A GEOMETRIC DESCRIPTION OF THE m - CLUSTER CATEGORIES OF TYPE

We show that the m-cluster category of type Dn is equivalent to a certain geometrically-defined category of arcs in a punctured regular nm − m + 1-gon. This generalises a result of Schiffler for m = 1. We use the notion of the mth power of a translation quiver to realise the m-cluster category in terms of the cluster category. Introduction Let k be a field and Q a quiver of Dynkin type ∆. Let D(kQ) denote the bounded derived category of finite dimensional kQ-modules. Let τ denote the Auslander-Reiten translate of D(kQ) and let S denote the shift. For m ∈ N the m-cluster category associated to kQ is the orbit category C ∆ := D(kQ) Smτ−1 . This category was introduced in [Kel] and has been studied by the authors [BaM], Thomas [Tho], Wralsen [Wra] and Zhu [Zhu]. It is known that C ∆ is triangulated [Kel], Krull-Schmidt and has almost split triangles [BMRRT, 1.2,1.3]. The m-cluster category is a generalisation of the cluster category. The cluster category was introduced in [CCS1] (for type A) and [BMRRT] (general hereditary case), and can be regarded as the case m = 1 of the m-cluster category. Keller has shown that the m-cluster category is Calabi-Yau of dimension m+1 [Kel]. We remark that such Calabi-Yau categories have also been studied in [KR]. One of the aims of the definition of the cluster category was to model the Fomin-Zelevinsky cluster algebra [FZ] representation-theoretically. We show that C Dn can be realised geometrically in terms of a category of arcs in a punctured polygon with nm − m + 1 vertices. This generalises a result of Schiffler [Sch], who considered the case m = 1. We remark that the punctured polygon model for the cluster algebra of typeDn appears in work of Fomin, Schapiro and Thurston [FST] as part of a more general set-up, building on [FG1, FG2, GSV1, GSV2] which consider links between cluster algebras and Teichmüller theory. Also, such a geometric realisation of a cluster category first appeared (with a construction for type An in the case m = 1) in [CCS1]. Our approach is based on the idea of the mth power of a translation quiver introduced in [BaM]. We show that, with a slight modification of the definition for m = 2, the Auslander-Reiten quiver of C Dn can be realised as a connected component of the mth power of the Auslander-Reiten quiver of C Dnm−m+1. In 2000 Mathematics Subject Classification. Primary: 16G20, 16G70, 18E30 Secondary: 05E15, 17B37.