The cluster category was introduced in (BMRRT, 2006) and also in (CCS1, 2006) for type A, as a categorical model to understand better the cluster algebras of Fomin and Zelevinsky (FZ, 2002). It is a quotient of the bounded derived category Db(modA) of the finitely generated modules over a finite dimensional hereditary algebra A. It was then natural to consider the endomorphism algebras of tilti...

Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object T in a hereditary abelian category H, we verify that the tilting functor HomH(T,−) induces a triangle equivalence from the cluster category C(H) to the cluster category C(A),...

Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. Some of them are already proved for hereditary abelian categories there. In the present paper, all basic results about tilting theory are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object T in a hereditary abelian category H, we verify that t...

In [3] and [13], the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the properties of 2–Calabi–Yau (Auslander–Reiten formula) and high order associativity. Introduction Cluster algebras were introduced by Fomin and Zelevinsky in [9]. By definition, the cluste...

We give a geometric realization of cluster categories of type Dn using a polygon with n vertices and one puncture is its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices. 0 Introduction Cluster categories were introduced in [BMRRT] and, independently, in [CCS1] for type An, as a means ...

The purpose of this chapter is to give an introduction to the theory of cluster categories and cluster-tilted algebras, with some background on the theory of cluster algebras, which motivated these topics. We will also discuss some of the interplay between cluster algebras on one side and cluster categories/cluster-tilted algebras on the other, as well as feedback from the latter theory to clus...

We associate to any object in the nilpotent module category of an algebra with the 2-Calabi-Yau property a character (in the sense of [11]) and prove a multiplication formula for the characters. This formula extends a multiplication formula for the evaluation forms (in particular, dual semicanonical basis) associated to modules over a preprojective algebra given by Geiss, Leclerc and Schröer [6...

in this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. we focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of fomin and reading, and colored quiver mutation.

Let D be a triangulated category with a cluster tilting subcategory U . The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to support tilting subcategories of D/U , and that, in turn, support tilting subcategories of D/U can be lifted uniquely to weak cluster tilting subcategories of D.