Preprojective algebra structure on skew-group algebras

We give a class of finite subgroups $G<SL(n,k)$ for which the skew-group algebra $k[x_1,\dots ,x_n]\mathrm{\#}G$ does not admit the grading structure of a higher preprojective algebra. Namely, we prove that if a finite group $G<SL(n, k)$isconjugatetoafinitesubgroupof$SL(n_1, k)\times SL(n_2, k)$, for some $n_1,n_2\ge 1$, then the skew-group algebra $k[x_1,\dots ,x_n]\mathrm{\#}G$ is not Morita equivalent to a higher preprojective algebra. This is related to the preprojective algebra structure on the tensor product of two Koszul bimodule Calabi-Yau algebras. We prove that such an algebra cannot be endowed with a grading structure as required for a higher preprojective algebra. Moreover, we construct explicitly the bound quiver of the higher preprojective algebra over a finite-dimensional Koszul algebra of finite global dimension. We show in addition that preprojective algebras over higher representation-infinite Koszul algebras are derivation-quotient algebras whose relations are given by a superpotential.