2-CY-tilted algebras that are not Jacobian

TL;DR

This paper constructs 2-CY-tilted algebras over fields of positive characteristic that are not Jacobian, introduces hyperpotentials to extend the notion of potentials, and computes fractional CY dimensions of orbit categories, including a G2-type cluster category.

Contribution

It introduces hyperpotentials to extend the concept of potentials, enabling the construction of 2-CY-tilted algebras outside the Jacobian framework in positive characteristic.

Findings

Constructed 2-CY-tilted algebras not arising from Jacobian potentials.

Defined hyperpotentials to generalize potentials in positive characteristic.

Computed fractional CY dimensions of orbit categories, including a G2 cluster category.

Abstract

Over any field of positive characteristic we construct 2-CY-tilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields of positive characteristic are 2-CY-tilted even if they do not arise from potentials. In another direction, we compute the fractionally Calabi-Yau dimensions of certain orbit categories of fractionally CY triangulated categories. As an application, we construct a cluster category of type $G_2$.