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Poisson cohomology, Koszul duality, and Batalin-Vilkovisky algebras Xiaojun Chen∗1, Youming Chen†2, Farkhod Eshmatov‡1, and Song Yang§3 1Department of Mathematics, Sichuan University, Chengdu 610064,P. R. China 2 School of Science, Chongqing University of Technology, Chongqing 400054,P. R. China 7 1 3Center for Applied Mathematics, Tianjin University, Tianjin 300072, P. R. China 0 2 n a J Abstract 2 2 We study the noncommutative Poincaré duality between the Poisson homology and co- ] h homology of a unimodular quadratic Poissonalgebraand its Koszul dual, and that between p the Hochschildhomologyandcohomologyoftheir deformationquantizations. We showthat - h Kontsevich’s deformation quantizationpreserves the corresponding Poincaréduality, and as t a a corollary, the Batalin-Vilkovisky algebra structures that naturally arise in these cases are m all isomorphic. [ 1 v Contents 2 1 1 1 Introduction 2 6 0 2 Preliminaries on Koszul algebras 6 . 1 2.1 Quadratic and Koszul algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0 7 2.2 Koszul duality for quadratic Poisson polynomial algebras . . . . . . . . . . . . . 7 1 : 2.3 Koszul algebra over a discrete evaluation ring . . . . . . . . . . . . . . . . . . . . 8 v i X 3 Poisson homology and cohomology 8 r a 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Isomorphism of the Poisson cohomology for Koszul pairs . . . . . . . . . . . . . . 9 4 Unimodular Poisson algebras and Koszul duality 11 4.1 Differential calculus of Tamarkin-Tsygan . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Unimodular Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Unimodular cyclic Poisson algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 Unimodular quadratic Poisson algebras . . . . . . . . . . . . . . . . . . . . . . . 19 ∗Email: [email protected] †Email: [email protected],[email protected] ‡Email: [email protected] §Email: [email protected],[email protected] 1 5 Poisson cohomology and the Batalin-Vilkovisky algebra 21 6 Calabi-Yau algebras 23 6.1 Calabi-Yau algebras and the Batalin-Vilkovisky algebra structure . . . . . . . . . 23 6.2 Cyclic algebras and the Batalin-Vilkovisky algebra structure . . . . . . . . . . . . 26 6.3 Koszul Calabi-Yau algebras and Rouquier’s conjecture . . . . . . . . . . . . . . . 26 7 Calabi-Yau/cyclic algebras and their deformations 28 7.1 Deformation quantization of Calabi-Yau Poisson algebras . . . . . . . . . . . . . 28 7.2 Deformation quantization of cyclic Poisson algebras. . . . . . . . . . . . . . . . . 29 7.3 Identification of two Poincaré dualities . . . . . . . . . . . . . . . . . . . . . . . . 31 7.4 Twisted Poincaré duality for Poisson algebras . . . . . . . . . . . . . . . . . . . . 31 1 Introduction Inthispaperwe studythenoncommutative Poincaré duality between thePoisson homology and cohomologyofaunimodularquadraticPoissonalgebraanditsKoszuldual,andthatbetweenthe Hochschild homology and cohomology of their deformation quantizations, following the works of Shoikhet [32] and Dolgushev [9]. Let A = C[x ,··· ,x ] be the complex polynomial algebra of n variables. Recall that a 1 n Poisson structure on A is a bivector π such that the Schouten bracket [π,π] = 0; it is called quadratic if the coefficients of π are quadratic; that is, ∂ ∂ π = c x x ∧ , c ∈ C. ij i1 i2∂x ∂x ij i,j j1 j2 X Quadratic Poisson algebras form an important class of Poisson algebras, and have been exten- sively studied in the past two decades. Several years