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

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.


Introduction
Pre-Calabi-Yau algebras were introduced in [8], and further studied in [2] and [3]. However, 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). 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 (see [5]). Following [7], a (compact) Calabi-Yau structure (of dimension n) on a compact A ∞ -algebra A is a nondegenerate cyclically invariant pairing on A of degree n. In the sense of formal noncommutative geometry, it is the analogue of a symplectic structure. The problem with this definition is that for applications related to path spaces, Fukaya categories, open Calabi-Yau manifolds or Fano manifolds, the hypothesis of compactness is too restrictive. This was the reason why pre-Calabi-Yau algebras were originally introduced in [8].
Roughly speaking, a pre-Calabi-Yau algebra can be regarded as a formal noncommutative Poisson structure on a non-compact algebra because it is a noncommutative analogue of a solution to the Maurer-Cartan equation for the Schouten bracket on polyvector fields. More precisely, let A be a Z-graded vector space, and let C (k) (A) := r≥0 Hom(A [1] ⊗r , A ⊗k ), for k ≥ 1. A pre-Calabi-Yau structure on A is a solution m = k≥0 m (k) , m (k) ∈ C (k) (A) of the Maurer-Cartan equation [m, m] gen.neckl = 0 (see [3], Def. 2.5). Here, [ , ] gen.neckl is the "generalized necklace bracket", which is a kind of graded commutator (see [3], Def. 2.4). Nevertheless, for our purposes, we will use a different but equivalent version of this notion (see [3], Prop. 2.7). A pre-Calabi-Yau algebra essentially is a cyclic A ∞ -algebra structure on A ⊕ A # [d − 1] for the natural bilinear form of degree d − 1 induced by evaluation such that A is an A ∞ -subalgebra (see Definition 4.2).
If pre-Calabi-Yau structures are regarded as noncommutative Poisson structures in the setting of formal noncommutative geometry, double Poisson algebras are the natural candidates for Poisson structures in the context of noncommutative differential geometry based on double derivations as developed in [1] and [15]. Indeed, let Der A = Der(A, A ⊗ A) be the A-bimodule of double derivations, and let DA = T A (Der A) be its tensor algebra. Roughly speaking, a double Poisson algebra is an algebra endowed with a bivector P ∈ (DA) 2 such that {P, P } = 0, where { , } is a kind of commutator in this context (see [15], Section 4.4). Besides their similarity with the commutative notion, double Poisson algebras turn out to be the appropriate noncommutative Poisson algebras in this setting because they satisfy the Kontsevich-Rosenberg principle (see [6] and [15], Section 7.5), whereby a structure on an associative algebra has geometric meaning if it induces standard geometric structures on its representation spaces.
Hence, since pre-Calabi-Yau algebras and double Poisson algebras can be regarded as noncommutative Poisson structures, one should expect some relationship between them. For instance, W.-K. Yeung [16] proved that double Poisson structures on dg categories provide examples of pre-Calabi-Yau structures. Furthermore, given an associative algebra A, N. Iyudu and M. Kontsevich showed that there exists an explicit one-to-one correspondence between the class of nongraded double Poisson algebras and that of pre-Calabi-Yau algebras whose multiplications m i vanish for i ∈ N \ {2, 3}, such that m 2 is the usual product of the square-zero extension A⊕A # [d −1], and m 3 sends A⊗A # ⊗A to A and A # ⊗A⊗A # to A # (see [3], Thm. 1.1).
The first main result of this article is an extension of this correspondence to the differential graded setting (see Theorem 5.2). Our second main result shows that such a correspondence satisfies a simple functorial property (see Theorems 5.6 and 5.9), for a suitable notion of morphism of d-pre-Calabi-Yau algebras (Definition 5.8). We remark that this notion does not define a category but a partial category of d-pre-Calabi-Yau algebras, since not all pairs (f, g) of morphisms such that the codomain of f is the domain of g are composable.
Moreover, T. Schedler [12] showed an interesting connection of the classical and associative Yang-Baxter equations with double Poisson algebras, that he generalized to L ∞ -algebras, giving rise to "infinity" versions of Yang-Baxter equations and double Poisson algebras. The latter arise by relaxing the (double) Jacobi identity up to homotopies, but not the associativity of the multiplication. We recall Schedler's definition of double P ∞ -algebras in Definition 6.1, which coincides with the usual notion of dg double Poisson algebras if the higher brackets vanish. The third main result of the article states that there is also a correspondence between certain pre-CY structures on (nonunitary) graded algebras A and double P ∞ -algebras, giving a different extension of Theorem 5.2 if d = 0 (see Theorem 6.3).
We believe that our results can be a powerful tool to define both new double Poisson and pre-Calabi-Yau structures. For example, the study of linear and quadratic double Poisson brackets on free associative algebras, as in [9] or [10], might be useful to better understand and extend the results obtained by N. Iyudu in [2], where pre-Calabi-Yau structures on path algebras of quivers with one vertex and a finite number of loops are studied. Moreover, 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, although transfer theorems for strongly homotopic structures over dioperads or properads are known to hold, they are not explicit. Indeed, as a major difference with the theory of (al)gebras over operads we can mention that there does not exist in general a Schur functor construction for dioperads/properads -so there is in particular no bar construction for (al)gebras over dioperads/properads-, the category of (al)gebras over dioperads/properads does not carry any natural model structure, etc.
The contents of the article are as follows. We begin in Section 2 by fixing our notations and conventions, and in Section 3 we review some known definitions and results related to double Poisson dg algebras. After reviewing the basic definitions and results on A ∞ -algebras in the first part of Section 4, we recall the crucial notion of a d-pre-Calabi-Yau structure as well as some additional conditions on A ∞ -algebras that we will need to prove our main results.
Section 5 is the core of the article. 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. In Subsection 5.2 we prove our second main result, namely the functoriality of the previous correspondence (see Theorems 5.6 and 5.9). Finally, in Section 6, we prove our last main result, Theorem 6.3, that extends the previous bijection in case d = 0 to include double P ∞ -algebras.

