scholarly journals Identifying SARS-CoV2 transmission cluster category: An analysis of country government database

2021 ◽  
Author(s):  
Basem Fouda ◽  
Ha P.B. Tram ◽  
Omar M. Makram ◽  
Abdelrahman S. Abdalla ◽  
Tushar Singh ◽  
...  
Keyword(s):  
Cluster Category

scholarly journals The cluster category of a surface with punctures via group actions

2021 ◽  
Vol 389 ◽  
pp. 107884
Author(s):  
Claire Amiot ◽  
Pierre-Guy Plamondon
Keyword(s):  
Group Actions ◽  
Cluster Category

scholarly journals The cluster category of a canonical algebra

2010 ◽  
Vol 362 (08) ◽  
pp. 4313-4330 ◽  
Author(s):  
M. Barot ◽  
D. Kussin ◽  
H. Lenzing
Keyword(s):  
Cluster Category ◽  
Canonical Algebra

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 Cotorsion pairs in the cluster category of a marked surface

2013 ◽  
Vol 391 ◽  
pp. 209-226 ◽  
Author(s):  
Jie Zhang ◽  
Yu Zhou ◽  
Bin Zhu
Keyword(s):  
Cluster Category ◽  
Cotorsion Pairs ◽  
Marked Surface

scholarly journals On the cluster category of a marked surface without punctures

2011 ◽  
Vol 5 (4) ◽  
pp. 529-566 ◽  
Author(s):  
Thomas Brüstle ◽  
Jie Zhang
Keyword(s):  
Cluster Category ◽  
Marked Surface

scholarly journals Algebras of acyclic cluster type: Tree type and type Ã

2013 ◽  
Vol 211 ◽  
pp. 1-50 ◽  
Author(s):  
Claire Amiot ◽  
Steffen Oppermann
Keyword(s):  
Equivalence Class ◽  
Complete Classification ◽  
Global Dimension ◽  
Path Algebra ◽  
Derived Category ◽  
Derived Equivalence ◽  
Cluster Category ◽  
Cluster Type ◽  
Tree Type

AbstractIn this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of typeÃ.We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster typeÃnfor each possible orientation ofÃn.We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

scholarly journals Density of g-Vector Cones From Triangulated Surfaces

2020 ◽  
Vol 2020 (21) ◽  
pp. 8081-8119
Author(s):  
Toshiya Yurikusa
Keyword(s):  
Asymptotic Behavior ◽  
Half Space ◽  
Closed Surface ◽  
Cluster Algebras ◽  
Connected Components ◽  
Cluster Category ◽  
Triangulated Surfaces ◽  
Cluster Tilting ◽  
Tilting Objects ◽  
Marked Surface

Abstract We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for a closed surface with exactly one puncture, in which case it is equal to the half space of a certain explicit hyperplane in $\mathbb{R}^n$. Our main ingredients are laminations on $(S,M)$, their shear coordinates, and their asymptotic behavior under Dehn twists. As an application, if $(S,M)$ is not a closed surface with exactly one puncture, the exchange graph of cluster tilting objects in the corresponding cluster category is connected. If $(S,M)$ is a closed surface with exactly one puncture, it has precisely two connected components.

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 Periodicities of T-systems and Y-systems

2010 ◽  
Vol 197 ◽  
pp. 59-174 ◽  
Author(s):  
Rei Inoue ◽  
Osamu Iyama ◽  
Atsuo Kuniba ◽  
Tomoki Nakanishi ◽  
Junji Suzuki
Keyword(s):  
Lie Algebra ◽  
Direct Method ◽  
Cluster Algebra ◽  
Quantum Affine Algebra ◽  
Cluster Category ◽  
Affine Algebra ◽  
Grothendieck Ring ◽  
Finite Dimensional ◽  
Level 2 ◽  
T System

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).

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