2005/12/13 by Pavel Etingof, Etingof, Pavel, Ching-Hwa Eu +1 · 2 citations
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0512287
8 pages, latex
arxiv created 2005/12/13 · openalex publication_date 2005/12/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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+t2), 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+DJt2), where DJ is the diagonal matrix with (DJ)ii=0 if i is in J and (DJ)ii=1 otherwise. Finally, we show that both results are valid in a slightly more general framework of modified preprojective algebras.