On cluster categories of weighted projective lines with at most three weights

Recollements, comma categories and morphic enhancements

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2021.8 ◽

2021 ◽

pp. 1-25

Author(s):

Xiao-Wu Chen ◽

Jue Le

Keyword(s):

Module Category ◽

Triangulated Category ◽

Triangulated Categories ◽

Comma Category ◽

The Right ◽

Projective Lines ◽

Triangle Functor

For each recollement of triangulated categories, there is an epivalence between the middle category and the comma category associated with a triangle functor from the category on the right to the category on the left. For a morphic enhancement of a triangulated category $\mathcal {T}$ , there are three explicit ideals of the enhancing category, whose corresponding factor categories are all equivalent to the module category over $\mathcal {T}$ . Examples related to inflation categories and weighted projective lines are discussed.

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Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories

Algebra Colloquium ◽

10.1142/s1005386713000114 ◽

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.

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NEW EXAMPLES OF CALABI–YAU 3-FOLDS AND GENUS ZERO SURFACES

Communications in Contemporary Mathematics ◽

10.1142/s0219199713500107 ◽

2014 ◽

Vol 16 (02) ◽

pp. 1350010 ◽

Cited By ~ 5

Author(s):

GILBERTO BINI ◽

FILIPPO F. FAVALE ◽

JORGE NEVES ◽

ROBERTO PIGNATELLI

Keyword(s):

Fundamental Group ◽

Moduli Space ◽

General Type ◽

Picard Number ◽

Surfaces Of General Type ◽

Geometric Genus ◽

Genus Zero ◽

New Family ◽

Hodge Numbers ◽

Projective Lines

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi–Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

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Cotorsion pairs in cluster categories of type A∞∞

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2017.12.003 ◽

2018 ◽

Vol 156 ◽

pp. 119-141 ◽

Cited By ~ 2

Author(s):

Huimin Chang ◽

Yu Zhou ◽

Bin Zhu

Keyword(s):

Type A ◽

Cluster Categories ◽

Cotorsion Pairs

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Quotients of cluster categories

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210508000425 ◽

2010 ◽

Vol 140 (1) ◽

pp. 65-81 ◽

Cited By ~ 12

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.

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