Repetitive higher cluster categories of type An

Abstract

We show that the repetitive higher cluster category of type An, defined as the orbit category Db(mod kAn)/(τ−1[m])p, is equivalent to a category defined on a subset of diagonals in a regular polygon. This generalizes the construction of Caldero–Chapoton–Schiffler [Quivers with relations arising from clusters (Ancase), Trans. Amer. Math.Soc. 358(3) (2006) 1347–1364], which we recover when p = m =1,and the work of Baur–Marsh, [A geometric description of the m-cluster categories, Trans. Amer. Math.Soc. 360(11) (2008) 5789–5803], treating the case p =1,m>1. Our approach also leads to a geometric model of the bounded derived category in type A.