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...

The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type An, by counting the mutation class of any quiver with underlying graph An. It will also follow that if T and T ′ are cluster-tilting objects in a cluster category C, then EndC(T ) is isomorphic to EndC(T ) if and only if T = τ T . 1. Cluster-tilted algebras The cluster categ...

We describe a natural $q$-deformation of Fock and Goncharov's canonical basis for the algebra regular functions on cluster variety associated to quiver type $A$. then an extension this construction involving called symplectic double.

Cluster algebras are a remarkable discovery of S. Fomin and A. Zelevinsky [FZI]. Cluster algebras are certain commutative algebras defined by a very simple and general data. Below we consider only cluster algebras of geometric origin, which are quite general and probably the most important examples of cluster algebras. We show that a cluster algebra of geometric origin is part of a richer struc...

Given a maximal rigid object T of the cluster tube, we determine the objects finitely presented by T . We then use the method of Keller and Reiten to show that the endomorphism algebra of T is Gorenstein and of finite representation type, as first shown by Vatne. This algebra turns out to be the Jacobian algebra of a certain quiver with potential, when the characteristic of the base field is no...

Let Λ be a preprojective algebra of Dynkin type, and let G be the corresponding semisim-ple simply connected complex algebraic group. We study rigid modules in subcategories Sub Q for Q an injective Λ-module, and we introduce a mutation operation between complete rigid modules in Sub Q. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

Let Λ be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories Sub Q for Q an injective Λ-module, and we introduce a mutation operation between complete rigid modules in Sub Q. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

We investigate the existence and non-existence of maximal green sequences for quivers arising from weighted projective lines. Let Q be Gabriel quiver endomorphism algebra a basic cluster-tilting object in cluster category \(\mathcal {C}_{\mathbb {X}}\) line \(\mathbb {X}\). It is proved that there exists \(Q^{\prime }\) mutation equivalence class Mut(Q) such admits sequence. Furthermore, which ...

We prove the existence of an m-cluster tilting object in a generalized m-cluster category which is (m+1)-Calabi–Yau andHom-finite, arising froman (m+2)-Calabi–Yau dg algebra. This is a generalization of the result for them = 1 case in Amiot’s Ph.D. thesis. Our results apply in particular to higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with s...