Cyclic $A_{\infty }$-algebras and double Poisson algebras

Abstract

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.