Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations

Jin, Haibo

doi:10.48550/arxiv.1812.03737

Cited by 3 publications

(2 citation statements)

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“…Extriangulated categories, recently introduced in [NP19], axiomatize extension-closed subcategories of triangulated categories in a (moderately) similar way that Quillen's exact categories axiomatize extension-closed subcategories of abelian categories. They appear in representation theory in relation with cotorsion pairs [CZZ18, ZH19, LN19, Liu17], with Auslander-Reiten theory [INP18], with cluster algebras, mutations, or cluster-tilting theory [CZZ18, Pre17, ZZ18, LZ18a, LZ18b, LZ19], with Cohen-Macaulay dgmodules in the remarkable [Jin18]. We also note the generalization, called n-exangulated categories [HLN17], to a version suited for higher homological algebra.…”

Section: Relations For G-vectors In Brick Algebras Via Extriangulated...mentioning

confidence: 99%

Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors

Padrol¹,

Palu²,

Pilaud³

et al. 2019

Preprint

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We show that the mesh mutations are the minimal relations among the g-vectors with respect to any initial seed in any finite type cluster algebra. We then use this algebraic result to derive geometric properties of the g-vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high dimensional positive orthant with well-chosen affine spaces. This sheds a new light on and extends earlier results of N. Arkani-Hamed, Y. Bai, S. He, and G. Yan in type A and of V. Bazier-Matte, G. Douville, K. Mousavand, H. Thomas and E. Yıldırım for acyclic initial seeds.Moreover, we use a similar approach to study the space of polytopal realizations of the g-vector fans of other generalizations of the associahedron. For non-kissing complexes (a.k.a. support τ -tilting complexes) of gentle algebras, we show that the space of realizations of the nonkissing fan is simplicial when the gentle bound quiver is brick and 2-acyclic, and we describe in this case its facet-defining inequalities in terms of mesh mutations. For graphical nested complexes, we show that the space of realizations of the nested fan is simplicial only in the case of the associahedron, and we describe its facet-defining inequalities in general.Along the way, we prove algebraic results on 2-Calabi-Yau triangulated categories, and on extriangulated categories that are of independent interest. In particular, we prove, in those two setups, an analogue of a result of M. Auslander on minimal relations for Grothendieck groups of module categories.The type cone of the cluster fan of type A 2 .

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Section: Relations For G-vectors In Brick Algebras Via Extriangulated...mentioning

confidence: 99%

Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors

Padrol¹,

Palu²,

Pilaud³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…On the other hand, Chan, Koenig and Liu [7] noticed that for a representation-finite self-injective algebra A, the simple-minded systems in A-mod correspond exactly to the combinatorial configurations in the Auslander-Reiten quiver of A, a key notion introduced by Riedtmann ( [24,26,25]) in the 1980's in her famous work on classification of representation-finite self-injective algebras. A similar notion in −d-Calabi-Yau triangulated categories is called d-Riedtmann configuration (see [10]) or −d-Calabi-Yau configuration (see [18]). The connection between simple-minded systems and combinatorial configurations is quite useful since the combinatorial configurations are often easier to handle.…”

Section: Introductionmentioning

confidence: 99%