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A (smooth) dg algebra is called (left) Calabi-Yau if (see for example here) $$ A^! = A[-n]$$ Here we use the inverse dualizing complex $A^!=\mathbf{R}\operatorname{Hom}_{(A^e)^{op}}(A,A^e)$. In ...

Let $\Pi:=\Pi(Q)$ be the preprojective algebra of a connected non-Dynkin quiver over an algebraically closed field of characteristic zero. In H. Minamoto "Ampleness of two-sided tilting complexes", ...