Levi and Malcev, the classification of complex semisimple Lie algebras and Serre's theorem. In the chapter on representations of semisimple Lie algebras we present the classification by highest weight, introducing also the universal enveloping algebra. We give several applications such as Weyl's character formula and Weyl's dimension formula.
Monday, 28th of January 2021, 15:00 - 16:15 Digital.