  In this paper, we show that for a Koszul $n$-homogeneous algebra $\Lambda$,
the quadratic dual of certain twisted trivial extension is the
$(n+1)$-preprojective algebra of its quadratic dual, that is, $
(\Delta_{\nu}\Lambda)^{!,op} \simeq\Pi( \Lambda^{ !, op })$. 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.