Generalities
In what follows, k will denote a field of characteristic zero. We recall that, if V = ⊕ n∈Z V n is a (cohomological) graded vector space (resp., dg vector space with differential ∂ V ), V [m] is the graded (resp., dg) vector space over k whose n-th homogeneous component V [m] n is given by V n+m , for all n, m ∈ Z (resp., and whose differential ∂ V [m] sends a homogeneous v ∈ V n+m to (−1) m ∂ V (v)). It is called the shift of V . Given a nonzero element v ∈ V n , we will denote |v| = n the degree of v. If we refer to the degree of an element, we will be implicitly assuming that it is nonzero and homogeneous.
We recall that a morphism f : V → W of graded (resp., dg) vector spaces of degree d ∈ Z is a homogeneous linear map of degree d, i.e. f (V n ) ⊆ W n+d for all n ∈ Z, (resp., satisfying that f • ∂ V = (−1) d ∂ W • f ). A morphism of degree zero will be called closed. Moreover, if f : V → W is a morphism of graded (resp., dg) vector spaces of degree d, is the morphism of degree d whose underlying set-theoretic map is (−1) md f . In this way, the shift (−)[m] defines an endofunctor on the category of graded (resp., dg) vector spaces provided with closed morphisms.
Given any d ∈ Z, we will denote by s d the suspension morphism, whose underlying map is the identity of V , and s 1 V will be denoted simply by s V . To simplify notation, we write sv instead of s V (v) for a homogeneous v ∈ V . All morphisms between vector spaces will be k-linear (satisfying further requirements if the spaces are further decorated). All unadorned tensor products ⊗ would be over k. Since graded vector spaces can be considered as dg vector spaces with trivial differentials, we will proceed to consider the case of dg vector spaces. We also remark that N will denote the set of positive integers, whereas N 0 will be the set of nonnegative integers.

