Hilbert Series Of Partial Period Preprojective Algebra

Letâ–³be a finite quiver without cycle. This paper introduces the concept of par-tial period preprojective algebra defined overâ–³, which is a algebra defined on stabletranslation quiver Zâ–³/(Ï„^P) of period p, denoted by∏_{(Q(â–³, p)),J}. when p=1, partialperiod preprojective algebra is just partial preprojective algebra, so partial periodpreprojective algebra is a generalization of partial preprojective algebra, partial pre-projective algebra is special case of partial period preprojective algebra. In thispaper we discribe Hilbert series of a partial period preprojective algebra over a starshaped quiver and we obtain a formula by graded algebra free product. Then wediscuss the Hilbert series of partial period preprojective algebra∏_{(Q(â–³,p)),J} decidedby a noempty white vertex set and obtain the formula h_{∏Q(â–³, p),J}(t)=1/(1-Ct+D_Jt²),for it, where C and D_J are pn matrices,A is the adjacent matrix of star shaped quiver, A' is the transpose of A, p is periodof partial period preprojective algebra.