Preprojective algebras and fractional Calabi-Yau algebras
Given a quiver we consider two algebras: its path algebra and its preprojective algebra. If the quiver is Dynkin (ADE) then both have nice properties: the path algebra is fractionally Calabi-Yau and the preprojective algebra has a Nakayama automorphism of finite order. I will explain what these words mean and how these properties are related, using 2-dimensional category theory. This gives a useful criterion to check if a d-representation finite algebra is fractionally Calabi-Yau.