Over an algebraically closed field, Gabriel’s theorem states that the path algebra kQ of a connected quiver is representation-finite if and only if the underlying graph of Q is an ADE Dynkin diagram. Equivalently, kQ is representation-finite precisely when the preprojective algebra of Q is finite-dimensional.

d-Representation-finite (d-RF) algebras, introduced by Iyama and Oppermann, are a generalisation of representation-finite path algebras. Attached to each d-RF algebra is a (d+1)-preprojective algebra. Grant showed that a d-RF algebra is fractional Calabi-Yau if and only if the Nakayama automorphism of its (d+1)-preprojective algebra has finite order.

In this talk, we present a family of algebras which arise from the well-studied 3-preprojective algebras of type A by “taking orbifolds”. We show that a subset of these are themselves 3-preprojective algebras (of type D). Thus we provide new examples of 2-RF algebras, which we show are also fractional Calabi-Yau.