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

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.

Hochschild cohomology of m-cluster tilted algebras of type 𝔸̃

2020 ◽  
pp. 2150018
Author(s):  
Viviana Gubitosi
Keyword(s):  
Hochschild Cohomology ◽  
Cohomology Groups ◽  
Tilted Algebra ◽  
Type Formula ◽  
Tilted Algebras ◽  
Gerstenhaber Algebra ◽  
Cluster Tilted Algebra ◽  
Multiplicative Structures

In this paper, we compute the dimension of the Hochschild cohomology groups of any [Formula: see text]-cluster tilted algebra of type [Formula: see text]. Moreover, we give conditions on the bounded quiver of an [Formula: see text]-cluster tilted algebra [Formula: see text] of type [Formula: see text] such that the Gerstenhaber algebra [Formula: see text] has nontrivial multiplicative structures. We also show that the derived class of gentle [Formula: see text]-cluster tilted algebras is not always completely determined by the dimension of the Hochschild cohomology.

scholarly journals On the Galois coverings of a cluster-tilted algebra

2009 ◽  
Vol 213 (7) ◽  
pp. 1450-1463 ◽  
Author(s):  
Ibrahim Assem ◽  
Thomas Brüstle ◽  
Ralf Schiffler
Keyword(s):  
Tilted Algebra ◽  
Galois Coverings ◽  
Cluster Tilted Algebra

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.

The Auslander–Reiten components of $\mathcal{K}^b({\rm proj}\, \Lambda)$ for a cluster-tilted algebra of type Ã

2014 ◽  
Vol 14 (01) ◽  
pp. 1550005 ◽  
Author(s):  
Kristin Krogh Arnesen ◽  
Yvonne Grimeland
Keyword(s):  
Main Tool ◽  
Type A ◽  
Tilted Algebra ◽  
Cluster Tilted Algebra

We classify the Auslander–Reiten components of [Formula: see text], where Λ is a cluster-tilted algebra of type Ã. The main tool is the combinatoric description of the indecomposable complexes in [Formula: see text] via homotopy strings and homotopy bands.

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