Permutations
Given n ∈ N, we will denote by S n the group of permutations of n elements {1, . . . , n}, and given any σ ∈ S n , sgn(σ) ∈ {±1} will denote its sign. Given two dg vector spaces V and W , we denote by τ V,W : V ⊗ W → W ⊗ V the closed morphism determined by v ⊗ w → (−1) |v||w| w ⊗ v, for all homogeneous elements v ∈ V and w ∈ W . Moreover, given any transposition ς = (ij) with i < j in the group of permutations S n of n ∈ N elements, it induces a unique closed morphism More generally, for any permutation σ ∈ S n , written as a composition of transpositions ς 1 • · · · • ς m , we define the closed morphism τ V,n (σ) : V ⊗n → V ⊗n given by τ V,n (ς 1 ) • · · · • τ V,n (ς m ). We leave to the reader the verification that this is independent of the choice of the transpositions used in the decomposition of σ. In fact, it is easy to check that τ V,n (σ) sendsv = v 1 ⊗ · · · ⊗ v n to We will usually write σ instead of τ V,n (σ) to simplify the notation.

The closed monoidal structure
Given two dg vector spaces V and W we will denote by Hom(V, W ) the dg vector space whose component of degree d is formed by all morphisms from V to W of degree d, and whose differential sends an homogeneous element f ∈ Hom(V, W ) and Hom(W, f ) : Hom(W, V ) → Hom(W, V ) are defined by Hom(f, W )(g) = (−1) |f ||g| g • f and Hom(W, f )(g) = f • g, respectively. If W = k, then Hom(f, k) will be denoted by f # .
It is easy to check that, given homogeneous morphisms f : V → V and g : V → V , then and The usual tensor product V ⊗ W of vector spaces is a dg vector space for the grading given by , for all homogeneous v ∈ V and w ∈ W . Given f ∈ Hom(V, W ) and g ∈ Hom(V , W ), the map . This gives a closed morphism Λ V,V ,W,W : Moreover, if it is clear from the context, we will denote Λ V,V ,W,W (f ⊗g) simply by f ⊗g. Note that, using this notation, the differential of V ⊗W is precisely It is easy to check that as well as for any homogeneous morphism h : V → U . For later use, we recall that, given v 1 , . . . , v n ∈ V homogeneous elements of a graded vector space, and f 1 , . . . , f n ∈ V # homogeneous elements, then (2.8)

The closed monoidal structure and the suspension
Given d ∈ Z, and V and W two dg vector spaces, define the closed isomor- Moreover, define also the closed isomorphisms L d V,W :

Double Poisson brackets on dg algebras
The definitions of dg algebras (possibly with unit) and dg (bi)modules are supposed to be well-known. We will recall however the definition of a double Poisson dg algebra, specially to avoid some imprecisions concerning signs that exist in the literature. The reader might check that the definition coincides with the one introduced in [15], Section 2.7, for the case where the differential vanishes.
(ii) for any a ∈ A, the homogeneous map AD(a) : Usually, the identity in (ii) is called the Leibniz property, and (iii) is the double Jacobi identity.

Remark 3.2.
Note that { { , } } A being a closed morphism of dg vector spaces means precisely that On the other hand, condition (ii) in the previous definition is tantamount to the following one.
of degree zero, which, together with the map

Cyclic A ∞ -algebras and pre-Calabi-Yau structures 4.1 A ∞ -algebras
We recall that a nonunitary A ∞ -algebra is a (cohomologically) graded vector space A = ⊕ n∈Z A n together with a collection of maps {m n } n∈N , where m n : A ⊗n → A is a homogeneous morphism of degree 2 − n, satisfying the equation Since we are going to deal exclusively with nonunitary A ∞ -algebras, from now on, A ∞ -algebras will always be nonunitary, unless otherwise stated.
(ii) small if the multiplications {m n } n∈N satisfy that m n = 0, for all n ≥ 4; (iii) essentially odd if m 2i = 0, for all i > 1.
In case (i) we also say that (A, m • ) is a fully manageable extension of the dg algebra (A, m 2 , m 1 ), if we want to emphasize the latter.
Note that given an essentially odd A ∞ -algebra, SI(2p) is equivalent to
We also introduce the following definition, that will be useful in the sequel. In order to do so, given n ∈ N consider the injective map n : S n → S 2n , sending ς ∈ S n to the permutation σ defined by σ(2i) = 2ς(i) and σ(2i for all n ∈ N and all permutations ς ∈ S n , we have that for all homogeneous a 1 , b 1 , . . . , a n , b n ∈ A, where (ς −1 ,ā,b) is the sign given in (2.2) for σ = n (ς −1 ) andv = a 1 ⊗ b 1 ⊗ · · · ⊗ a n ⊗ b n . As before, if we do not assume that γ is nondegenerate in the previous definition, we will say that A is a degenerate d-ultracyclic (nonunitary) A ∞ -algebra.

