The classification of 3-Calabi–Yau algebras with 3 generators and 3 quadratic relations

Abstract

Let k be an algebraically closed field of characteristic not 2 or 3, V a 3-dimensional vector space over k, R a 3-dimensional subspace of $V \otimes V$, and $\textit{TV}/(R)$ the quotient of the tensor algebra on V by the ideal generated by R. Raf Bocklandt proved that if $\textit{TV}/(R)$ is 3-Calabi–Yau, then it is isomorphic to $J(\mathsf{w})$, the “Jacobian algebra” of some $\mathsf{w}\in V^{\otimes 3}$. This paper classifies the $\mathsf{w}\in V^{\otimes 3}$ such that $J(\mathsf{w})$ is 3-Calabi–Yau. The classification depends on how $\mathsf{w}$ transforms under the action of the symmetric group $S_3$ on $V^{\otimes 3}$ and on the nature of the subscheme $\{\overline{\mathsf{w}}=0\} \subseteq {{\mathbb {P}}}^2$ where $\overline{\mathsf{w}}$ denotes the image of $\mathsf{w}$ in the symmetric algebra $\textit{SV}$.