Quadratic algebras associated with exterior 3-forms

TL;DR

This paper explores quadratic algebras generated by exterior 3-forms, establishing conditions for regularity and Calabi-Yau properties in low dimensions, and providing counterexamples in higher dimensions.

Contribution

It characterizes when such algebras are regular and 3-Calabi-Yau, especially in dimensions up to 7, and presents a counterexample for dimension 8.

Findings

All algebras with 3-regular exterior 3-forms in dimensions ≤7 are regular and 3-Calabi-Yau.

Regularity fails in dimension 8, with a specific counterexample provided.

The results depend on the algebraically closed field assumption.

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 $n=8$.