Abstract: The usual setting for questions in representation theory are abelian categories (of modules or sheaves), but In recent years a lot of attention has been shifted to triangulated categories (as they arise as stable module categories for group algebras, or as the derived categories of an abelian category). Under suitable finiteness conditions such triangulated categories are Krull-Remak-Schmidt categories with Auslander-Reiten triangles, and one may look at the shape of the corresponding Auslander-Reiten components. Of special interest seem to be triangulated categories which are tubular (so that all AR-components are tubes), since they occur quite frequently on the border line between finite (or discrete) type and wild type.
The lecture will focus the attention to a typical class of examples, namely the category of modules over the preprojective algebras of type An which are cogenerated by a fixed indecomposable injective module. According to Geiss-Leclerc-Schroeer, these categories play an important role when studying the cluster structure on the coordinate ring of some Grassmannians. Note that here we deal with what is called an ADE-chain. For n = 8 one obtains a tubular triangulated category and we will outline in which way a classification of all the indecomposables can be achieved.
(This is a report on joint investigations with B. Leclerc and M. Schmidmeier.)