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

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Abstract

We categorify various finite-type cluster algebras with coefficients using completed orbit categories associated to Frobenius categories. Namely, the Frobenius categories we consider are the categories of finitely generated Gorenstein projective modules over the singular Nakajima category associated to a Dynkin diagram and their standard Frobenius quotients. In particular, we are able to categorify all finite-type skew-symmetric cluster algebras with universal coefficients and finite-type Grassmannian cluster algebras. Along the way, we classify the standard Frobenius models of a certain family of triangulated orbit categories which include all finite-type n-cluster categories, for all integers \(n\ge 1\).

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  1. For the definition of quiver mutation and generalities on cluster algebras the reader can consult [14].

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  1. CONACyT-Instituto de matemáticas UNAM, León 2, altos, 68000, Oaxaca, Oaxaca, Mexico

    Alfredo Nájera Chávez

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  1. Alfredo Nájera Chávez
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Correspondence to Alfredo Nájera Chávez.

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This work was partially supported by CONACyT grant CB2016 no. 284621.

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Nájera Chávez, A. A 2-Calabi–Yau realization of finite-type cluster algebras with universal coefficients. Math. Z. 291, 1495–1523 (2019). https://doi.org/10.1007/s00209-019-02261-5

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