The seminar will give an introduction to homological methods in commutative algebra. The study of local intersection multiplicities will be a central application of the theory.
Lie algebras are ubiquitous in algebra, but also in many other areas of mathe- matic. They usually arise as linearization (i.e. as the tangent space) of Lie groups or algebraic groups, but can also be studied as objects of intrinsic interest. By def- inition they are given by a vector space g over a field k together with a Lie bracket [−, −] : g×g → g that behaves like the commutator XY −Y X on gln = Matn×n(k). In the first part of the seminar we will study Lie algebras and in particular semi- simple Lie algebras. It turns out that these objects admit a beautiful classification in terms of root systems.
In the second part of the seminar we will study representations of semi-simple Lie algebras. The category of finite dimensional representations turns out to be semi- simple and there is a complete classification of the irreducible representations. The last three talks will present an introduction to further and more advanced topics about certain infinite dimensional representations.
- Lehrende/r: Eugen Hellmann