On the noncommutative geometry of square superpotential algebras

TL;DR

This paper explores the noncommutative geometry of square superpotential algebras, revealing their structure via embeddings into a torus, classifying simple modules, and establishing their properties as noncommutative crepant resolutions and Calabi-Yau algebras.

Contribution

It introduces the concept of an impression for analyzing square superpotential algebras and provides a classification of simple modules, along with geometric and homological characterizations.

Findings

Z is a 3-dimensional normal toric domain.

A_m is a noncommutative crepant resolution of Z_m.

Y^{p,q} algebras have coinciding Azumaya and smooth loci.

Abstract

A superpotential algebra is square if its quiver admits an embedding into a two-torus such that the image of its underlying graph is a square grid, possibly with diagonal edges in the unit squares; examples are provided by dimer models in physics. Such an embedding reveals much of the algebras representation theory through a device we introduce called an impression. Let A be a square superpotential algebra, Z its center, and \mathfrak{m} the maximal ideal at the origin of Spec(Z). Using an impression, we give a classification of all simple A-modules up to isomorphism, and give algebraic and homological characterizations of the simple A-modules of maximal k-dimension; show that Z is a 3-dimensional normal toric domain and Z_{\mathfrak{m}} is Gorenstein, by determining transcendence bases and Z-regular sequences; and show that A_{\mathfrak{m}} is a noncommutative crepant resolution of Z_{\mathfrak{m}}, and thus a local Calabi-Yau algebra. A particular class of square superpotential algebras, the Y^{p,q} algebras, is considered in detail. We show that the Azumaya and smooth loci of the centers coincide, and propose that each ramified maximal ideal sitting over the singular locus is the exceptional locus of a blowup shrunk to zero size.