Poisson cohomology, Koszul duality, and Batalin-Vilkovisky algebras

We study the noncommutative Poincar\'e duality between the Poisson homology and cohomology of unimodular Poisson algebras, and show that Kontsevich's deformation quantization as well as Koszul duality preserve the corresponding Poincar\'e duality. As a corollary, the Batalin-Vilkovisky algebra structures that naturally arise in these cases are all isomorphic.


Introduction
In this paper we study the noncommutative Poincaré duality between the Poisson homology and cohomology of unimodular Poisson algebras, and show that Kontsevich's deformation quantization as well as Koszul duality preserve the corresponding Poincaré duality, following the works of Shoikhet [27] and Dolgushev [7] among others.
Let A = R[x 1 , · · · , x n ] be the real polynomial algebra in n variables. A Poisson bivector on A, say π, is called quadratic if it is in the form (1.1) Several years ago, Shoikhet [27] observed that if π is quadratic, then the Koszul dual algebra A ! of A, namely, the graded symmetric algebra Λ(ξ 1 , · · · , ξ n ) generated by n elements of degree −1, has a Poisson structure (let us call it the Koszul dual of π), given by and proved that Kontsevich's deformation quantization preserves this type of Koszul duality. Shoikhet's result motivates us to study some other properties of a Poisson algebra under Koszul duality. First, the following theorem is clear from Shoikhet's article, once we explicitly write down the corresponding complexes. Historically, the Poisson homology and cohomology were introduced by Koszul [16] and Lichnerowicz [20] respectively. In 1997 Weinstein [33] introduced the notion of unimodular Poisson manifolds, and two years later Xu [36] proved that in this case, there is a Poincaré duality between the Poisson cohomology and homology of M . A purely algebraic version of Weinstein's notion was later formulated by Dolgushev in [7] (see also [18,22]), and in this case we also have HP • (A) ∼ = HP n−• (A), (1.4) for some n depending on A.
For a finite dimensional algebra such as A ! above, Zhu, Van Oystaeyen and Zhang introduced in [37] the notion of Frobenius Poisson algebras (in the rest of the paper, we shall use the word symmetric instead of Frobenius, just to be consistent with other references), and proved that if they are unimodular in some sense (to be recalled below), then there also exists a version of The main technique to prove the above theorem is the so-called "differential calculus", a notion introduced by Tamarkin and Tsygan in [28]. Later, Lambre [17] used the terminology "differential calculus with duality" to study the "noncommutative Poincaré duality" in these cases.
In the above-mentioned two references [36,37], the authors also proved that the Poisson cohomology of a unimodular Poisson algebra (in both cases) has a Batalin-Vilkovisky algebra structure. The Batalin-Vilkovisky structure is a very important algebraic structure that has appeared in, for example, mathematical physics, Calabi-Yau geometry and string topology. For unimodular quadratic Poisson algebras, we have the following: 1 , · · · , x n ] is a unimodular quadratic Poisson algebra. Denote by A ! its Koszul dual. Then is an isomorphism of Batalin-Vilkovisky algebras.
The above three theorems have some analogy to the case of Calabi-Yau algebras, which were introduced by Ginzburg [13] in 2006. Suppose a Calabi-Yau algebra, say A, is Koszul, then its Koszul dual is a symmetric algebra. In [13, §5.4] Ginzburg stated a conjecture, which he attributed to R. Rouquier, saying that for a Koszul Calabi-Yau algebra, say A, its Hochschild cohomology is isomorphic to the Hochschild cohomology of its Koszul dual A !
