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

TL;DR

This paper constructs a categorical framework to realize finite-type cluster algebras with universal coefficients using Frobenius categories and orbit categories, advancing the understanding of their algebraic and geometric structures.

Contribution

It introduces a new categorification approach for finite-type cluster algebras with universal coefficients via completed orbit categories of Frobenius categories.

Findings

Categorified all finite-type skew-symmetric cluster algebras with universal coefficients

Classified Frobenius models of certain triangulated orbit categories

Connected cluster categories with Gorenstein projective modules over Nakajima categories

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 $n\geq 1$.