A 2-Calabi–Yau realization of finite-type cluster algebras with universal coefficients

Chávez, Alfredo Nájera

doi:10.1007/s00209-019-02261-5

Cited by 2 publications

(4 citation statements)

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“…It was shown in [25] that finite type cluster algebras can be endowed with universal coefficients. This construction was categorified in [42] and the categorification was used to perform various computations in the following sections. To further clarify the preceding discussion let us first recall the notion of finite type cluster algebras with universal coefficients.…”

Section: Preliminaries On Cluster Algebrasmentioning

confidence: 99%

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Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

Bossinger

Mohammadi

Chávez

2021

SIGMA

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Let V be the weighted projective variety defined by a weighted homogeneous ideal J and C a maximal cone in the Gröbner fan of J with m rays. We construct a flat family over A m that assembles the Gröbner degenerations of V associated with all faces of C. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base X C (the toric variety associated to C) along the universal torsor A m → X C . We apply this construction to the Grassmannians Gr(2, C n ) with their Plücker embeddings and the Grassmannian Gr 3, C 6 with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated to this cone. Further, for Gr(2, C n ) we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.

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Section: Preliminaries On Cluster Algebrasmentioning

confidence: 99%

“…Remark 4.54. There are various methods in cluster theory to compute the exchange relations for A univ 3,6 and M 3,6 , e.g., one can use the categorification of finite type cluster algebras with universal coefficients introduced in [42]. To compute M 3,6 one can use the compatibility degree of cluster variables from [24].…”

Section: The Grassmannian Gr 3 Cmentioning

confidence: 99%

Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

Bossinger

Mohammadi

Chávez

2021

SIGMA

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“…This has been achieved for a number of families of cluster algebras [16], [17], [24], [32], [39] using suitable stably -Calabi–Yau Frobenius categories in place of the -Calabi–Yau triangulated cluster categories appearing in the case of no frozen variables. A Frobenius category is, by definition, an exact category with enough projective objects and enough injective objects, such that these two classes of objects coincide.…”

Section: Introductionmentioning

confidence: 99%

“…For example, Geiß–Leclerc–Schröer [24], Demonet–Luo [17], Jensen–King–Su [32], and Demonet–Iyama [16] construct Frobenius categorifications for cluster algebra structures on coordinate rings of partial flag varieties by exploiting the fact that these rings are well understood geometrically. The Frobenius categorifications of universal coefficient cluster algebras by Nájera Chávez [39] are restricted to finite type, again making the global combinatorics of the cluster algebra (such as its exchange graph and cluster complex) more tractable. By contrast, our approach is more akin to the work of Buan–Marsh–Reineke–Reiten–Todorov [10] and Amiot [1], in which categorifications are constructed from the local data defining the cluster algebra, namely the initial seed (enhanced in Amiot’s case by choosing the extra data of a potential on the quiver).…”

Section: Introductionmentioning

confidence: 99%

A Categorification of Acyclic Principal Coefficient Cluster Algebras

Pressland

2023

Nagoya Math. J.

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In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi–Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi–Yau algebras for cluster algebras with polarized principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables) and obtain Frobenius categorifications in the acyclic case. Via partial stabilization, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh’s result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$ -Calabi–Yau.

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A 2-Calabi–Yau realization of finite-type cluster algebras with universal coefficients

Cited by 2 publications

References 42 publications

Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

A Categorification of Acyclic Principal Coefficient Cluster Algebras

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