In this thesis we study partial cluster-tilted algebras. These algebras are opposite endomorphism rings of rigid objects in cluster categories, and they are a generalisation of cluster-tilted algebras. The key motivation for the work we present here is to understand the representation theory of a partial cluster-tilted algebra. In our study of how the Auslander-Reiten theory of a partial cluster-tilted algebra is induced by the Auslander-Reiten theory of the corresponding cluster category, we use twin cotorsion pairs on triangulated categories to extract quasi-abelian categories from cluster categories, and develop Auslander-Reiten theory in quasi-abelian and Krull-Schmidt 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 quasi-abelian category. If C is also Krull-Schmidt and T=U, we show that the heart of the cotorsion pair (S,T) is equivalent to the Gabriel-Zisman 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 quasi-abelian 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 semi-abelian and quasi-abelian categories. In addition, we generalise some Auslander-Reiten theory results of S. Liu for Hom-finite, Krull-Schmidt categories by removing the Hom-finite and indecomposability restrictions. As a main result, we give equivalent characterisations of Auslander-Reiten sequences in a skeletally small, quasi-abelian, Krull-Schmidt category.
Lastly, we construct partial cluster-tilted algebras of arbitrarily large finite global dimension coming from cluster categories associated to Dynkin-type A quivers. In particular, this shows that there is an infinite family of partial cluster-tilted algebras that are not cluster-tilted. Then we consider how the Auslander-Reiten theory of the algebra (End R)^op, where R is a basic rigid object of a Hom-finite, Krull-Schmidt, triangulated k-category C with Serre duality, is induced by the Auslander-Reiten 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).
