Local forms for the double $A_n$ quiver

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.