as Batalin-Vilkovisky algebras. This conjecture is recently proved by two authors of the current paper together with G. Zhou in [4]. In fact, Theorem 1.3 may be viewed as a generalization of Rouquier's conjecture in Poisson geometry, which has been a folklore for several years. More than just being an analogy, in [7, Theorem 3], Dolgushev proved that for the coordinate ring A of an affine Calabi-Yau Poisson variety, its deformation quantization in the sense of Kontsevich, say A , is Calabi-Yau if and only if A is unimodular. Similarly Felder and Shoikhet ([10]) and later ) proved that, for a symmetric Poisson algebra, its deformation quantization is again symmetric if and only if it is unimodular. Based on these results, Dolgushev asked two questions in [7, §7] (see also [8]). The first question is whether there is any relationship between the roles that the unimodularity plays in these two types of deformation quantizations. The following theorem partially answers his question, although both cases that Dolgushev and Felder-Shoikhet considered are more general (i.e., not necessarily Koszul): x n ] is a quadratic Poisson algebra. Denote by A ! the Koszul dual algebra of A, and by A and A ! the Kontsevich deformation quantization of A and A ! respectively. If A is unimodular (and by Theorem 1.2 A ! is unimodular symmetric), then A is Calabi-Yau and A ! is symmetric, and the following diagram In other words, the first half of the theorem says that, the unimodularity that appears in the deformation quantization of Poisson Calabi-Yau algebras and the one that appears in the deformation quantization of Poisson symmetric algebras are related by Koszul duality. Note that in the theorem, A and A ! are Koszul dual to each other by Shoikhet [27].
The second question that Dolgushev asked in [7, §7] is whether there exists a relationship between the Poincaré duality of the Poisson (co)homology of A and the Poincaré duality of the Hochschild (co)homology of A . The following theorem, on which the proof of the second half of Theorem 1.4 is based, answers this question: x n ] is a unimodular Poisson algebra. Let A be its deformation quantization. Then the following diagram commutes.
In other words, the two versions of Poincaré duality, one between the Poisson cohomology and homology, and the other between the Hochschild cohomology and homology, are preserved under Kontsevich's deformation quantization. Thus as a corollary, one obtains that if A = R[x 1 , · · · , x n ] is a unimodular quadratic Poisson algebra, then all the homology and cohomology groups (Poisson and Hochschild) in Theorems 1.4 and 1.5 are isomorphic.
The rest of the paper is devoted to the proof of the above theorems. It is organized as follows: in §2 we collect several facts on Koszul algebras, and their application to quadratic Poisson polynomials; in §3 we first recall the definition of Poisson homology and cohomology, and then prove Theorem 1.1; in §4 we study unimodular quadratic Poisson algebras and their Koszul dual, and prove Theorem 1.2; in §5 we prove Theorem 1.3 by means of the so-called "differential calculus with duality"; in §6 we discuss Calabi-Yau algebras, their Koszul duality and the Batalin-Vilkovisky algebras associated to them; and at last, in §7 we discuss the deformation quantization of Poisson algebras and prove Theorems 1.4 and 1.5.
Acknowledgements. This work is inspired by several interesting conversations of the authors with P. Smith, S.-Q. Wang and C. Zhu, to whom we express our gratitude, during the Noncommutative Algebraic Geometry Workshop 2014 held at Fudan University. It is partially supported by NSFC (No. 11271269) and RFDP (No. 20120181120090).
Convention. Throughout the paper, k is a field of characteristic zero, which we may assume to be R as in §1. All tensors and morphisms are graded over k unless otherwise specified. For a chain complex, its homology is denoted by H • (−), and its cohomology is H • (−) := H −• (−).