ago, Shoikhet [32] observed that if π is quadratic, then the Koszul dual algebra A! of A, namely A! = Λ(ξ ,··· ,ξ ) the graded sym- 1 n metric algebra generated by n elements of degree −1, has a canonical Poisson structure π! (let us call it the Koszul dual of π), given by ∂ ∂ π! = c ξ ξ ∧ , ij j1 j2∂ξ ∂ξ i,j i1 i2 X and proved that Kontsevich’s deformation quantization preserves this type of Koszul duality. Shoikhet’s result invokes the following naturalquestion: Does there exist any other property of a Poisson algebra that preserves or is preserved by Koszul duality? This is the primary motivation of our study, and in this paper, we are mainly concerned with two important invariants of Poisson algebras, namely, the Poisson homology and cohomology. The following theorem is clear from Shoikhet’s article, once we explicitly write down the corresponding complexes. Theorem 1.1 (Theorem 3.4). Let A = C[x ,··· ,x ] be a quadratic Poisson algebra. Denote 1 n by A! the Koszul dual Poisson algebra of A. Then there are isomorphisms HP (A) ∼= HP−•(A!;A¡) and HP•(A) ∼= HP•(A!), • 2 where A¡ := HomC(A!,C) is the linear dual of A!. In the above theorem, HP (−) is the Poisson homology, HP•(−) is the Poisson cohomology, • and HP•(A!;A¡) is the Poisson cohomology of A! with values in its dual space. Historically, the Poisson homology and cohomology were introduced by Koszul [20] and Lichnerowicz [24] respectively. They were further studied by Brylinski [3], Huebschmann [18], Weinstein and his school, and many others. In particular, in 1997 Weinstein [38] introduced the notion of unimodular Poisson manifolds, and two years later Xu [41] proved that in this case, there is a version of Poincaré duality HP•(M) ∼= HP (M), (1) n−• where M is an n-dimensional oriented Poisson manifold. A purely algebraic version of Wein- stein’s notion was formulated later by Dolgushev in [9]; see also [22, 27], etc. There is another version of Poincaré duality for the Poisson homology and cohomology of finite dimensional Poisson algebras such as A! above. This was studied by Zhu, Van Oystaeyen and Zhang in [42]. More precisely, a finite dimensional (possibly graded) Poisson algebra, say A!, is called Frobenius if it is equipped with a cyclically invariant non-degenerate pairing. In the rest of the paper, we shall use the word cyclic instead of Frobenius, just to be consistent with other references. They proved that if a cyclic Poisson algebras is unimodular in some sense (to be recalled below), then there exists Poincaré duality in this case: HP•(A!)∼= HP•−n(A!;A¡), (2) where n is the degree of the pairing. Combining the two versions of Poincaré duality (1) and (2) as well as Theorem 1.1, we have the following: Theorem 1.2 (Theorems 4.16+4.17). Let A = C[x ,··· ,x ] be a quadratic Poisson algebra. 1 n Then (A,π) is unimodular if and only if its Koszul dual (A!,π!) is unimodular cyclic. In this case, we have the following commutative diagram: HP•(A) ∼= //HPn−•(A) ∼= ∼= (cid:15)(cid:15) (cid:15)(cid:15) HP•(A!) ∼= // HP•−n(A!;A¡). The main technique to prove the above theorem is the so-called “differential calculus”, a notion introduced by Tamarkin and Tsygan in [33]. Later, Lambre [21] used the terminology “differentialcalculus withduality”tostudythenoncommutative Poincaré duality inthesecases. In the above-mentioned two references [41, 42], the authors also proved that the Poisson cohomology of a unimodular Poisson algebra (in both cases) has a Batalin-Vilkovisky algebra structure. 