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Self-Injective Jacobian Algebras from Postnikov Diagrams

  • Andrea PasqualiEmail author
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Abstract

We study a finite-dimensional algebra Λ from a Postnikov diagram D in a disk, obtained from the dimer algebra of Baur-King-Marsh by factoring out the ideal generated by the boundary idempotent. Thus, Λ is isomorphic to the stable endomorphism algebra of a cluster tilting module T ∈CM(B) introduced by Jensen-King-Su in order to categorify the cluster algebra structure of \(\mathbb {C}[\text {Gr}_{k}(\mathbb {C}^{n})]\). We show that Λ is self-injective if and only if D has a certain rotational symmetry. In this case, Λ is the Jacobian algebra of a self-injective quiver with potential, which implies that its truncated Jacobian algebras in the sense of Herschend-Iyama are 2-representation finite. We study cuts and mutations of such quivers with potential leading to some new 2-representation finite algebras.

Keywords

Dimer model Postnikov diagram Self-injective algebra Jacobian algebra Preprojective algebra Higher dimensional Auslander-Reiten theory Grassmannian cluster algebra 

Notes

Acknowledgements

I am thankful to my advisor Martin Herschend for the many helpful discussions and comments. I would like to thank Jakob Zimmermann for his help with the computational aspects of determining self-injectivity, and both him and Laertis Vaso for suggestions about the manuscript. I also thank the anonymous referees for spotting issues and suggesting improvements to previous versions of the paper. Finally, my thanks go to Karin Baur, Alastair King and Robert Marsh, for their helpful comments.

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© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

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