Preliminaries on Koszul algebras
In this section, we collect some necessary facts about Koszul algebras. The interested reader may refer to Loday-Vallette [21, Chapter 3] for some more details.
Let V be a finite-dimensional vector space over k. Denote by T V the free (tensor) algebra generated by V over k. Suppose R is a subspace of V ⊗ V , and let (R) be the two-sided ideal generated by R in T V , then the quotient algebra A := T V /(R) is called a quadratic algebra.
Consider the subspace of T V , then U is a coalgebra whose coproduct is induced from the de-concatenation of the tensor products. The Koszul dual coalgebra of A, denoted by A ¡ , is where Σ is the degree shifting-up (suspension) functor. A ¡ has a graded coalgebra structure induced from that of U with The Koszul dual algebra of A, denoted by A ! , is just the linear dual space of A ¡ , which is then a graded algebra. More precisely, Let V * = Hom(V, k) be the linear dual space of V , and let R ⊥ denote the space of annihilators of R in V * ⊗ V * . Shift the grading of V * down by one, denoted by Σ −1 V * , then Choose a set of basis {e i } for V , and let {e * i } be their duals in V * . There is a chain complex associated to A, called the Koszul complex:

Definition 2.1 (Koszul algebra). A quadratic algebra
x n ] be the space of polynomials (the symmetric tensor algebra) with n generators. Then A is a Koszul algebra, and its Koszul dual algebra A ! is the graded symmetric algebra Λ(ξ 1 , ξ 2 , · · · , ξ n ), with grading |ξ i | = −1. So far, we have assumed that V is a k-linear space. In §7, we will study the deformed algebras, which are algebras over k[[ ]]. In [27], Shoikhet proved that the definitions and results in above subsections remain to hold for algebras over a discrete evaluation ring, such as k

Poisson homology and cohomology
The notions of Poisson homology and cohomology were introduced by Koszul [16] and Lichnerowicz [20] respectively. Later Huebschmann [14] studied both of them from purely algebraic perspective.
For an commutative algebra A, in the following we denote by Ω p (A) the set of p-th Kähler differential forms of A, and by X −p A (M ) (or simply X −p (M ) if A is clear from the context) the space of skew-symmetric multilinear maps A ⊗p → M that are derivations in each argument. Note that from the universal property of Kähler differentials, there is an identity of left A-modules Definition 3.1 (Koszul [16]). Suppose (A, π) is a Poisson algebra. Then the Poisson chain complex of A, denoted by CP • (A), is where ∂ is given by The associated homology is called the Poisson homology of A, and is denoted by HP • (A).

Definition 3.2 (Lichnerowicz [20]). Suppose (A, π) is a Poisson algebra and M is a left Poisson
where δ is given by The associated cohomology is called the Poisson cohomology of A with values in M , and is denoted by HP • (A; M ). In particular, if M = A, then the cohomology is just called the Poisson cohomology of A, and is simply denoted by HP • (A).
Note that in the above definition, the Poisson cochain complex, viewed as a chain complex, is negatively graded, and the coboundary δ has degree −1. However, by our convention, the Poisson cohomology are positively graded.
The boundary maps are completely analogous to those of Poisson chain and cochain complexes (with Koszul's sign convention counted).
we have an explicit expression for Ω • (A), which is where Λ means the graded symmetric tensor product, and |x i | = 0 and |dx i | = 1, for i = 1, · · · , n. Similarly, where |ξ i | = −1 and |dξ i | = 0 for i = 1, · · · , n, and therefore Thus from (3.3) and (3.4) there is a canonical grading preserving isomorphism of vector spaces: It is a direct check that Φ is a chain map, and thus we obtain an isomorphism of Poisson complexes which then induces an isomorphism on the homology.
(2) We now show the second isomorphism in (1.3). Similarly to the above argument, we have and Under the identity we again obtain an isomorphism of chain complexes This completes the proof.

Unimodular Poisson algebras and Koszul duality
In this section, we study unimodular Poisson algebras. We are particularly interested in the algebraic structures on their Poisson cohomology and homology groups, which are summarized by differential calculus, a notion introduced by Tamarkin and Tsygan in [28].
Definition 4.1 (Differential calculus; Tamarkin-Tsygan [28]). Let H • and H • be graded vector spaces. A differential calculus is the sextuple satisfying the following conditions: is a degree 1 or −1 graded Lie algebra, and the product and Lie bracket are compatible in the following sense for homogeneous P, Q, R ∈ V of degree p, q, r, respectively; for any f ∈ H n and α ∈ H m ; (3) There is a map d : In the following, if ∪, ι, [−, −] and d are clear from the context, we will simply write a differential calculus by (H • , H • ) for short.

Differential calculus on Poisson (co)homology
Suppose A is a commutative algebra. We have the following operations on X • (A) and Ω • (A): (1) Wedge (cup) product: suppose P ∈ X −p (A) and Q ∈ X −q (A), then the wedge product of P and Q, denoted by P ∪ Q, is a polyvector in X −p−q (A) defined by where σ runs over all (p, q)-shuffles of (1, 2, · · · , p + q).
(2) Schouten bracket: suppose P ∈ X −p (A) and Q ∈ X −q (A), then their Schouten bracket, denoted by [P, Q], is an element in X −p−q+1 (A) given by (3) Contraction (inner product): suppose P ∈ X −p (A) and ω = df 1 ∧ · · · ∧ df n ∈ Ω n (A), then the contraction of P with ω, denoted by ι P (ω), is an A-linear map with values in Ω n−p (A) given by (4) Lie derivative: the Lie derivative is given by the Cartan formula, namely for P ∈ X −p (A) and ω ∈ Ω n (A), the Lie derivative of ω with respect to P is given by where d is the de Rham differential. Then where d is the de Rham differential, is a differential calculus.
Proof. We only have to show the operations listed above respect the Poisson boundary and coboundary. It is a direct check and can be found in [19,Chapter 3].

