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.

Double quasi-Poisson algebras are pre-Calabi-Yau

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In this article we prove that double quasi-Poisson algebras, which are noncommutative analogues of quasi-Poisson manifolds, naturally give rise to pre-Calabi-Yau algebras. This extends one of the main results in [11] (see also [10]), where a relationship between pre-Calabi-Yau algebras and double Poisson algebras was found. However, a major difference between the pre-Calabi-Yau algebra constructed in the mentioned articles and the one constructed in this work is that the higher multiplications indexed by even integers of the underlying A∞-algebra structure of the pre-Calabi-Yau algebra associated to a double quasi-Poisson algebra do not vanish, but are given by nice cyclic expressions multiplied by explicitly determined coefficients involving the Bernoulli numbers.

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“…In this section we recall the basic definitions of cyclic A ∞ -algebras and pre-Calabi-Yau structures. Most of this material can be found in [11,12,14] (see also [10]). We also recall and extend some technical terminology on cyclic A ∞ -algebras and pre-Calabi-Yau structures from [10], §4.…”

“…Quite strikingly, N. Iyudu and M. Kontsevich showed in [11] (see also [12]) that there exists an explicit one-to-one correspondence between a particular class of pre-Calabi-Yau algebras and that of non-graded double Poisson algebras. This bijection was further extended to the differential graded setting in [10], Thm. 5.1.…”

“…We will use the same notations and conventions as in [10], but for the reader's convenience we recall the most important ones. In what follows, k will denote a field of characteristic zero.…”

Noncommutative derived Poisson reduction

“…. , a n−1 , −}} : A → A ⊗n is a graded derivation (for the outer bimodule structure on A ⊗n ) of degree p 2),(3) (see [13]). All the results in this Section hold also for m-shifted brackets, however, in this paper we do not need these structures, therefore we discuss only the 0-shifted (homogeneous) case, which shortens the length of the signs involved in the formulas.…”