This paper studies the noncommutative singularity theory of the double <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> quiver <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$Q_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> (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 <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$Q_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$cA_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <msub> <mi>A</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> singularities, (3) solve the Realisation Conjecture of Brown–Wemyss in this setting. For <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n \le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, 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.

Local forms for the double $A_n$ quiver

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2026年3月26日

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Hao Zhang

摘要

This paper studies the noncommutative singularity theory of the double $A_n$ $A_{n}$ quiver $Q_n$ $Q_{n}$ (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 $Q_n$ $Q_{n}$ , 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 $cA_n$ $c A_{n}$ singularities, (3) solve the Realisation Conjecture of Brown–Wemyss in this setting. For $n \le 3$ $n \leq 3$ , 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.

Local forms for the double \(A_n\) quiver

摘要