Natural bilinear forms and pre-Calabi-Yau structures
Moreover, as it will be useful later, given a cyclic A ∞ -algebra (A, m • ) with a nondegenerate bilinear form γ and n ∈ N, we will define the linear map SI(n) γ : Note that the (A, m • ) being a cyclic A ∞ -algebra is equivalent to the vanishing of SI(n) γ , for all n ∈ N. For the following definition, we first recall the definition of the natural bilinear form of degree d ∈ Z associated with any (cohomologically) graded vector space . For clarity, we will denote the suspension map s d simply by t, and any element of A # [d] will be thus denoted by tf , for f ∈ A # . Define now the bilinear form for all homogeneous a, b ∈ A and f, g ∈ A # . Note that A has degree d. If there is no risk of confusion, we shall denote A simply by . 2 We recall the following crucial definition from [8].
A 0-pre-Calabi-Yau algebra will be simply called a pre-Calabi-Yau algebra.
This implies in particular that the maps {m n | A ⊗n } n∈N define an A ∞ -algebra structure on A such that its canonical inclusion into ∂ d−1 A is a strict morphism of A ∞ -algebras.

Good and nice A ∞ -algebras
We will now introduce the following terminology that will be useful in the sequel. Let us first fix some notation. Assume that there is a decomposition B 0 ⊕ B 1 of a graded vector space B. In many of our examples, B 0 will be a graded vector space A and B 1 will be A # [d − 1]. Then, for any odd integer n ∈ N, the decomposition B = B 0 ⊕ B 1 induces a canonical decomposition and 2. The symbol was downloaded from https://www.charbase.com.

Definition 4.3.
Let B be an A ∞ -algebra provided with an extra decomposition B = B 0 ⊕ B 1 . We say that B is (i) good if the A ∞ -algebra structure is essentially odd and for every odd integer n ∈ N the multiplication map m n is good; (ii) special if the A ∞ -algebra structure is essentially odd and (d − 1)-ultracyclic; (iii) nice if it is good and small (see Definition 4.1, (ii)).
All these definitions apply in particular to a d-pre-Calabi-Yau structure on A, 5 Nice pre-Calabi-Yau structures and double Poisson dg algebras

