A categorification of acyclic principal coefficient cluster algebras

Pressland, M. (2023) A categorification of acyclic principal coefficient cluster algebras. Nagoya Mathematical Journal, (doi: 10.1017/nmj.2023.6) (Early Online Publication)

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.

• Item Type: Articles
• Status: Early Online Publication
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Pressland, Dr Matthew
• Authors: Pressland, M.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Nagoya Mathematical Journal
• Publisher: Cambridge University Press
• ISSN: 0027-7630
• ISSN (Online): 2152-6842
• Published Online: 22 February 2023
• Copyright Holders: Copyright © 2023 The Author
• First Published: First published in Nagoya Mathematical Journal 2023
• Publisher Policy: Reproduced under a Creative Commons licence