On trivial extensions and higher preprojective algebras

For a Kosuzl n-properly-graded algebra, we prove that the (n+1)-preprojective algebra of its quadratic dual is isomorphic to the quadratic dual of a twisted trivial extension. This is applied to the tau-slice algebras of stable n-translation algebras and gives a noncommutative version of Bernstein-Gelfand-Gelfand correspondence for such algebras. (C) 2019 Elsevier Inc. All rights reserved.