The Koszul dual of an exact Hochschild extension of a dg algebra is isomorphic to the deformed Calabi-Yau completion of the Koszul dual.
On Trivial Extensions and Higher Preprojective Algebras
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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.
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Exact Hochschild extensions and deformed Calabi-Yau completions
The Koszul dual of an exact Hochschild extension of a dg algebra is isomorphic to the deformed Calabi-Yau completion of the Koszul dual.