Title:

Partial clustertilted algebras via twin cotorsion pairs, quasiabelian categories and AuslanderReiten theory

In this thesis we study partial clustertilted algebras. These algebras are opposite endomorphism rings of rigid objects in cluster categories, and they are a generalisation of clustertilted algebras. The key motivation for the work we present here is to understand the representation theory of a partial clustertilted algebra. In our study of how the AuslanderReiten theory of a partial clustertilted algebra is induced by the AuslanderReiten theory of the corresponding cluster category, we use twin cotorsion pairs on triangulated categories to extract quasiabelian categories from cluster categories, and develop AuslanderReiten theory in quasiabelian and KrullSchmidt categories. We prove that, under a mild assumption, the heart H of a twin cotorsion pair ((S,T),(U,V)) on a triangulated category C is a quasiabelian category. If C is also KrullSchmidt and T=U, we show that the heart of the cotorsion pair (S,T) is equivalent to the GabrielZisman localisation of H at the class of its regular morphisms. In particular, suppose C is a cluster category with a rigid object R and let X(R) denote the ideal of morphisms factoring through X(R)=Ker(Hom_(R,)). Then applications of our results show that C/X(R) is a quasiabelian category. We also obtain a new proof of an equivalence between the localisation of this category at its class of regular morphisms and a certain subfactor category of C. We generalise some of the theory developed for abelian categories in papers of Auslander and Reiten to semiabelian and quasiabelian categories. In addition, we generalise some AuslanderReiten theory results of S. Liu for Homfinite, KrullSchmidt categories by removing the Homfinite and indecomposability restrictions. As a main result, we give equivalent characterisations of AuslanderReiten sequences in a skeletally small, quasiabelian, KrullSchmidt category. Lastly, we construct partial clustertilted algebras of arbitrarily large finite global dimension coming from cluster categories associated to Dynkintype A quivers. In particular, this shows that there is an infinite family of partial clustertilted algebras that are not clustertilted. Then we consider how the AuslanderReiten theory of the algebra (End R)^op, where R is a basic rigid object of a Homfinite, KrullSchmidt, triangulated kcategory C with Serre duality, is induced by the AuslanderReiten theory of C via the functor Hom(R,). Let C(R) denote the subcategory of C consisting of objects X for which there is a triangle R_{0} > R_{1} > X > R_{0}[1] with R_{i} in add(R). We show that if f: X>Y is an irreducible morphism in C with X in C(R), then Hom(R,f) is irreducible if Y also lies in C(R), or Hom(R,f) is split otherwise. If X does not lie in C(R), we provide partial results dependent on properties of the morphism f+X(R)(X,Y) in the quotient C/X(R).
