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

TL;DR

This paper classifies 3-Calabi-Yau algebras with three generators and quadratic relations by analyzing the tensor ${f w}$ in $V^{ ensor 3}$, revealing only nine non-isomorphic classes among all possible Jacobian algebras.

Contribution

It provides a complete classification of 3-Calabi-Yau Jacobian algebras based on the tensor ${f w}$ and its symmetry properties, extending previous results on their structure.

Findings

Only nine isomorphism classes of non-3-Calabi-Yau Jacobian algebras exist.

The classification depends on the symmetry group action on ${f w}$.

The geometric properties of the subscheme $ ext{Proj}( ext{Sym}(V)/(ar{f w}))$ influence the Calabi-Yau property.

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 $TV/(R)$ the quotient of the tensor algebra on $V$ by the ideal generated by $R$. Raf Bocklandt proved that if $TV/(R)$ is 3-Calabi-Yau, then it is isomorphic to $J({\sf{w}})$, the "Jacobian algebra" of some ${\sf{w}} \in V^{\otimes 3}$. This paper classifies the ${\sf{w}}\in V^{\otimes 3}$ such that $J({\sf{w}})$ is 3-Calabi-Yau. The classification depends on how ${\sf{w}}$ transforms under the action of the symmetric group $S_3$ on $V^{\otimes 3}$ and on the nature of the subscheme $\{\overline{{\sf{w}}}=0\} \subseteq \mathbb{P}^2$ where $\overline{{\sf{w}}}$ denotes the image of ${\sf{w}}$ in the symmetric algebra $SV$. Surprisingly, as ${\sf{w}}$ ranges over $V^{\otimes 3}-\{0\}$, only nine isomorphism classes of algebras appear as non-3-Calabi-Yau $J({\sf{w}})$'s.