This course can be used as a specialized course ("Vertiefungslehrveranstaltung") in the areas "algebra, number theory and discrete mathematics" and "geometry and topology" of the master programs in mathematics.

The concept of symmetries is among the most fundamental ideas of mathematics. Symmetries always form groups, and in many situations these symmetry groups natural carry Lie group structures which brings analytical methods into the game. To any Lie group, one naturally associates a Lie algebra, which is a finite dimensional vector space endowed with a bilinar operation with certain properties. Lie algebras are much simpler objects than Lie groups and can be studied using linear algebra. Still a lot of information about a Lie group can be recovered from its Lie algebra.

In the course, we will discuss the general theory of Lie algebras as well as the basics of structure theory and representation theory of complex semisimple Lie algbras. Depending on the available time, I will also discuss connections to representations of finite groups, in particular of permutation groups. We will not formally need Lie groups and their relations to Lie algebras in the main part of the course. Since these concepts however provide the main motivation for studying Lie algebras, I will briefly and informally discuss them in the beginning of the course.

To follow the course, one mainly needs a good background in linear algebra. To appreciate the connection to Lie groups, basic knowledge of differential geometry will be helpful.

I will provide lecture notes in English for the course, which will be available online via http://www.mat.univie.ac.at/~cap/lectnotes.html. (The version from spring term 2009 which is currently online there, will probably be changed slightly.)