A Categorification of Acyclic Principal Coefficient Cluster Algebras

Abstract

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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“…□ Remark 3.6. When each cycle of ḡ is supported on an interval, the positroid variety Π ∘ 𝑔 decomposes as a product of positroid varieties for each cycle of ḡ , implying the product formula (3.8); see [Pre23,Proposition 4.4].…”

Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors

“…Extriangulated categories, recently introduced in [72], 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 [29, 58, 59, 104], with Auslander–Reiten theory [51], with cluster algebras, mutations, or cluster‐tilting theory [29, 63–65, 83, 106], with Cohen–Macaulay dg‐modules in the remarkable [53]. We also note the generalization, called $$n$$‐exangulated categories [47, 48], to a version suited for higher homological algebra.…”