Koszulity and the Hilbert series of preprojective algebras

TL;DR

This paper proves that preprojective algebras of connected non-Dynkin quivers are Koszul and computes their Hilbert series, extending known results to partial and modified preprojective algebras.

Contribution

It establishes the Koszulity and Hilbert series formulas for preprojective and partial preprojective algebras of connected non-Dynkin quivers, generalizing previous findings.

Findings

Preprojective algebras of non-Dynkin quivers are Koszul.

Hilbert series of these algebras are explicitly computed.

Results extend to partial and modified 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.