We prove a structure theorem for triangulated Calabi-Yau categories: An algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category iff it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable categor...

We show that an algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category if it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As a first application, we show that the stable category of maximal Cohen-Macaulay modules over a certain isolated singularity of ...

This thesis is concerned with higher cluster tilting objects in generalized higher cluster categories and tropical friezes associated with Dynkin diagrams. The generalized cluster category arising from a suitable 3-Calabi-Yau differential graded algebra was introduced by C. Amiot. It is Hom-finite, 2-Calabi-Yau and admits a canonical cluster-tilting object. In this thesis, we extend these resul...

We show that the m-cluster category of type An−1 is equivalent to a certain geometrically-defined category of diagonals of a regular nm+ 2-gon. This generalises a result of Caldero, Chapoton and Schiffler for m = 1. The approach uses the theory of translation quivers and their corresponding mesh categories. We also introduce the notion of the mth power of a translation quiver and show how it ca...

This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On one hand, the d-cluster tilting subcategories of C have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of C which lack functorial ...

Let $${\cal M}$$ be an n-cluster tilting subcategory of mod-Λ, where Λ is Artin algebra. S}({\cal M})$$ denote the full S}(\Lambda )$$ , submodule category Λ, consisting all monomorphisms in . We construct two functors from to $$\bmod - \underline {\cal M} $$ finitely presented additive contravariant on stable show that these are full, dense and objective hence provide equivalences between quot...

Abstract. We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and provide combinatorial methods for constructing special classes of monomorphisms and epimorphisms. In the case of cluster algebras from surfaces, we de...

We define an analogue of the Caldero-Chapoton map ([CC]) for the cluster category of finite dimensional nilpotent representations over a cyclic quiver. We prove that it is a cluster character (in the sense of [Pa]) and satisfies some inductive formulas for the multiplication between the generalized cluster variables (the images of objects of the cluster category under the map). Moreover, we con...

We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism [T ] : A → B is an equivalence class of rigid objects T in the cluster category of A so that B is the right hom-ext perpendicular category of the underlying object |T | ∈ A. Facto...