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Quadratic algebras associated with exterior 3-forms

Michel Dubois-Violette, Blas Torrecillas

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 $n=8$.

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 with . 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 with : we describe a counter example in dimension .

Paper Structure

This paper contains 7 sections, 16 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Exterior 3-forms
  3. Exterior 3-forms as superpotentials
  4. Exterior 3-forms in $\mathbb K^n$ with $3\leq n\leq 7$
  5. Regularity
  6. Exterior 3-forms in $\mathbb K^n$ with $n\geq 8$
  7. Conclusion and further prospects

Key Result

Proposition 1

Let ${\mathcal{A}}$ be a regular algebra of global dimension $D$. $(\imath)$ If $D=2$ then ${\mathcal{A}}$ is quadratic and Koszul. $(\imath\imath)$ If $D=3$ then ${\mathcal{A}}$ is $N$-homogeneous with $N\geq 2$ and Koszul.

Theorems & Definitions (16)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • Proposition 5
  • Proposition 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 6 more