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Title: Cluster combinatorics and derived equivalences for m-cluster tilted algebras
Author: Murphy, Graham James
ISNI:       0000 0001 3433 5405
Awarding Body: University of Leeds
Current Institution: University of Leeds
Date of Award: 2008
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Using the polygonal models for the m-cluster complexes developed in [25] we classify maximal m-orthogonal subsets, defined in [38], of the translation quiver Z~ where t1 is a Dynkin diagram of type Am En/en or Dn by describing explicitly a bijective correspondence between the maximal facets of the m-cluster complexes and the maximal m-orthogonal subsets of Z~. This generalizes results appearing in [38]. We then apply our result to classify the blocks of group algebras whose defect group is not quatemion and for which a maximal m-orthogonal module exists. We also describe the m-cluster tilted algebras of type An using the facets of the m-cluster complex. We introduce tilting complexes which correspond in a natural way to the algebra mutations induced by the exchange relation between facets of the m-cluster complex of type An- We prove that known necessary conditions on the Cartan matrix for the derived equivalence of two m-cluster tilted algebras of type An [17], are in fact sufficient, thereby classifying up to derived equivalence the m-cluster tilted algebras of type An. To achieve this we provide an algorithm which uses the tilting complexes associated with the algebra mutations in type An in order to reduce connected components of m-cluster tilted algebras of type An with the same number of simple modules and cycles in their quivers to a normal form. This generalizes a result of Buan and Vatne [22] which classifies the I-cluster tilted algebras of type An up to derived equivalence.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available