scholarly journals Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object

2015 ◽  
Vol 218 ◽  
pp. 101-124 ◽  
Author(s):  
Thorsten Holm ◽  
Peter Jørgensen
Keyword(s):  
Cluster Algebra ◽  
Cluster Category ◽  
Rigid Object ◽  
Cluster Tilting Object ◽  
Dynkin Type ◽  
Cluster Categories ◽  
Tilting Object ◽  
Definition Of ◽  
Cluster Tilting ◽  
Laurent Polynomial Ring

AbstractThe (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it mapsreachableindecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is acategorificationof the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-calledfriezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we callgeneralized friezesand that, for cluster categories of Dynkin typeA, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

scholarly journals Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object

2015 ◽  
Vol 218 ◽  
pp. 101-124 ◽  
Author(s):  
Thorsten Holm ◽  
Peter Jørgensen
Keyword(s):  
Cluster Algebra ◽  
Cluster Category ◽  
Rigid Object ◽  
Cluster Tilting Object ◽  
Dynkin Type ◽  
Cluster Categories ◽  
Tilting Object ◽  
Definition Of ◽  
Cluster Tilting ◽  
Laurent Polynomial Ring

AbstractThe (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

scholarly journals Maximal rigid objects without loops in connected 2-CY categories are cluster-tilting objects

2015 ◽  
Vol 14 (05) ◽  
pp. 1550071 ◽  
Author(s):  
Jinde Xu ◽  
Baiyu Ouyang
Keyword(s):  
Rigid Object ◽  
Rigid Objects ◽  
Unipotent Groups ◽  
Cluster Tilting Object ◽  
Tilting Object ◽  
Cluster Tilting ◽  
Maximal Rigid Object ◽  
Tilting Objects

In this paper, we study Conjecture II.1.9 of [A. B. Buan, O. Iyama, I. Reiten and J. Scott, Cluster structures for 2-Calabi–Yau categories and unipotent groups, Compos. Math. 145(4) (2009) 1035–1079], which said that any maximal rigid object without loops or 2-cycles in its quiver is a cluster-tilting object in a connected Hom-finite triangulated 2-CY category [Formula: see text]. We obtain some conditions equivalent to the conjecture, and by using them we prove the conjecture.

scholarly journals Lifting to Cluster-tilting Objects in Higher Cluster Categories

2012 ◽  
Vol 19 (04) ◽  
pp. 707-712
Author(s):  
Pin Liu
Keyword(s):  
Positive Integer ◽  
Cluster Category ◽  
Tilting Modules ◽  
Endomorphism Rings ◽  
Tilted Algebras ◽  
Cluster Categories ◽  
Cluster Tilting ◽  
Tilting Objects

Let d > 1 be a positive integer. In this note, we consider the d-cluster-tilted algebras, i.e., algebras which appear as endomorphism rings of d-cluster-tilting objects in higher cluster categories (d-cluster categories). We show that tilting modules over such algebras lift to d-cluster-tilting objects in the corresponding higher cluster category.

scholarly journals Tilting and Cluster Tilting for Preprojective Algebras and Coxeter Groups

2017 ◽  
Vol 2019 (18) ◽  
pp. 5597-5634 ◽  
Author(s):  
Yuta Kimura
Keyword(s):  
Coxeter Groups ◽  
Cluster Category ◽  
Preprojective Algebra ◽  
Endomorphism Algebra ◽  
Factor Algebra ◽  
Stable Category ◽  
Preprojective Algebras ◽  
Tilting Object ◽  
Cluster Tilting ◽  
Silting Object

AbstractWe study the stable category of the graded Cohen–Macaulay modules of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\boldsymbol{w})$ of this category associated with each reduced expression $\boldsymbol{w}$ of $w$ and give a sufficient condition on $\boldsymbol{w}$ such that $M(\boldsymbol{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\boldsymbol{w})$. Moreover, we compare it with a triangle equivalence given by Amiot–Reiten–Todorov for a cluster category.

scholarly journals Finding a cluster-tilting object for a representation finite cluster-tilted algebra

2010 ◽  
Vol 121 (2) ◽  
pp. 249-263 ◽  
Author(s):  
M. A. Bertani-Økland ◽  
S. Oppermann ◽  
A. Wrålsen
Keyword(s):  
Tilted Algebra ◽  
Cluster Tilting Object ◽  
Tilting Object ◽  
Cluster Tilting ◽  
Cluster Tilted Algebra

Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories

2013 ◽  
Vol 20 (01) ◽  
pp. 123-140
Author(s):  
Teng Zou ◽  
Bin Zhu
Keyword(s):  
Simplicial Complex ◽  
Root System ◽  
Cluster Complex ◽  
Cluster Complexes ◽  
Endomorphism Algebras ◽  
Tilted Algebras ◽  
Cluster Categories ◽  
Hereditary Algebras ◽  
Cluster Tilting ◽  
Tilting Objects

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.

scholarly journals Quotients of cluster categories

2010 ◽  
Vol 140 (1) ◽  
pp. 65-81 ◽  
Author(s):  
Peter Jørgensen
Keyword(s):  
The Other ◽  
Dynkin Type ◽  
Cluster Categories

Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories can be obtained as quotients of cluster categories. The other half are quotients of 2-cluster categories, the ‘lowest’ type of higher cluster categories. Hence, in Dynkin types A and D, all higher cluster phenomena are implicit in cluster categories and 2-cluster categories. In contrast, the same is not true in Dynkin type E.

scholarly journals Graded mutation in cluster categories coming from hereditary categories with a tilting object

2012 ◽  
Vol 216 (12) ◽  
pp. 2783-2799
Author(s):  
Marco Angel Bertani-Økland ◽  
Steffen Oppermann ◽  
Anette Wrålsen
Keyword(s):  
Cluster Categories ◽  
Tilting Object

scholarly journals CLUSTER AUTOMORPHISMS AND COMPATIBILITY OF CLUSTER VARIABLES

2014 ◽  
Vol 56 (3) ◽  
pp. 705-720 ◽  
Author(s):  
IBRAHIM ASSEM ◽  
VASILISA SHRAMCHENKO ◽  
RALF SCHIFFLER
Keyword(s):  
Cluster Algebra ◽  
The Other ◽  
Cluster Algebras ◽  
Dynkin Type ◽  
Rank 2

AbstractIn this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters, as well as the notion of weak unistructural cluster algebras, for which the set of cluster variables determines the clusters, provided that the type of the cluster algebra is fixed. We prove that, for cluster algebras of the Dynkin type, the two notions of unistructural and weakly unistructural coincide, and that cluster algebras of rank 2 are always unistructural. We then prove that a cluster algebra $\mathcal A$ is weakly unistructural if and only if any automorphism of the ambient field, which restricts to a permutation of cluster variables of $\mathcal A$, is a cluster automorphism. We also investigate the Fomin-Zelevinsky conjecture that two cluster variables are compatible if and only if one does not appear in the denominator of the Laurent expansions of the other.

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