Relation between objects
We first recall that, given a (nonunitary) dg algebra A with product µ A and differential ∂ A , then A # is naturally a dg bimodule over A via For simplicity, we will write the product of A and its action on any dg bimodule M by juxtaposition, or a small dot.
Moreover, given a dg bimodule M over a (nonunitary) dg algebra A, consider the dg vector space A ⊕ M with the product (a, m) · (a , m ) = (aa , m · a + a · m ). It is easy to verify that the dg vector space A ⊕ M provided with the previous product is a (nonunitary) dg algebra. In particular, we see that is a (nonunitary) dg algebra. We leave to the reader to verify the easy assertion that this dg algebra together with the natural bilinear form of degree d − 1 defined in (4.5) is in fact a d-pre-Calabi-Yau structure, by taking m 1 to be the differential of , m 2 its product, and m n = 0, for all n ≥ 3.
Theorem 5.2. Let d ∈ Z, and let A = ⊕ n∈Z A n be a (nonunitary) dg algebra with product µ A and differential ∂ A . Consider the dg algebra algebra structure on A ⊕ A # [d − 1] explained above, with product m 2 and differential m 1 , as well as the natural bilinear form on it of degree d−1 defined in (4.5). Given any nice and fully manageable d-pre-Calabi-Yau given by sending m 3 to the double Poisson bracket determined by (5.1) is a bijection.
Proof. We are only going to consider s a,b f,g when (5.1) is not trivially zero, i.e. if |f | + |g| = |a| + |b| + d. Note that this last identity implies also that s a,b f,g = (−1) |b|(|f |+d) . We will first prove that { { , } }, as defined in (5.1), is a double Poisson bracket on the dg algebra A. We remark that we will be using Convention 3.3. Let us start with the antisymmetric property (i) in Definition 3.1, i.e.
for all homogeneous a, b ∈ A and f, g ∈ A # . Using (5.1) on each side, we obtain that (5.4) is equivalent to On the left-hand side, using the cyclicity property of , we obtain that Hence, comparing (5.5) and (5.6), we see that (5.3) holds if and only if where we have used that |a| + |b| + |f | + |g| + d = 0 (mod 2). Replacing s a,b g,f by its definition, we see that (5.7) holds, so (5.3) does it as well, as was to be shown.
Let us now prove the Leibniz property (ii) in Definition 3.1, i.e.
for all homogeneous a, b, c ∈ A. In order to do so, consider the identity (SI(n)) for where a, b, c ∈ A and f ∈ A # are homogeneous elements. This gives for all homogeneous a, b, c ∈ A and f ∈ A # . By applying (−, tg), for a general homogeneous g ∈ A # , we see that (5.9) is tantamount to Using definition (5.1), we see that the first term of (5.10) is precisely Similarly, using the identity b.(tf ) = (−1) |b|(d−1) t(b.f ), for all homogeneous b ∈ A and f ∈ A # , and (5.1), the second term of (5.10) becomes , for all homogeneous b, v, w ∈ A and f, g ∈ A # , and the fact that |{ {c, a} }| = |c| + |a| − d, we conclude that (5.12) is equal to Finally, using the cyclicity of we see that the third term of (5.10) is where we used the super symmetry of in the second identity, and tg.a = t(g.a) in the last equality. Using the identity for all homogeneous a, v, w ∈ A and f, g ∈ A # , we conclude that (5.14) is equal to Then, multiplying (5.10) by s c,ab g,f and replacing the corresponding terms by the ones given by (5.11), (5.13) and (5.15), we get g,f is just a function of the degrees |a|, |b|, |f | and |g| (satisfying that |a| + |b| + |f | + |g| + d = 0 (mod 2)), one can in fact show that our choice for s a,b g,f is the unique solution of (5.7) and (5.17), up to multiplicative constant ±1. This is in fact how we found such an expression.
Let us now show that { { , } } is a closed morphism of dg vector spaces, i.e.
for all homogeneous a, b ∈ A. In order to prove this, consider the identity (SI(n)) for n = 3 for the A ∞ -algebra structure of where a, b ∈ A and g ∈ A # are homogeneous elements, which gives Applying (−, tf ), for an arbitrary homogeneous f ∈ A # , we see that (5.19) is tantamount to Applying the cyclicity and super symmetry properties of in the first term, as well as the fact that m 1 (th) = (−1) |h|+d t(h • ∂ A ) for any homogeneous element h in A # in the first and third terms, we see that (5.20) is equivalent to (5.21) By (5.1) and multiplying (5.21) by (−1) |b|(|a|+|g|)+|g| , we see that (5.21) is tantamount to where we have used that m 1 | A = ∂ A , and |a| + |b| + |f | + |g| + d + 1 = 0 (mod 2). Now, using the Koszul sign rule, we obtain that (5.22) is precisely 23) for all for all homogeneous a, b ∈ A and f, g ∈ A # , which is clearly equivalent to (5.18).
We shall now prove the double Jacobi identity (see (iii) in Definition 3.1), which can be explicitly written as for arbitrary homogeneous elements a, b, c ∈ A, where σ ∈ S 3 is the unique cyclic permutation sending 1 to 2. In order to do so, consider (SI(n)) for n = 5 evaluated at a ⊗ tf ⊗ b ⊗ tg ⊗ c, where a, b, c ∈ A and f, g ∈ A # are homogeneous elements. It gives  The following result will be essential to prove the double Jacobi identity.
Fact 5.4. Let a, b, c ∈ A and f, g, h ∈ A # be homogeneous elements. Then, which is equivalent to for all l ∈ A # homogeneous of degree |c| + |b| + |h| − d. The left member of (5.31) is given by whereas, on the right member, by the super symmetry of . By (5.1) and (5.33), the right member of (5.31) gives which coincides with (5.32), proving (5.31), as was to be shown.
By Fact 5.4, the first term of the left member of (5.26) is precisely or, more explicitly, Next, using the cyclicity of twice and the fact that |a|+|b|+|c|+|f |+|g|+|h| = 0 (mod 2), we see that the second term of the left member of (5.26) is where we have used the cyclicity in the second line. Fact 5.4 tells us finally that the second term of the left member of (5.26) is which, by (2.8), is equal to (5.36) Using the definition of 2 b,a,c g,f,h as well as |a| + |b| + |c| + f | + |g| + |h| = 0 (mod 2) and |x| 2 = |x|(mod 2), one obtains that (5.36) is given by Finally, using the cyclicity of twice and |a|+|b|+|c|+|f |+|g|+|h| = 0 (mod 2), we see that the third term of the right member of (5.26) is given by which, by Fact 5.4, coincides with By (2.8), we see that (5.38) coincides with which can be further reduced to the form Since |a| + |b| + |c| + |f | + |g| + |h| = 0 (mod 2), it is fairly easy to prove that the product of the sign appearing in (5.34) and the one in (5.37) is precisely (−1) (|c|+d)(|a|+|b|) , whereas the product of the sign appearing in (5.34) and the one in (5.39) is (−1) (|a|+d)(|b|+|c|) . Using these results, plugging (5.34), (5.37) and (5.39) in (5.26), and multiplying the latter by the sign appearing in (5.34), we obtain precisely (5.24), as was to be shown.
To sum up, we have proved that, via (5.1), (A, µ A , ∂ A ) is endowed with a double Poisson dg structure, which in turn means that the map (5.2) is well-defined. Furthermore, notice that, if {m • } •∈N is a small and fully manageable d-pre-Calabi-Yau structure on A and { { , } } is the obtained double Poisson bracket on A, in the paragraph including (5.18) we have showed that (SI(3)) for {m • } •∈N is indeed equivalent to the fact that { { , } } is a closed morphism of dg vector spaces. Similarly, the equivalent version of the Leibniz property given by (5.9) shows that the latter is in fact tantamount to the vanishing of SI(4) . Finally, the double Jacobi identity expressed by (5.24) shows that it is in fact equivalent to the vanishing of SI(5) | A⊗tA # ⊗A⊗tA # ⊗A⊗tA # . Moreover, we remark that, since A is a dg algebra, it is easy to see that the family of Stasheff identities (SI(n)) for the multiplications {m • } •∈N on ∂ d−1 A is equivalent to just (SI(n)) for n ∈ {3, 4, 5}, since (SI(1)) is equivalent to ∂ A • ∂ A = 0, (SI(2)) is the Leibniz property of ∂ A with respect to the product µ A , and (SI(n)) trivially vanishes for n > 5.
We will finally show that (5.2) is bijective. In order to do so, we first note that, given any good, small and fully manageable d-pre-Calabi-Yau structure {m • } •∈N on A, it is uniquely determined by m 3 | A⊗A # [d−1]⊗A . Indeed, the fact that the d-pre-Calabi-Yau structure on A is good tells us that the full m 3 on ∂ d−1 A is unique, the manageability hypothesis implies that m 1 and m 2 are uniquely determined by the dg algebra structure of A, whereas the smallness assumption tells us m i = 0, for all i > 3. As a consequence, and using that the identity (5.1) implies that the associated double bracket { { , } } completely determines m 3 | A⊗A # [d−1]⊗A , we conclude that (5.2) is injective.

