A criterion for homogeneous potentials to be 3-Calabi-Yau

TL;DR

This paper establishes a precise criterion for when homogeneous potentials generate 3-Calabi-Yau algebras, and applies it to recover known examples through skew polynomial algebra constructions.

Contribution

It provides a necessary and sufficient condition for N-Koszul algebras from homogeneous potentials to be 3-Calabi-Yau, linking algebraic properties to geometric structures.

Findings

Derived a criterion for 3-Calabi-Yau property in N-Koszul algebras

Recovered known 3-Calabi-Yau algebras via skew polynomial methods

Connected algebraic conditions with geometric interpretations

Abstract

We give a necessary and sufficient condition for an N-Koszul algebra defined by a homogeneous potential, to be 3-Calabi-Yau. As an application, we recover two families of 3-Calabi-Yau algebras recently appeared in the literature, by studying skew polynomial algebras over non-commutative quadrics.