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: 3 Theorem 1.3 (Theorem 5.5). Suppose A = C[x ,··· ,x ] is unimodular quadratic Poisson 1 n algebra. Denote by A! the Koszul dual Poisson algebra of A. Then HP•(A) ∼= HP•(A!) is an isomorphism of Batalin-Vilkovisky algebras. The above theorems have some analogy to the case of Calabi-Yau algebras. Calabi-Yau algebras were introduced by Ginzburg [16] in 2006. Ever since then, they have been widely studied by many mathematicians from various directions. In the past decade an important class of Calabi-Yau that are well studied are those which are Koszul. In this case, the Koszul dual of a Koszul Calabi-Yau algebra is a cyclic algebra. In [16, §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! HH•(A) ∼= HH•(A!) (3) as Batalin-Vilkovisky algebras. This conjecture is recently proved by two authors of the current paper together with G. Zhou in [6]. In fact, Theorem 1.3 may be viewed as a generalization of Rouquier’s conjecture in Poisson geometry, which has been folklore for several years. Morethanbeingjustananalogy, in[9,Theorem3],Dolgushevprovedthatforthecoordinate ring A of an affine Calabi-Yau Poisson variety, its deformation quantization in the sense of Kontsevich [19], say A~, is Calabi-Yau if and only if A is unimodular. Similarly in the paper [13], Felder and Shoikhet proved that for a cyclic Poisson algebra, its deformation quantization is again cyclic if and only if it is unimodular (see loc. cit. Corollary in page 77 and also Willwacher-Calaque [40, Theorem 37]). Dolgushev then asked two questions in [9, §7]. The first question is whether there is any relationship between these two types of unimodularity appeared in Kontsevich’s deformation quantization. If we generalize Kontsevich’s deformation quantizationtothegradedPoissonalgebracase(ashasbeendonebyCattaneoandFelderin[5]), then the following theorem partially answers his question, although both cases that Dolgushev and Felder-Shoikhet considered are more general: Theorem 1.4 (Theorem 7.6). Let A = C[x ,··· ,x ] be a quadratic Poisson algebra. Denote 1 n by A! the Koszul dual Poisson algebra of A, and by A~ and A!~ the Kontsevich deformation quantization of A and A! respectively. If A is unimodular (recall that by Theorem 1.2 A! is unimodular cyclic), then A~ is Calabi-Yau and A!~ is cyclic, and the following diagram HP•(A[[~]]) ∼= //HP•(A![[~]]) (4) ∼= ∼= HH•((cid:15)(cid:15) A~) ∼= // HH•((cid:15)(cid:15)A!~). is commutative as Batalin-Vilkovisky algebra isomorphisms, where A[[~]] and A![[~]] are equipped with the Poisson bivectors ~π and ~π! respectively. 4 In other words, the first half of the theorem says that, under Koszul duality, the unimodularity that guarantees the deformation quantization of a Poisson Calabi-Yau algebra to be again Calabi-Yau becomes the unimodularity that guarantees the deformation quantization of a Pois- son cyclic algebra to be again cyclic. More explicitly, in Kontsevich’s deformation quantization of Poisson algebras, the two versions of unimodularity that appeared in [9] and [13] are related by Koszul duality. Note that in the above theorem, A~ and A!~ are Koszul dual to each other by Shoikhet [32, Theorem 0.3]. The second question that Dolgushev asked in [9, §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: Theorem 1.5 (Theorems 7.2+7.5). (1) Suppose A = C[x ,··· ,x ] is a unimodular Poisson 1 n algebra. Let A~ be its deformation quantization. Then the following diagram HH•(A~) ∼= //HHn−•(A~) ∼= ∼= HP•(A(cid:15)(cid:15) [[~]]) ∼= // HPn−•((cid:15)(cid:15)A[[~]]) is commutative. (2) Similarly, suppose A! = Λ(ξ ,··· ,ξ ) is a unimodular cyclic Poisson algebra. Let A! be 1 n ~ its deformation quantization. Then the following diagram HH•(A!