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Title: On a triangulated category which models positive noncrossing partitions
Author: Coelho Guardado Simões, Raquel
Awarding Body: University of Leeds
Current Institution: University of Leeds
Date of Award: 2012
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Let Q be a simply laced Dynkin quiver, Db(Q) the bounded derived category of the path algebra associated to Q and C(Q) the TΣ2-orbit category ofDb( Q), where T is the Auslander- Reiten-translation and Σ is the shift functor. Note that C( Q) is a triangulated category [57]. In this thesis we study two classes of representation-theoretic objects in C ( Q): maximal Hom- free objects, also known as Horn-configurations, and maximal rigid (i.e. Ext-free) objects. Horn-configurations (in the derived category) were used by Riedtrnann [65] in order to classify selfinjective algebras of finite-representation type. Riedtmann proved that these objects are invariant under the autoequivalence TΣ2 of Db(Q). This is the reason why we consider the orbit category C( Q). We establish a bijection between Horn-configurations and noncrossing partitions of the Coxeter group associated to Q which are not contained in any proper standard parabolic subgroup. These noncrossing partitions are said to be positive, because they are proved to be in 1-1 correspondence with the positive clusters in the corresponding cluster algebra (cf. [62]). The bijection between Horn-configurations and positive noncrossing partitions generalizes a result of Riedtmann which states that Horn-configurations in Db (Q), where Q is of type An' are in bijection with classical noncrossing partitions of the set {1, ... , n}. Riedtmann's bijection allows us to construct a geometrical model for C(Q) in type A. Using this geometrical setup, and inspired by the classification of the cluster-tilting objects in the cluster category of type An in terms of triangulations of a polygon, we classify, also in type An, the maximal rigid objects in C(Q) in terms of certain noncrossing bipartite graphs. In addition, we describe the corresponding endomorphism algebras in terms of quivers with relations. We also give a natural notion of mutation of Horn-configurations in C(Q) and we present some partial results and conjectures about the mutation graph and the representation-theoretic description of these mutations
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available