Local forms for the double $A_n$ quiver

TL;DR

This paper explores the noncommutative singularity theory of the double $A_n$ quiver, establishing connections to algebraic geometry, classifying potentials, and solving conjectures related to crepant resolutions.

Contribution

It introduces intrinsic definitions of Type A potentials, proves a monomialization result, classifies potentials for small n, and solves the Realisation Conjecture in this context.

Findings

Type A potentials correspond to crepant resolutions of cAn singularities.

Full classification of Type A potentials for n ≤ 3.

Resolution of the Realisation Conjecture of Brown-Wemyss.

Abstract

This paper studies the noncommutative singularity theory of the double $A_n$ quiver $Q_n$ (with a single loop at each vertex), with applications to algebraic geometry and representation theory. We give various intrinsic definitions of a Type A potential on $Q_n$, then via coordinate changes we (1) prove a monomialization result that expresses these potentials in a particularly nice form, (2) prove that Type A potentials precisely correspond to crepant resolutions of cAn singularities, (3) solve the Realisation Conjecture of Brown-Wemyss in this setting. For $n \leq 3$, we furthermore give a full classification of Type A potentials (without loops) up to isomorphism, and those with finite-dimensional Jacobi algebras up to derived equivalence. There are various algebraic corollaries, including to certain tame algebras of quaternion type due to Erdmann, where we describe all basic algebras in the derived equivalence class.