In this paper, starting with a simply laced root system, we define a triangulated category which we call the m-cluster category, and we show that it encodes the combinatorics of the m-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of Fomin and Zelevinsky. For Φ any root sys...

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...

This thesis is concerned with the development and application of categorical tools in the study of the cluster algebras of S. Fomin and A. Zelevinsky. C. Amiot’s generalized cluster category is a triangulated category which has been used, in the case where it is Hom-finite, to categorify a certain class of cluster algebras, using cluster characters in the sense of Y. Palu. In this thesis, we ge...

Cyclic posets are generalizations of cyclically ordered sets. In this paper we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The stable category of a Frobenius category is always triangulated and has a cluster structure in many cases. The continuous cluster categories of [14], the infinity-gon of [12], the m-cluster category of type A∞ (m ≥ 3)...

The (usual) Caldero-Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps “reachable” indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalises the idea that the cluster category is a “categorification” of the cluster algebra. The definition of the Caldero-C...

This is a concise introduction to Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category.