The classification of 3-dimensional noetherian cubic Calabi-Yau algebras

Abstract

It is known that every 3-dimensional noetherian Calabi-Yau algebra generated in degree 1 is isomorphic to a Jacobian algebra of a superpotential. Recently, S. P. Smith and the first author classified all superpotentials whose Jacobian algebras are 3-dimensional noetherian quadratic Calabi-Yau algebras. The main result of this paper is to classify all superpotentials whose Jacobian algebras are 3-dimensional noetherian cubic Calabi-Yau algebras. As an application, we show that if $S$ is a 3-dimensional noetherian cubic Calabi-Yau algebra and $\sigma$ is a graded algebra automorphism of $S$, then the homological determinant of $\sigma$ can be calculated by the formula $\operatorname{hdet} \sigma=(\operatorname{det} \sigma)^2$ with one exception.