On Trivial Extensions and Higher Preprojective Algebras

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