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 $TV/(R)$ 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 $J(\mathsf{w})$, 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 $SV$. Surprisingly, as $\mathsf{w}$ ranges over $V^{\otimes 3}-\{0\}$, only nine isomorphism classes of algebras appear as non-3-Calabi-Yau $J(\mathsf{w})$'s.

26 pages v2. Minor changes after the referee's report

References in corpus (2)

• Skew Calabi-Yau Algebras and Homological Identities
• A criterion for homogeneous potentials to be 3-Calabi-Yau