We show that the Koszul calculus of a preprojective algebra, whose graph is distinct from A ${}_1$ and A ${}_2$, vanishes in any (co)homological degree $p>2$. Moreover, its (higher) cohomological calculus is isomorphic as a bimodule to its (higher) homological calculus, by exchanging degrees $p$ and $2-p$, and we prove a generalised version of the 2‐Calabi–Yau property. For the ADE Dynkin graphs, the preprojective algebras are not Koszul and they are not Calabi–Yau in the sense of Ginzburg's definition, but they satisfy our generalised Calabi–Yau property and we say that they are Koszul complex Calabi–Yau (Kc–Calabi–Yau) of dimension 2. For Kc–Calabi–Yau (quadratic) algebras of any dimension, defined in terms of derived categories, we prove a Poincaré Van den Bergh duality theorem. We compute explicitly the Koszul calculus of preprojective algebras for the ADE Dynkin graphs.

中文翻译：

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投影前代数的Koszul演算

我们证明了预投影代数的Koszul演算，其图不同于A ${}_{\mathrm{1个}}$ 和一个 ${}_2$，以任何（同）谐度消失 $p>2$。此外，通过交换度数，其（较高）同调微积分与（较高）同质微积分是同构双模的$p$ 和 $2-p$，我们证明了2–Calabi–Yau属性的广义版本。对于ADE Dynkin图，根据Ginzburg的定义，预射代数不是Koszul，也不是Calabi–Yau，但是它们满足我们广义的Calabi–Yau性质，我们说它们是Koszul复数Calabi–Yau（Kc–Calabi维度2的-Yau）。对于任意维度的Kc-Calabi-Yau（二次）代数（根据派生类别定义），我们证明了PoincaréVan den Bergh对偶定理。我们为ADE Dynkin图显式计算预投影代数的Koszul演算。