Relation between morphisms
has a dg algebra structure, as explained in the first two paragraphs of Subsection 5.1. Moreover, ∂ d−1 φ is naturally endowed with a super symmetric bilinear form ) and φ (a, b) = φ (tf, tg) = 0, (5.40) for all homogeneous a, b ∈ A and f, g ∈ B # .
recalled before, whose product and differential will be denoted by m φ 2 and m φ 1 , respectively. Then, ∂ d−1 φ provided with φ defined in (5.40) is a degenerate d-cyclic dg algebra.
Proof. We first remark that, by definition, ∂ d−1 φ provided with φ is a degenerate d-cyclic dg algebra if and only if (4.3) is verified for n = 1, 2, i.e.
for all homogeneous x, y, z ∈ ∂ d−1 φ. The first equation is trivially verified for x, y ∈ A or x, y ∈ B # [d − 1], and using a symmetry argument it suffices to consider the case x = a ∈ A and y = tf , with Since φ is a morphism of dg algebras, we conclude that (4.3) for n = 1 is always verified.
The definition of φ tells us that the second equation in (5.41) trivially holds if x, y, z ∈ A, or if there are at least two arguments among x, y, z that belong to B # [d − 1]. Finally, the three cases where two arguments of (5.41) are elements of A and the other is in · a, b), for all homogeneous a, b ∈ A and f ∈ B # . The latter is tantamount to f •φ(ab) = f (φ(a)φ(b)), which is trivially verified for φ, since it is a morphism of dg algebras.
Remarkably, the construction provided in Theorem 5.2 is functorial in the following sense: and (a, tf ) → (φ(a), tf ) for all a ∈ A and f ∈ B # , respectively, are strict morphisms of A ∞ -algebras preserving the corresponding bilinear forms.
Proof. We first remark that the fact that for all homogeneous a, b ∈ A and f, g ∈ B # . Indeed, the first identity follows from whereas the second follows from We now show that ∂ d−1 φ is an A ∞ -algebra. The first two Stasheff identities (SI(n)) are clearly verified, since they only involve the differential and the product of the dg algebras A and B. Moreover, the Stasheff identity (SI(n)) for n = 3 is also trivially verified. Indeed, since ∂ d−1 φ provided with m φ 1 and m φ 2 is a dg algebra, it suffices to show that the contribution of the terms involving m φ 3 in the Stasheff identity for n = 3 at an element are homogeneous elements, vanish. It is easy to see that in this case the Stasheff identity for n = 3 of ∂ d−1 φ is a consequence of the corresponding Stasheff identity for n = 3 of either ∂ d−1 A or ∂ d−1 B.
We now prove the Stasheff identity (SI(n)) for n = 4 at an element are homogeneous elements. It is easy to see that the only cases where there is at least one possibly nonvanishing term are the following

