scholarly journals The dual actions, equivariant autoequivalences and stable tilting objects

2021 ◽  
Vol 70 (6) ◽  
pp. 2677-2736
Author(s):  
Jianmin Chen ◽  
Xiao-Wu Chen ◽  
Shiquan Ruan
Keyword(s):  
Tilting Objects

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 Recollements and tilting objects

2011 ◽  
Vol 215 (4) ◽  
pp. 420-438 ◽  
Author(s):  
Lidia Angeleri Hügel ◽  
Steffen Koenig ◽  
Qunhua Liu
Keyword(s):  
Tilting Objects

Tilting objects in triangulated categories

2019 ◽  
Vol 48 (1) ◽  
pp. 410-429
Author(s):  
Yonggang Hu ◽  
Hailou Yao ◽  
Xuerong Fu
Keyword(s):  
Triangulated Categories ◽  
Tilting Objects

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 Cluster structures for 2-Calabi–Yau categories and unipotent groups

2009 ◽  
Vol 145 (4) ◽  
pp. 1035-1079 ◽  
Author(s):  
A. B. Buan ◽  
O. Iyama ◽  
I. Reiten ◽  
J. Scott
Keyword(s):  
Cluster Structure ◽  
Cluster Algebras ◽  
New Class ◽  
Preprojective Algebras ◽  
Unipotent Groups ◽  
Special Cases ◽  
Dynkin Quivers ◽  
Cluster Tilting ◽  
Stable Categories ◽  
Tilting Objects

AbstractWe investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau and related categories. In particular, we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi–Yau categories contains, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. For these 2-Calabi–Yau categories, we construct cluster-tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.

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 On Tubular Tilting Objects in the Stable Category of Vector Bundles

2018 ◽  
Vol 35 (4) ◽  
pp. 494-512
Author(s):  
Jian Min Chen ◽  
Ya Nan Lin ◽  
Shi Quan Ruan
Keyword(s):  
Vector Bundles ◽  
Stable Category ◽  
Tilting Objects
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