Please use this identifier to cite or link to this item: http://theses.ncl.ac.uk/jspui/handle/10443/4257
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dc.contributor.authorPescod, David-
dc.date.accessioned2019-04-10T14:31:22Z-
dc.date.available2019-04-10T14:31:22Z-
dc.date.issued2018-
dc.identifier.urihttp://hdl.handle.net/10443/4257-
dc.descriptionPhD Thesisen_US
dc.description.abstractOver 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)=Nen_US
dc.language.isoenen_US
dc.publisherNewcastle Universityen_US
dc.titleHomological algebra and friezesen_US
dc.typeThesisen_US
Appears in Collections:School of Mathematics and Statistics

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