3-Preprojective Algebras of Type D

We present a family of selfinjective algebras of type D, which arise from the 3-preprojective algebras of type A by taking a \(\mathbb {Z}_3\) ${\mathbb{Z}}_3$ -quotient. We show that a subset of these are themselves 3-preprojective algebras, and that the associated 2-representation-finite algebras are fractional Calabi-Yau. In addition, we show our work is connected to modular invariants for SU(3).