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 $n\leq 7$. 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 $n\geq 8$ : we describe a counter example in dimension $n=8$.