Based on paper:
For an element w of the simply-laced Weyl group, Buan-Iyama-Reiten-Scott introduced a subcategory F(w) of a module category over a preprojective algebra. This category plays an important role in the representation theory of algebras (torsion-free classes), as well as in the Lie theory (a categorification of the cluster structure on the coordinate ring of the unipotent cell). In this talk, I will study the structure of F(w) as an exact category. I will explain the classification of simple objects in F(w) in terms of the root system, and give some applications. If time permits, I will discuss how to classify simple objects in torsion-free classes over general finite-dimensional algebras.