  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 \cite{Kel11} and for
preprojective algebras of higher representation finite algebras using
\cite{Dugas}.
