Lie groups are essentially matrix groups such as group of orthogonal or unitary resp sympleczic matrices. Compact Lie groups can be classified and the theory of group representations can be developed. With this abstract aproach the characteristic properties of these groups evolve to a common theory of the structure of these groups. The aim of this lecture ist to develope the fundamental concepts of this theory.

Differential manifolds, Liegroups and Liealgebras, Haar measure, Theorem of Peter-Weyl, Structure theorems, Main Theorem of maximal tori, Weyl's integral formula, root systems, Weyl's character formula, classification of simple Lie groups.