Preprojective algebra structure on skew-group algebras

Abstract

We give a class of finite subgroups G<SL(n,k) for which the skew-group algebra k[x1,…,xn]#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(n1,k)×SL(n2,k), for some n1,n2≥1, then the skew-group algebra k[x1,…,xn]#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.