Unimodular Poisson algebras
Suppose A is a commutative algebra, and η ∈ Ω n (A). We say η is a volume form if X • (A) is an isomorphism of vector spaces. Now suppose A is Poisson, then we have the following diagram which may not be commutative, i.e., η may not be a Poisson cycle. We say A is unimodular if there exists a volume form η such that (4.2) commutes. This following is now immediate.

Unimodular symmetric Poisson algebras
Now, we go to unimodular symmetric Poisson algebras, a notion introduced by Zhu, Van Oystaeyen and Zhang in [37].
Suppose A ! is a finite dimensional graded commutative algebra. A ! is called symmetric if it is equipped with a bilinear, non-degenerate symmetric pairing −, − : A ! ⊗ A ! → k of degree n which is cyclically invariant, that is, a, b · c = (−1) (|a|+|b|)|c| c, a · b , for all homogeneous a, b, c ∈ A ! . This is equivalent to saying that there is an A ! -bimodule isomorphism where A ¡ = (A ! ) * . In this case, we may view η ! as an element in Hom According to Zhu-Van Oystaeyen-Zhang [37], if there exists η ! ∈ X • A ! (A ¡ ) such that ι * (−) η ! is an isomorphism, then η ! is called a volume form, and if furthermore, the digram (4.3) commutes, then A ! is called a unimodular symmetric Poisson algebra of degree n. From the definition, we immediately have: In this paper, since we are interested in A = k[x 1 , · · · , x n ] or A ! = Λ(ξ 1 , · · · , ξ n ), we always assume the volume form is constant.
Proof of Theorem 1.2. First, we show that a quadratic Poisson algebra (A = k[x 1 , · · · , x n ], π) is unimodular if and only if (A ! , π ! ) is unimodular symmetric. In fact, recall that for A = k[x 1 , · · · , x n ], Let η = dx 1 dx 2 · · · dx n and η ! = ξ * 1 ξ * 2 · · · ξ * n , where η ! is understood as contraction, namely, then under the identification commutes. This means η is a Poisson cycle for A if and only if η ! is a Poisson cocycle for A ! , which proves the claim. Second, for A as above, we show the following diagram (4.5) commutes. In fact, the two vertical isomorphisms are given by Theorem 1.1, and the two horizontal isomorphisms are given by Theorems 4.4 and 4.5 respectively. The commutativity of the diagram (4.5) follows from the chain level commutative diagram (4.4).

Poisson cohomology and the Batalin-Vilkovisky algebra
The purpose of this section is to show that for unimodular quadratic Poisson polynomial algebras, the horizontal isomorphisms in (4.5) naturally induce on HP • (A) and HP • (A ! ) a Batalin-Vilkovisky algebra structure, and the vertical isomorphisms in (4.5) are isomorphisms of Batalin-Vilkovisky algebras. We start with the notion of differential calculus with duality. is an isomorphism.
Such isomorphism PD is called the Van den Bergh duality (also called the noncommutative Poincaré duality), and η is called the volume form. (1) ∆ : V i → V i−1 is a differential, that is, ∆ 2 = 0; and (2) ∆ is second order operator, that is, Equivalently, if we define the bracket  The proof can be found in Lambre ([17, Théorème 1.6]); however, since some details in loc. cit. are omitted, we give a proof here for completeness.
Proof. Since (H • , ∪, [−, −]) is a Gerstenhaber algebra, we only need to show that the Gerstenhaber bracket is compatible with the operator ∆ in (5.3); that is, equation (5.2) holds. For any homogeneous elements f, g ∈ H • , by the definition of Poincaré duality PD (5.1) and the Cartan formulae (Lemma 6.3), we have Since PD is an isomorphism, we thus have Corollary 5.4 (see also Xu [36] and Zhu-Van Oystaeyen-Zhang [37]). Suppose A is a unimodular Poisson or unimodular symmetric Poisson algebra. Then HP • (A) admits a Batalin-Vilkovisky algebra structure. Proof of Theorem 1.3. Note that in Theorem 1.2, the right vertical isomorphism preserves the Kähler differential as well as the volume form, that is, the two differential calculus with duality  for some φ ∈ A (taking φ to be cubic then the Poisson structure is quadratic); for A = C[x 1 , x 2 , x 3 , x 4 ], Pym [25, §3] showed that any unimodular quadratic Poisson bracket on A may be written uniquely in the following form