The cases (a) and (b) only involve m A
3 and m B 3 , respectively, so they follow from the Stasheff identity (SI(n)) for n = 4 of ∂ d−1 A and ∂ d−1 B, respectively. The proof for cases (c) and (d) uses the precise same arguments, so we focus on the latter. We write x 1 = tf , x 3 = tg with f, g ∈ B # , and x 2 = a and x 4 = b. It is easy to see that the fourth Stasheff identity evaluated at tf ⊗ a ⊗ tg ⊗ b is precisely 45) where we have used the first identity of (5.43) in the last term of (5.44). Since (5.45) is precisely a particular instance of the Stasheff identity for n = 4 of ∂ d−1 B, the Stasheff identity for n = 4 of ∂ d−1 φ in the case (c) follows.
We finally prove the Stasheff identity (SI(n)) for n = 5 at an element are homogeneous elements. It is easy to see that the only cases where there are possibly nonvanishing terms are either if Since the arguments for both cases are the same, we only consider the former case, for which we write The corresponding Stasheff identity then reads or, equivalently, where we have used the second identity of (5.43) in the second term of (5.46). Since (5.47) is precisely a particular case of the Stasheff identity of ∂ d−1 A for n = 5, the Stasheff identity of ∂ d−1 φ for n = 5 follows. Taking into account that the Stasheff identities for n ≥ 6 trivially vanish, because the higher products m φ n of ∂ d−1 φ vanish for n ≥ 4, we conclude that ∂ d−1 φ is an A ∞ -algebra. Moreover, ∂ d−1 φ is by definition small, and it is clearly fully manageable with respect to the dg algebra structure fixed at the beginning of this subsection.
We now show that φ satisfies the cyclicity property (4.3) with respect to m φ 3 , i.e. , y), z , for all homogeneous w, x, y, z ∈ ∂ d−1 φ. It is easy to see that the only nontrivial cases are either if x, z ∈ A and w, y for all homogeneous a, b ∈ A and f, g ∈ B # . By definition, the left member of (5.48) is precisely (−1) |a|(|b|+|f |+1) ((g • φ) ⊗ (f • φ))({ {b, a} }), whereas the right member is They clearly coincide, since φ is a morphism of double Poisson dg algebras. Combining this with the previous lemma we see that φ satisfies the cyclicity property (4.3) with respect to m φ n , for all n ∈ N. It is easy to verify that Φ A and Φ B commute with the corresponding differentials and the corresponding products, since φ is a morphism of dg algebras. To prove that they are strict morphisms A ∞ -algebras, it suffices to show that Since both identities are proved by the same arguments, we will only consider the former, evaluated at x 1 ⊗ x 2 ⊗ x 3 , for homogeneous x 1 , x 2 , x 3 ∈ ∂ d−1 φ. By definition of m φ 3 and m A 3 , the only nontrivial cases are either when and x 2 ∈ A. The first case is direct from the definition of Φ A , whereas the second follows from the second identity in (5.43).
It remains to show that Φ A and Φ B commute with the corresponding bilinear B , but this is straightforward. The theorem is thus proved.
A direct consequence of the previous theorem is the following result. Corollary 5.7. Assume the same hypotheses as in Theorem 5.6. If the morphism of double Poisson dg algebras φ : A → B is a quasi-isomorphism and B is locally finite dimensional, then the strict morphisms of degenerate cyclic A ∞ -algebras Φ A : Motivated by the previous theorem, we introduce the following.
Definition 5.8. Let A and A be two d-pre-Calabi-Yau algebra structures on the graded vector spaces A and A , respectively. A morphism from A to A is a triple (C, Φ, Ψ), where C is a degenerate (d − 1)-ultracyclic A ∞ -algebra, and Φ : C → A and Ψ : C → A are strict morphisms of A ∞ -algebras that preserve the corresponding bilinear forms.
We say that a morphism (C, Φ, Ψ) from A to A and a morphism (C , Φ , Ψ ) from A to A are composable if there exists a triple (C , Φ , Ψ ), where C is a degenerate (d − 1)-ultracyclic A ∞ -algebra, Φ : C → C and Ψ : C → C are strict morphisms of A ∞ -algebras that preserve the corresponding bilinear forms and Ψ • Φ = Φ • Ψ . The composition of (C, Φ, Ψ) and The proof of the following result follows exactly the same pattern as the (last part of the proof of) Theorem 5.6, so we leave it to the reader. Consider the fully manageable nice degenerate d-cyclic and (a, tf ) → (φ(a), tf ) for all a ∈ A and f ∈ C # , respectively, are strict morphisms of A ∞ -algebras preserving the corresponding bilinear forms, and satisfying that This result tells us that the constructions in Theorems 5.2 and 5.6 define a (partial) functor from the category of d-double Poisson dg algebras to the partial category of d-pre-Calabi-Yau algebras provided with the morphisms introduced in Definition 5.8, that preserves quasi-isomorphisms, under some mild assumptions.