~) ∼= //HH•−n(A!~;A¡~) ∼= ∼= (cid:15)(cid:15) (cid:15)(cid:15) HP•(A![[~]]) ∼= //HP•−n(A![[~]];A¡[[~]]) is commutative. The theorem says that, 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 = C[x ,··· ,x ] is a unimodular quadratic Poisson algebra, then all the homology and 1 n cohomology groups (Poisson and Hochschild) appeared in Theorems 1.4 and 1.5 are isomorphic. The rest of the paper is devoted to the proof of 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 calculuswithduality”; in§6wediscussCalabi-Yaualgebra,theirKoszuldualityandtheBatalin- Vilkovisky algebras associated to them; at last, in §7 we discuss the deformation quantization of Poisson algebras and prove Theorems 1.4 and 1.5. 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 Non- commutative Algebraic Geometry Workshop 2014 held at Fudan University. It is partially supported by NSFC (grant No. 11271269) and RFDP (grant No. 20120181120090). Convention. Throughout the paper, k is an algebraically closed field of characteristic zero, which we may assume to be C. All tensors and morphisms are graded over k unless other- wise specified. For a chain complex, its homology is denoted by H (−), and its cohomology is • H•(−) := H (−). −• 2 Preliminaries on Koszul algebras In this section, we collect some necessary facts about Koszul algebras. The best reference is the monograph of Loday-Vallette [26]. Themain result of this section is Lemma 2.3, which is dueto Shoikhet [32], saying that for a quadratic Poisson polynomial algebra, its Koszul dual algebra admits a canonical Poisson structure. 2.1 Quadratic and Koszul algebras Let V be a finite-dimensional (possibly graded) vector space over k. Denote by TV the free algebra generated by V over k; that is, TV is the tensor algebra generated by V. Suppose R is a subspce of V ⊗V, and let (R) be the two-sided ideal generated by R in TV, then the quotient algebra A := TV/(R) is called a quadratic algebra. There are two concepts associated to a quadratic algebra, namely, its Koszul dual coalgebra and Koszul dual algebra, which are given as follows: (1) Consider the subspace ∞ ∞ U = U := V⊗i⊗R⊗V⊗j n n=0 n=0i+j+2=n M M \ of TV, then U is not an algebra, but a coalgebra, whose coproduct is induced from the de- concatenation of the tensor products. The Koszul dual coalgebra of A, denoted by A¡, is ∞ A¡ = Σ⊗n(U ), n n=0 M where Σ is the degree shifting-up (suspension) functor. A¡ naturally has a graded coalgebra structure induced from that of U; for example, if all elements of V have degree zero, then (A¡) =k, (A¡) = V, (A¡) = R, ······ 0 1 2 (2) 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−1V∗, then A! = T(Σ−1V∗)/(Σ−1⊗Σ−1◦R⊥). 6 Choose a set of basis {e } for V, and let {e∗} be their duals in V∗. There is a natural chain i i complex associated to A, called the Koszul complex: δ ¡ δ ¡ δ ¡ δ ··· // A⊗A // A⊗A //··· // A⊗A // k, (5) i+1 i 0 where for any r⊗f ∈ A⊗A¡, δ(r⊗f)= e r⊗Σ−1e∗f. i i i X Definition 2.1 (Koszul algebra). A quadratic algebra A = TV/(R) is called Koszul if the Koszul chain complex (5) is acyclic. Example 2.2 (Polynomials). Let A = k[x ,x ,··· ,x ] be the space of polynomials (the sym- 1 2 n metric tensor algebra) with n generators, with each x having degree zero. It is well-known (cf. i [26]) that A is a Koszul algebra, and its Koszul dual algebra A! is the graded symmetric algebra Λ(ξ ,ξ ,··· ,ξ ), with grading |ξ |= −1. 