Quivers supporting twisted Calabi-Yau algebras

Abstract

We consider graded twisted Calabi-Yau algebras of dimension 3 which are derivation-quotient algebras of the form A = k 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.