Quadratic algebras associated with exterior 3-forms

Abstract

This paper is devoted to the study of the quadratic algebras with relations generated by superpotentials which are exterior 3-forms. Such an algebra is regular if and only if it is Koszul and is then a 3-Calabi–Yau domain. After some general results, we investigate the case of the algebras generated in low dimensions $n$ with $n\le 7$. We show that whenever the ground field is algebraically closed, all these algebras associated with 3-regular exterior 3-forms are regular and are thus 3-Calabi–Yau domains. This result does not generalize to dimensions $n$ with $n\ge 8$: we describe a counterexample in dimension $n=8$.