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...

Y. Palu has generalized the cluster multiplication formulas to 2Calabi-Yau categories with cluster tilting objects ([Pa2]). The aim of this note is to construct a variant of Y. Palu’s formula and deduce a new version of the cluster multiplication formula ([XX]) for acyclic quivers in the context of cluster categories. Introduction Cluster algebras were introduced by S. Fomin and A. Zelevinsky [...

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...

In this article, we study the geometric realizations of m-cluster categories Dynkin types A, D, A˜ and D˜. We show, in those four cases, that there is a bijection between (m+2)-angulations isoclasses basic tilting objects. Under these bijections, flips correspond to mutations Our strategy consists showing certain Iyama-Yoshino reductions under consideration can be described terms cutting along ...

We study monoidal categorifications of certain subcategories C J $\mathcal {C}_J$ finite-dimensional modules over quantum affine algebras, whose cluster algebra structures on their Grothendieck rings K ( ) $K(\mathcal {C}_J)$ are closely related to the category quiver Hecke type A ∞ $A_\infty$ via generalized Schur–Weyl duality functors. In particular, when is $A$ or B $B$ , subcategory coincid...