# A categorification of acyclic principal coefficient cluster algebras

## TL;DR

This paper develops Frobenius categorifications for acyclic principal coefficient cluster algebras using internally Calabi-Yau algebras and partial stabilisation, extending previous methods to more complex frozen variable configurations.

## Contribution

It constructs appropriate internally Calabi-Yau algebras for cluster algebras with polarized principal coefficients and defines extriangulated categories to categorify cases where Frobenius categorifications are not possible.

## Key findings

- Constructed internally Calabi-Yau algebras for polarized principal coefficients
- Defined extriangulated categories for acyclic principal coefficient cluster algebras
- Provided a Frobenius version of Van den Bergh's bimodule 3-Calabi-Yau result

## 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 polarised 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 stabilisation, 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.