Categories and functors.Free, projective, injective, flat modules. Snake Lemma.Chain complexes, (co-)homology, homotopy, projective resolutions.Derived functors; Ext-groups.Blocks of an algebra.Representations of finite groups in positive characteristic.Triangulated categories.

Semisimple and nilpotent adjoint orbits. Dynkin-Kostant classification of nilpotent orbits with the theorem of Jacobson-Mozorov. Classical cases (partitions). Weighted Dynkin diagrams. The exceptional Lie algebras.

Definition of Lie algebras, Properties. Theorems of Engel, Lie. Weyls Theorem. Representation theory of sl_2. Root systems, root space decomposition. Classification of Lie simple algebras. Dynkin diagrams. PBW theorem. Generators and Relations. Representation theory of Lie algebras. Weights. Highest weight modules: Existence and Uniqueness. Examples (sl_3).

Quivers, path algebras, representations of quivers. Indecomposable modules. Reflection functors, Coxeter functor. (Extended) Dynkin diagrams. Gabriel's theorem on finite representation type.