Koszulity and the Hilbert Series of Preprojective Algebras

The goal of this paper is to prove that if Q is a connected non-Dynkin quiver then the preprojective algebra ΠQ(k) of Q over any field k is Koszul, and has Hilbert series (1 − Ct+ t2)−1, where C is the adjacency matrix of the double Q̄ of Q. We also prove a similar result for the partial preprojective algebra ΠQ,J(k) of any connected quiver Q, where J ⊂ I is a subset of the set I of vertices of Q. By definition, ΠQ,J(k) is the quotient of the path algebra of kQ̄ by the preprojective algebra relations imposed only at vertices not contained in J . We show that if J 6= ∅ then ΠQ,J(k) is Koszul, and its Hilbert series is (1 − Ct + DJ t 2)−1, where DJ is the diagonal matrix with (DJ )ii = 0 if i ∈ J and (DJ )ii = 1, i / ∈ J . Moreover, we show that both results are valid in a slightly more general framework of modified preprojective algebras, considered in [K]. We note that the first result is known in most cases [MV, MOV, O]. In particular, it is known in general in characteristic zero ([MOV]), and in most positive characteristic cases [MV, O]. Our argument, however, is elementary, and different from the arguments of [MOV, O], which are based on the theory of tensor categories. Acknowledgments. P.E. is grateful to V. Ostrik for a useful discussion. The work of P.E. was partially supported by the NSF grant DMS-0504847 and by the CRDF grant RM1-2545-MO-03.