scholarly journals Tilting objects on tubular weighted projective lines: A cluster tilting approach

2019 ◽  
Author(s):  
Jianmin Chen ◽  
Yanan Lin ◽  
Pin Liu ◽  
Shiquan Ruan
Keyword(s):  
Cluster Tilting ◽  
Projective Lines ◽  
Tilting Objects

Tilting and cluster-tilting objects on weighted projective lines

2018 ◽  
Vol 48 (11) ◽  
pp. 1665
Author(s):  
Ruan Shiquan ◽  
Chen Jianmin ◽  
Lin Ya'nan ◽  
Liu Pin
Keyword(s):  
Cluster Tilting ◽  
Projective Lines ◽  
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 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 -tilting theory

2013 ◽  
Vol 150 (3) ◽  
pp. 415-452 ◽  
Author(s):  
Takahide Adachi ◽  
Osamu Iyama ◽  
Idun Reiten
Keyword(s):  
Direct Summand ◽  
Triangulated Category ◽  
Finite Dimensional Algebra ◽  
Tilting Module ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Tilting Theory ◽  
Finite Dimensional ◽  
Cluster Tilting ◽  
Tilting Objects

AbstractThe aim of this paper is to introduce $\tau $-tilting theory, which ‘completes’ (classical) tilting theory from the viewpoint of mutation. It is well known in tilting theory that an almost complete tilting module for any finite-dimensional algebra over a field $k$ is a direct summand of exactly one or two tilting modules. An important property in cluster-tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly two cluster-tilting objects. Reformulated for path algebras $kQ$, this says that an almost complete support tilting module has exactly two complements. We generalize (support) tilting modules to what we call (support) $\tau $-tilting modules, and show that an almost complete support $\tau $-tilting module has exactly two complements for any finite-dimensional algebra. For a finite-dimensional $k$-algebra $\Lambda $, we establish bijections between functorially finite torsion classes in $ \mathsf{mod} \hspace{0.167em} \Lambda $, support $\tau $-tilting modules and two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$. Moreover, these objects correspond bijectively to cluster-tilting objects in $ \mathcal{C} $ if $\Lambda $ is a 2-CY tilted algebra associated with a 2-CY triangulated category $ \mathcal{C} $. As an application, we show that the property of having two complements holds also for two-term silting complexes in ${ \mathsf{K} }^{\mathrm{b} } ( \mathsf{proj} \hspace{0.167em} \Lambda )$.

scholarly journals Mutation of cluster-tilting objects and potentials

2011 ◽  
Vol 133 (4) ◽  
pp. 835-887 ◽  
Author(s):  
A. B. Buan ◽  
O. Iyama ◽  
I. Reiten ◽  
D. Smith
Keyword(s):  
Cluster Tilting ◽  
Tilting Objects

scholarly journals Noncrossing partitions and representations of quivers

2009 ◽  
Vol 145 (6) ◽  
pp. 1533-1562 ◽  
Author(s):  
Colin Ingalls ◽  
Hugh Thomas
Keyword(s):  
Coxeter Group ◽  
Stability Condition ◽  
Quiver Representations ◽  
Finitely Generated ◽  
Cluster Category ◽  
Noncrossing Partitions ◽  
Finite Coxeter Group ◽  
Group Form ◽  
Cluster Tilting ◽  
Tilting Objects

AbstractWe situate the noncrossing partitions associated with a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated with a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show that these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition.

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