In this article we prove that there exists an explicit bijection between nice $d$-pre-Calabi–Yau algebras and $d$-double Poisson differential graded algebras, where $d \in \mathbb{Z}$, extending a result proved by N. Iyudu and M. Kontsevich. We also show that this correspondence is functorial in a quite satisfactory way, giving rise to a (partial) functor from the category of $d$-double Poisson dg algebras to the partial category of $d$-pre-Calabi–Yau algebras. Finally, we further generalize it to include double $P_{\infty}$-algebras, introduced by T. Schedler.

中文翻译：

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循环$ A_ \ infty $-代数和双Poisson代数

在本文中，我们证明了不错的$ d $ -Calabi-Yau前代数与$ d-双泊松微分渐变代数之间存在明确的双射，其中$ d \ in \ mathbb {Z} $的结果得到扩展由N. Iyudu和M. Kontsevich撰写。我们还表明，这种函式以令人满意的方式是泛函的，从而产生了（部分）函子，从$ d $-双重Poisson dg代数的类别到$ d $-前Calabi–Yau代数的部分类别。最后，我们进一步推广它，以包括T. Schedler引入的双$ P _ {\ infty} $代数。