Higher preprojective algebras and stably Calabi-Yau properties

In this paper, we give sufficient properties for a finite-dimensional graded algebra to be a higher preprojective algebra. These properties are of homological nature, they use Gorensteiness and bimodule isomorphisms in the stable category of Cohen-Macaulay modules. We prove that these properties are also necessary for 3-preprojective algebras using [18] and for preprojective algebras of higher representation finite algebras using [5].