Homological algebra and friezes

Abstract

Over the last decade frieze patterns, as introduced by Conway and Coxeter in the 1970's, have been generalised in many ways. One such exciting development is a homological interpretation of frieze patterns, which we call friezes. A frieze in the modern sense is a map from a triangulated category C to some ring. A frieze X is characterised by the propety that if x ! y ! x is an Auslander-Reiten triangle in C, then X( x)X(x)􀀀X(y) = 1. A canonical example of a frieze is the Caldero-Chapoton map. The more general notion of a generalised frieze was introduced by Holm and J rgensen in [25] and [26]. A generalised frieze X0 carries the more general property that X0( x)X0(x)􀀀 X0(y) 2 f0; 1g. In [25] and [26] Holm and J rgensen also introduced a modi ed Caldero- Chapoton map, which satis es the properties of a generalised frieze. This thesis consists of six chapters. The rst chapter provides a detailed outline of the thesis, whilst setting some of the main results in context and explaining their signi cance. The second chapter provides a necessary background to the notions used throughout the remaining four chapters. We introduce triangulated categories, the derived category, quivers and path algebras, Auslander-Reiten theory and cluster categories, including the polygonal models associated to the cluster categories of Dynkin types An and Dn. The third chapter is based around the proof of a multiplication formula for the modi ed Caldero-Chapoton map, which signi cantly simpli es its computation in practice. We de ne Condition F for two maps and , and show that when our category is 2-Calabi- Yau, Condition F implies that the modi ed Caldero-Chapoton map is a generalised frieze. We then use this to prove our multiplication formula. The de nition of the modi ed Caldero-Chapoton map requires a rigid subcategory R that sits inside a cluster tilting subcategory T. Chapter 4 proves several results showing that in the case of the cluster category of Dynkin type An, the modi ed Caldero-Chapoton map depends only on the rigid subcategory R. These results then allow us to prove a general formula for the group Ksplit 0 (T)=N, which is used in the de nition of the modi ed Caldero-Chapoton map. Chapter 5 provides a comprehensive list of exchange triangles in the cluster category of Dynkin type Dn. Chapter 6 then proves several similar results to Chapter 4 in the case of the cluster category of Dynkin type Dn. We prove that the modi ed Caldero-Chapoton map depends only on the rigid subcategory R before again producing a general formula for Ksplit 0 (T)=N