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, as introduced by T. Schedler.
Highlights
Pre-Calabi-Yau algebras were introduced in [8], and further studied in [2] and [3]. These structures have appeared in other works under different names, such as V∞-algebras in [11], A∞-algebras with boundary in [13], noncommutative divisors in Remark 2.11 in [14], or weak Calabi-Yau structures (see [5] for the case of algebras, [16] for differential graded categories and [4] for linear ∞-categories). These references show that pre-Calabi-Yau structures play an important role in homological algebra, symplectic geometry, string topology, noncommutative geometry and even in Topological Quantum Field Theory
In the sense of formal noncommutative geometry, it is the analogue of a symplectic structure
The results obtained in this article give rise to a more natural study of quasi-isomorphism classes of dg double Poisson algebras by considering the associated pre-Calabi-Yau A∞-algebras. We remark that the former problem is in principle specially difficult, as it is usually the case when dealing with double structures, since, transfer theorems for strongly homotopic structures over dioperads or properads are known to hold, they are not explicit
Summary
Pre-Calabi-Yau algebras were introduced in [8], and further studied in [2] and [3]. these structures (or equivalent ones) have appeared in other works under different names, such as V∞-algebras in [11], A∞-algebras with boundary in [13], noncommutative divisors in Remark 2.11 in [14], or weak Calabi-Yau structures (see [5] for the case of algebras, [16] for differential graded (dg) categories and [4] for linear ∞-categories). The results obtained in this article give rise to a more natural study of quasi-isomorphism classes of dg double Poisson algebras by considering the associated pre-Calabi-Yau A∞-algebras. We remark that the former problem is in principle specially difficult, as it is usually the case when dealing with double structures (e.g. double associative algebras, double Poisson algebras), since, transfer theorems for strongly homotopic structures over dioperads or properads are known to hold, they are not explicit. Subsection 5.1 is devoted to prove the first main result of our article, Theorem 5.2, that establishes the bijection between fully manageable nice d-pre-Calabi-Yau structures and double Poisson brackets of degree −d.
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