Calabi-Yau algebras
At the end of §1 we sketched some analogy between unimodular Poisson algebras and Calabi-Yau algebras. In the following sections, we study their relationships in more detail.
6.1 Calabi-Yau algebras and the Batalin-Vilkovisky algebra structure Definition 6.1 (Calabi-Yau algebra; Ginzburg [13]). Let A be an associative algebra over k. A is called a Calabi-Yau algebra of dimension n if (1) A is homologically smooth, that is, A, viewed as an A e -module, has a finite-long resolution of finitely generated projective A e -modules, and (2) there is an isomorphism in the derived category D(A e ) of A e -modules.
In the above definition, A e is the enveloping algebra of A, namely A e := A ⊗ A op . There are a lot of examples of Calabi-Yau algebras, such as the universal enveloping algebra of semi-simple Lie algebras, the skew-product of complex polynomials with a finite subgroup of SL(n, R), the Yang-Mills algebras, etc.
We next study Van den Bergh's noncommutative Poincaré duality for Calabi-Yau algebras ( [32]). To this end, we first recall the differential calculus structure for associative algebras.
In [6, Proposition 5.5], de Thanhoffer de Völcsey and Van den Bergh proved that, for a Calabi-Yau algebra A of dimension n, there exists a class η ∈ HH n (A) such that the contraction is an isomorphism. This immediately implies the following: 13,17]). Suppose A is a Calabi-Yau algebra A of dimension n. Then is a differential calculus with duality, and in particular, (HH • (A), ∪, ∆) is a Batalin-Vilkovisky algebra.

Symmetric algebras and the Batalin-Vilkovisky algebra structure
We now recall a differential calculus structure on the Hochschild complexes of symmetric algebras.
for any homogeneouss f ∈C • (A; A) and α ∈C • (A; A * ). We have the following.
Theorem 6.6. Let A be an associative algebra. Then is a differential calculus.
Proof. By the definition of differential calculus, we only need to show the last two equalities given in Definition 4.1.
(1) By the definition of ι * and Lemma 6.2 (1), one has for any homogenous elements f, g ∈ HH • (A) and α ∈ HH • (A; A * ). This means that the cap product is a left module action.
Now suppose A ! is symmetric. Recall that the existence of the degree n cyclic pairing is equivalent to an isomorphism η : A ! ∼ = Σ −n A ¡ as A ! -bimodules. Such η may be viewed as an element inC −n (A ! ; A ¡ ), which is a cocycle, and hence represents a cohomology class. By abuse of notation, this class is also denoted by η. The following map where η • − means composing with η, gives an isomorphism on the cohomology (due to Tradler [29]). Thus we have the following.
is a differential calculus with duality, and in particular, HH • (A ! ) is a Batalin-Vilkovisky algebra.

