Define a \(\mathcal V^{(d)}\)-algebra as an associative algebra with a symmetric and invariant co-inner product of degree d. Here, we consider \(\mathcal V^{(d)}\) as a dioperad which includes operations with zero inputs. We show that the quadratic dual of \(\mathcal V^{(d)}\) is \((\mathcal V^{(d)})^!=\mathcal V^{(-d)}\) and prove that \(\mathcal V^{(d)}\) is Koszul. We also show that the corresponding properad is not Koszul contractible.

中文翻译：

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$$ \ mathcal V ^ {（d）} $$ V（d）dioperad的Koszuality

将\（\ mathcal V ^ {（d）} \）-代数定义为具有度d的对称且不变的共内积的关联代数。在这里，我们将\（\ mathcal V ^ {（d）} \）视为一个dioperad，其中包括具有零输入的运算。我们证明\（\ mathcal V ^ {（d）} \）的二次对​​数是\（（\ mathcal V ^ {（d）}）^！= \ mathcal V ^ {（-d）} \）和证明\（\ mathcal V ^ {（d）} \）是Koszul。我们还显示了相应的propadad不是Koszul可收缩的。