Categorical construction of A,D,E root systems

Abstract

Let \Gamma be a Dynkin diagram of type A,D,E and let R denote the corresponding root system. In this paper we give a categorical construction of R from \Gamma. Instead of choosing an orientation of \Gamma and studying representations of the associated quiver, we study representations of a canonical quiver \Gammahat associated to \Gamma . This construction is very closely related to the preprojective algebra of \Gamma . In particular, the construction gives a certain periodicity result about the preprojective algebra.