1 2 n i 2.2 Koszul duality for quadratic Poisson polynomial algebras Recall that a Poisson bracket on a (possibly graded) commutative algebra A is a Lie bracket [−,−] :A⊗A−→ A, which is a derivation for each argument. Take A = k[x ,x ,··· ,x ] with each |x | = 0. The Poisson bracket is given in the form of 1 2 n i the Poisson bivector ∂ ∂ π = φ (x ,··· ,x ) ∧ , ij 1 n ∂x ∂x i j i,j X where φ (x ,··· ,x ) are polynomials. The Poisson algebra (A,π) is called quadratic if φ are ij 1 n ij all quadratics, that is, ∂ ∂ π = c x x ∧ , c ∈ k. (6) ij i1 i2∂x ∂x ij i,j j1 j2 X In this case, the Poisson bivector π gives a bivector π! on A!, which we would call the Koszul dual of π, and is given by ∂ ∂ π! := c ξ ξ ∧ (7) ij j1 j2∂ξ ∂ξ i,j i1 i2 X Lemma 2.3 (Shoikhet [32]). Let A = k[x ,··· ,x ] with a bivector π in the form (6). Then 1 n (A,π) is quadratic Poisson if and only if (A!,π!) is quadratic Poisson. Proof. This is a key observation in Shoikhet [32]. Since it is so important, we give a proof here for completeness. A bivector π defines a Poisson structure if and only if [π,π] = 0. Plugging (6) in [π,π] we have ∂ ∂ ∂ [π,π] = c c x x δ x +δ x ∧ ∧ (8a) ij kℓ i1 i2 k1j2 k2 k2j2 k1 ∂x ∂x ∂x i1,iX2,j1,j2k1,kX2,ℓ1,ℓ2(cid:16) (cid:0) (cid:1) j1 ℓ1 ℓ2 ∂ ∂ ∂ −c c x x δ x +δ x ∧ ∧ (8b) ij kℓ i1 i2 k1j1 k2 k2j1 k1 ∂x ∂x ∂x j2 ℓ1 ℓ2 (cid:0) (cid:1) 7 ∂ ∂ ∂ +c c x x δ x +δ x ∧ ∧ (8c) kℓ ij k1 k2 i1ℓ2 i2 i2ℓ2 i1 ∂x ∂x ∂x ℓ1 j1 j2 (cid:0) (cid:1) ∂ ∂ ∂ −c c x x δ x +δ x ∧ ∧ , (8d) kℓ ij k1 k2 i1ℓ1 i2 i2ℓ1 i1 ∂x ∂x ∂x ℓ2 j1 j2 (cid:17) (cid:0) (cid:1) where δ is the Kronecker symbol. Similarly, plugging the formula (7) into [π!,π!], we obtain •• ∂ ∂ ∂ [π!,π!] = c c ξ ξ δ ξ −δ ξ ∧ ∧ (9a) ij kℓ j1 j2 ℓ1i2 ℓ2 ℓ2i2 ℓ1 ∂ξ ∂ξ ∂ξ i1,iX2,j1,j2k1,kX2,ℓ1,ℓ2(cid:16) (cid:0) (cid:1) i1 k1 k2 ∂ ∂ ∂ +c c ξ ξ δ ξ −δ ξ ∧ ∧ (9b) ij kℓ j1 j2 ℓ1i1 ℓ2 ℓ2i1 ℓ1 ∂ξ ∂ξ ∂ξ i2 k1 k2 (cid:0) (cid:1) ∂ ∂ ∂ +c c ξ ξ δ ξ −δ ξ ∧ ∧ (9c) kℓ ij ℓ1 ℓ2 j1k2 j2 j2k2 j1 ∂ξ ∂ξ ∂ξ k1 i1 i2 (cid:0) (cid:1) ∂ ∂ ∂ +c c ξ ξ δ ξ −δ ξ ∧ ∧ . (9d) kℓ ij ℓ1 ℓ2 j1k1 j2 j2k1 j1 ∂ξ ∂ξ ∂ξ k2 i1 i2 (cid:17) (cid:0) (cid:1) Under the correspondence ∂ ∂ ∂ ∂ x 7→ , x 7→ , 7→ ξ , 7→ ξ , (10) i1,2 ∂ξ k1,2 ∂ξ ∂x j1,2 ∂x ℓ1,2 i1,2 k1,2 j1,2 ℓ1,2 we see that the summands (8a)+(8b)= (9c)+(9d) and (8c)+(8d)= (9a)+(9b), and therefore, [π,π] = 0 if and only if [π!,π!]= 0. This completes the proof. 2.3 Koszul algebra over a discrete evaluation ring 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 [32], Shoikhet proved that results in above subsections remain to holdfor algebras over a discrete evaluation ring, such as k[[~]]. For example, k[x ,··· ,x ][[~]] 1 n is Koszul dual to Λ(ξ ,··· ,ξ )[[~]] as graded algebras over k[[~]] (see [32, Theorem 0.3]). 1 n 3 Poisson homology and cohomology The notions of Poisson homology and cohomology were introduced by Koszul [20] and Lich- nerowicz [24] respectively. For more details of these concepts, we recommend the new classic [23] of Laurent-Gengoux, Pichereau and Vanhaecke. 3.1 Definitions Definition 3.1 (Poisson homology; Koszul [20]). Suppose (A,π) is a Poisson algebra. Denote by Ωp(A) the set of p-th Ka¨hler differential forms of A. Then the Poisson chain complex of A, denoted by CP (A,π), is • ··· //Ωp+1(A) ∂ //Ωp(A) ∂ //Ωp−1(A) ∂ // ··· //Ω0(A) = A, (11) 8 where ∂ is given by p ∂(f df ∧···∧df ) = (−1)i−1{f ,f }df ∧···df ···∧df 0 1 p 0 i 1 i p i=1 X + (−1)j−if d{f ,f }∧cdf ∧···df ···df ···∧df . 