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Title: Homological algebra and friezes
Author: Pescod, David
ISNI:       0000 0004 7961 0169
Awarding Body: Newcastle University
Current Institution: University of Newcastle upon Tyne
Date of Award: 2018
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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 τx→y→x is an Auslander-Reiten triangle in C, then X(τx)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 Jørgensen in [25] and [26]. A generalised frieze X' carries the more general property that X'(τx)X'(x)-X'(y)Ε{0,1}. In [25] and [26] Holm and Jørgensen also introduced a modified Caldero-Chapoton map, which satisfies the properties of a generalised frieze. This thesis consists of six chapters. The first chapter provides a detailed outline of the thesis, whilst setting some of the main results in context and explaining their significance. 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 modified Caldero-Chapoton map, which significantly simplifies its computation in practice. We define Condition F for two maps α and β, and show that when our category is 2-Calabi- Yau, Condition F implies that the modified Caldero-Chapoton map is a generalised frieze. We then use this to prove our multiplication formula. The definition of the modified 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 modified Caldero-Chapoton map depends only on the rigid subcategory R. These results then allow us to prove a general formula for the group K₀split(T)/N, which is used in the definition of the modified 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 modified Caldero-Chapoton map depends only on the rigid subcategory R before again producing a general formula for K₀split(T)/N.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available