Koszul Calabi-Yau algebras and Rouquier's conjecture
Analogously to the quadratic Poisson algebra case, the Koszul dual of a Koszul Calabi-Yau algebra is symmetric (chronologically the latter is discovered first), and we have the following theorem due to Van den Bergh (see [31,Theorem 9.2] as Gerstenhaber algebras, and Rouquier conjectured (it is stated in Ginzburg [13]) that, for a Koszul Calabi-Yau algebra, the above two Batalin-Vilkovisky are isomorphic, which turns out to be true (see [4, Theorem A] for a proof): are isomorphic as differential calculus with duality. In particular, HH • (A) and HH • (A ! ) are isomorphic as Batalin-Vilkovisky algebras.
The key point of the proof is that, with the differentials properly assigned on A ⊗ A ! and A ⊗ A ¡ respectively, then and via these quasi-isomorphisms, the volume forms as well as the contractions given by (6.2) and (6.5) are identical on the above middle terms (compare with the proof of Theorem 1.2). Example 6.9 (The polynomial case). Let A = R[x 1 , x 2 , · · · , x n ], which is n-Calabi-Yau. Its Koszul dual algebra A ! = Λ(ξ 1 , ξ 2 , · · · , ξ n ) is symmetric. As in the Poisson case, the volume forms on HH • (A) and HH • (A ! ; A ¡ ) are, via the above quasiisomorphisms, represented by 1 ⊗ ξ * 1 · · · ξ * n in A ⊗ A ¡ .

Calabi-Yau/symmetric algebras and their deformations
In this section, we take k to be R. Dolgushev [7, Theorem 3] (respectively Felder and Shoikhet [10] and Willwacher-Calaque [35,Theorem 37]) proved that for a Calabi-Yau algebra (respectively symmetric algebra), if it is unimodular Poisson (respectively unimodular symmetric Poisson), then its deformation quantization is again Calabi-Yau (respectively symmetric). We use their results to prove Theorems 1.4 and 1.5.

Deformation quantization of Calabi-Yau Poisson algebras
Recall that for a Poisson algebra A with bracket {−, −}, its deformation quantization, denoted by A , is a k[[ ]]-linear associative product (called the star-product) on where is the formal parameter and µ i are bilinear operators, satisfying In [15], Kontsevich showed that there is a one-to-one correspondence between the equivalence classes of the star-products and the equivalence classes of  (A; A) and Ω • (A). This is known as Tsygan's Formality Conjecture for chains, and is proved by Shoikhet in [26, Theorem 1.3.1]. Shoikhet also conjectured that such L ∞ -morphism is also compatible with the cup product, which was later proved by Calaque and Rossi in [2,Theorem A].
Recall that Ω • (A) andC • (A; A), we have the de Rham differential operator and the Connes boundary operator respectively. One naturally expects the L ∞ -quasiisomorphism constructed above respects these two operators. This is known as the Cyclic Formality Conjecture for chains, and is proved by Willwacher in [
Proof. Kontsevich's L ∞ -quasiisomorphism holds for the supermanifold case, as has been shown in Cattaneo and Felder [3,Appendix] (it has also been used by Shoikhet [27]). The theorem follows verbatim from [35], in particular, Theorem 37 therein.
According to the fact that the set of Poisson structures on A ! and the set of star products on A ! [[ ]] are one-to-one correspondent, (7.2) also induces a quasiisomorphism of mixed complexes (called the tangent homomorphism) where δ π ! is the Poisson coboundary operator with respect to π ! . In particular,

Twisted Poincaré duality for Poisson algebras
For a general associative algebra, say A, it may not be Calabi-Yau, and therefore there may not exist any Poincaré duality between HH • (A) and HH • (A). In [1], Brown and Zhang introduced the so-called "twisted Poincaré duality" for associative algebras. That is, for such A, keeping its left A-module structure (the multiplication) as usual, the right A-module structure of A is the multiplication composed with an automorphism σ : A → A. Denote such A-bimodule by A σ , then Brown and Zhang showed that for a lot of algebras, there exists a twisted Poincaré duality HH • (A) ∼ = HH n−• (A; A σ ) for some n ∈ N (cf. [1,Corollary 5.2]). In this case A is called a twisted Calabi-Yau algebra of dimension n.
Such phenomenon also occurs for Poisson algebras. Namely, not all Poisson algebras are unimodular, and hence there may not exist an isomorphism between HP • (A) and HP • (A). In [18,22,37,38], the authors studied the so-called twisted Poincaré duality for Poisson algebras, similarly to that of associative algebras. They also studied some comparisons with twisted Calabi-Yau algebras. However, it would be very interesting to study the relationships between the deformation quantization of twisted unimodular Poisson algebras and twisted Calabi-Yau algebras, and obtain a theorem similar to Theorem 1.4 in this twisted case.