0 i j 1 i j p 1≤i<j≤p X c c The associated homology is called the Poisson homology of A, and is denoted by HP (A,π). • Definition 3.2 (Poisson cohomology; Lichnerowicz [24]). Suppose (A,π) is a Poisson algebra and M is a left Poisson A-module. Let X−p(M) be the space of skew-symmetric multilinear A maps A⊗p → M that are derivations in each argument. The Poisson cochain complex of A with values in M, denoted by CP•(A,π;M), is the cochain complex M = X0(M) δ //··· // X−p+1(M) δ // X−p(M) δ //X−p−1(M) δ // ··· A A A A where δ is given by δ(P)(f ,f ,··· ,f ) := (−1)i{f ,P(f ,··· ,f ,··· ,f )} 0 1 p i 0 i p 0≤i≤p X + (−1)i+jP({f ,f },bf ,··· ,f ,··· ,f ,··· ,f ). i j 1 i j p 0≤i<j≤p X b b The associated cohomology is called the Poisson cohomology of A with values in M, and is denoted by HP•(A,π;M). In particular, if M = A, then the cohomology is just called the Poisson cohomology of A, and is simply denoted by HP•(A,π). 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 arepositively graded. In thefollowing, if thePoisson structureis clear from the context, we often omit π in the notations of Poisson homology and cohomology. From the universal property of Ka¨hler differentials (cf. Loday [25, Proposition 1.3.9]), there is an identity of left A-modules X−p(M) = Hom (Ωp(A),M). (12) A A Remark 3.3 (The graded case). The Poisson homology and cohomology can be defined for graded Poisson algebras as well. In this case, Ωp(A) = f df ∧···∧df f ∈A,|f |+|f |+···+|f |+n = p 0 1 n i 0 1 n Mn∈Zn (cid:12) o (cid:12) (cid:12) and X−p(M) is again given by Hom (Ωp(A),M). Theboundarymaps are completely analogous A A to those of Poisson chain and cochain complexes (with Koszul’s sign convention counted). 3.2 Isomorphism of the Poisson cohomology for Koszul pairs We are now ready to prove Theorem 1.1. 9 Theorem 3.4 (Theorem 1.1). Let A = k[x ,··· ,x ]. Suppose (A,π) is a quadratic Poisson 1 n algebra, and denote by (A!,π!) its Koszul dual. Then there are isomorphisms HP (A,π) ∼= HP−•(A!,π!;A¡), HP•(A,π) ∼= HP•(A!,π!). (13) • Proof. (1) We first show the first isomorphism in (13). Since A = k[x ,··· ,x ], we have an 1 n explicit expression for Ω•(A), which is Ω•(A) = Λ(x ,··· ,x ,dx ,··· ,dx ), (14) 1 n 1 n whereΛmeansthegradedsymmetrictensorproduct,and|x | = 0and|dx | = 1,fori = 1,··· ,n. i i Similarly, Ω•(A!) = Λ(ξ ,··· ,ξ ,dξ ,··· ,dξ ), 1 n 1 n where |ξ |= −1 and |dξ |= 0 for i = 1,··· ,n, and therefore i i X• (A¡) = Hom (Ω•(A!),A¡) A! A! = HomΛ(ξ1,···,ξn)(Λ(ξ1,··· ,ξn,dξ1,··· ,dξn),Hom(Λ(ξ1,··· ,ξn),k)) = HomΛ(ξ1,···,ξn)(Λ(ξ1,··· ,ξn)⊗Λ(dξ1,··· ,dξn),Hom(Λ(ξ1,··· ,ξn),k)) = Hom(Λ(dξ ,··· ,dξ ),Hom(Λ(ξ ,··· ,ξ ),k)) 1 n 1 n = Hom(Λ(dξ ,··· ,dξ )⊗Λ(ξ ,··· ,ξ ),k) 1 n 1 n = Hom(Λ(dξ ,··· ,dξ ,ξ ,··· ,ξ ),k) 1 n 1 n ∂ ∂ = Λ ,··· , ,ξ∗,··· ,ξ∗ . (15) ∂ξ ∂ξ 1 n 1 n (cid:16) (cid:17) Thus from (14) and (15) there is a canonical grading preserving isomorphism of vector spaces: Φ : Ω•(A) −→ X•(A¡) x 7−→ ∂ i ∂ξi dx 7−→ ξ∗, i= 1,··· ,n. i i It is a direct check that Φ is a chain map, and thus we obtain a quasiisomorphism of Poisson complexes Φ :CP (A,π) ∼= CP−•(A!,π!;A¡), (16) • which then induces an isomorphism on homology. (2) We now show the second isomorphism in (13). Similar to the above argument, we have CP•(A) = Hom (Ω•(A),A) A = HomΛ(x1,···,xn)(Λ(x1,··· ,xn,dx1,··· ,dxn),Λ(x1,··· ,xn)) = HomΛ(x1,···,xn)(Λ(x1,··· ,xn)⊗Λ(dx1,··· ,dxn),Λ(x1,··· ,xn)) = Hom(Λ(dx ,··· ,dx ),Λ(x ,··· ,x )) 1 n 1 n ∂ ∂ = Λ ,··· , ⊗Λ(x ,··· ,x ) (17) 1 n ∂x ∂x 1 n (cid:16) (cid:17) and similarly, CP•(A!) = Hom (Ω•(A!),A!) A! 10