Koszulity and the Hilbert series of preprojective algebras

Abstract

The goal of this paper is to prove that if Q is a connected non-Dynkin quiver then the preprojective algebra of Q over any field k is Koszul, and has Hilbert series 1/(1-Ct+t^2), where C is the adjacency matrix of the double of Q. (This result, in somewhat less general formulations, was previously obtained by Martinez-Villa and Malkin-Ostrik-Vybornov). We also prove a similar result for the partial preprojective algebra of any connected quiver Q, associated to a subset J of the set I of vertices of Q (by definition, this is the quotient of the path algebra of the double by the preprojective algebra relations imposed only at vertices not contained in J). Namely, we show that if J is not empty then this algebra is Koszul, and its Hilbert series is 1/(1-Ct+D_Jt^2), where D_J is the diagonal matrix with (D_J)_{ii}=0 if i is in J and (D_J)_{ii}=1 otherwise. Finally, we show that both results are valid in a slightly more general framework of modified preprojective algebras.