Quivers supporting twisted Calabi-Yau algebras

TL;DR

This paper classifies graded twisted Calabi-Yau algebras of dimension 3 derived from quivers with polynomial growth, focusing on their types characterized by incidence, automorphism, and superpotential degree, especially for quivers with up to 3 vertices.

Contribution

It introduces a classification framework for these algebras based on their type and provides a nearly complete classification for quivers with at most 3 vertices.

Findings

Classified possible types of such algebras with polynomial growth.

Provided a nearly complete classification for quivers with up to 3 vertices.

Connected algebraic properties to quiver structures and superpotential degrees.

Abstract

We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form $A = \kk Q/I$, where $Q$ is a quiver and $I$ is an ideal of relations coming from taking partial derivatives of a twisted superpotential on $Q$. We define the type $(M, P, d)$ of such an algebra $A$, where $M$ is the incidence matrix of the quiver, $P$ is the permutation matrix giving the action of the Nakayama automorphism of $A$ on the vertices of the quiver, and $d$ is the degree of the superpotential. We study the question of what possible types can occur under the additional assumption that $A$ has polynomial growth. In particular, we are able to give a nearly complete answer to this question when $Q$ has at most 3 vertices.