On the global dimension and Koszul property for preprojective algebras

We study preprojective algebras associated to either finite dimensional hereditary algebras, or locally finite hereditary tensor algebras, and in particular show that they have global dimension two in non-Dynkin type. Moreover, starting from a locally finite hereditary tensor algebra of non-Dynkin type, we show that the corresponding preprojective algebra is Koszul, and compute both its Hilbert polynomial and its Koszul dual. We finish by looking at preprojective algebras of Dynkin type, and show how certain properties of the Weyl group arise in the structure of the preprojective algebra.