Growth of Graded Twisted Calabi-Yau Algebras

TL;DR

This paper explores the growth and Hilbert series of graded twisted Calabi-Yau algebras derived from weighted quivers, providing new classifications and descriptions for dimensions up to three.

Contribution

It introduces techniques to analyze growth and Hilbert series of twisted Calabi-Yau algebras without restrictions on generators or relations, extending previous methods.

Findings

For dimension 2, algebras have mesh relations.

For dimension 3, algebras are derivation-quotient types from twisted superpotentials.

Partial classification of quivers with finite GK-dimension.

Abstract

We initiate a study of the growth and matrix-valued Hilbert series of non-negatively graded twisted Calabi-Yau algebras that are homomorphic images of path algebras of weighted quivers, generalizing techniques previously used to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras. Several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras. We give particular attention to twisted Calabi-Yau algebras of dimension d at most 3, giving precise descriptions of their matrix-valued Hilbert series and partial results describing which underlying quivers yield algebras of finite GK-dimension. For d = 2, we show that these are algebras with mesh relations. For d = 3, we show that the resulting algebras are a kind of derivation-quotient algebra arising from an element that is similar to a twisted superpotential.