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),...,x(n)]#G does not admit the grading structure of a higher preprojective algebra. Namely, we prove that if a finite group G < SL(n,k) is conjugate to a subgroup of SL(n(1), k) x SL(n(2), k), for some n(2) >= 1, then the skew-group algebra k[x(1), ..., x(n)]#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. (C) 2020 Elsevier Inc. All rights reserved.