Pre-Calabi-Yau structures and double P ∞ -algebras
We now introduce the definition of a double P ∞ -algebra. It is essentially the same as the one presented in [12], Def. 4.1, up to some sign differences. Definition 6.1. A double P ∞ -algebra is a (nonunitary) graded algebra A = ⊕ n∈Z A n provided with a family of homogeneous maps (ii) for all p ∈ N and homogeneous elements a 1 , . . . , a p−1 ∈ A, the homogeneous map (iii) for all p ∈ N, Proof. We will first prove that the family of brackets {{ {. . .} } p } p∈N defined by (6.1) gives indeed a double P ∞ -algebra structure on the graded algebra A. In other words, we shall prove that this bracket satisfies the conditions of Definition 6.1.
As explained in the first paragraph of the proof of Theorem 5.2, we can assume without loss of generality that p j=1 |a j |+2−p = p j=1 |f j | in (6.2), else the identity (6.1) trivially holds.
We will first prove the antisymmetric condition (i) given in Definition 6.1, i.e.
where f a = f 1 ⊗ a 1 ⊗ · · · ⊗ f p ⊗ a p . Hence, comparing (6.6) and (6.7), we see that Replacing s a1,...,ap f1,...,fp by its definition and considering the case where σ is any transposition of two successive elements, it is easy but lengthy to show that the antisymmetric condition (6.8) holds, which in turn implies that (6.4) holds, as was to be shown.