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In earlier work, the author introduced a method for constructing a Frobeniuscategorification of a cluster algebra with frozen variables by starting fromthe data of an internally Calabi-Yau algebra, which becomes the endomorphismalgebra of a cluster-tilting object in the resulting category. In this paper,we construct appropriate internally Calabi-Yau algebras for cluster algebraswith polarised principal coefficients (which differ from those with principalcoefficients by the addition of more frozen variables), and obtain Frobeniuscategorifications in the acyclic case. Via partial stabilisation, we thendefine extriangulated categories, in the sense of Nakaoka and Palu,categorifying acyclic principal coefficient cluster algebras, for whichFrobenius categorifications do not exist in general. Many of the intermediateresults used to obtain these categorifications remain valid without theacyclicity assumption, as we will indicate, and are interesting in their ownright. Most notably, we provide a Frobenius version of Van den Bergh's resultthat the Ginzburg dg-algebra of a quiver with potential is bimodule3-Calabi-Yau.

A categorification of acyclic principal coefficient cluster algebras