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Poisson (co)homology of truncated polynomial algebras in two variables

We study the Poisson (co)homology of the algebra of truncated polynomials in two variables viewed as the semi-classical limit of a quantum complete intersection studied by Bergh and Erdmann. We show in particular that the Poisson cohomology ring of such a Poisson algebra is isomorphic to the Hochschild cohomology ring of the corresponding quantum complete intersection.


Introduction
Given a Poisson algebra, its Poisson cohomology-as introduced by Lichnerowicz in [13]-provides important informations about the Poisson structure (the Casimir elements are reflected by the degree zero cohomology, Poisson derivations modulo Hamiltonian derivations by the degree one,...). It also plays a crucial role in the study of deformations of the Poisson structure. A classical problem in algebraic deformation is to compare the Poisson (co)homology of a Poisson algebra with the Hochschild (co)homology of its deformation. Although these homologies are known to behave similarly in smooth cases (see e.g. [7,9,10,12]), the singular case seems more complicated to deal with. There are many examples in which the trace groups already do not match in [1]. Even though the dimension of the homology spaces in degree zero match, as for Kleinian singularities in [2], it might not be the case in higher degree (see [3] and [14]).
Our aim in this paper is to provide an example of a singular Poisson algebra such that its Poisson cohomology ring is isomorphic as a graded commutative algebra to the Hochschild cohomology ring of a natural deformation. Namely for two integers a, b ≥ 2 we consider the truncated polynomial algebra Λ(a, b) := C[X, Y ]/(X a , Y b ) with the Poisson bracket given by {X, Y } = XY . This algebra is the semi-classical limit of the quantum complete intersection Λ q (a, b) studied by Bergh and Erdmann in [5]. The homological properties of this class of noncommutative finite-dimensional algebras have been extensively studied recently (see for instance [8,4,5,6]). In particular it is proved in [8] that the generic quantum exterior algebra (corresponding to the case a = b = 2) provides a counter-example to Happel's question. That is, the global dimension of Λ q (2, 2) is infinite whereas its Hochschild cohomology groups vanish in degree greater than 2. Recently the Hochschild cohomology ring of Λ q (a, b) has been described in [5] when q is not a root of unity. It is a five-dimensional graded algebra isomorphic to the fibre product F := C[U ]/(U 2 ) × C C V, W /(V 2 , V W + W V, W 2 ), with U in degree zero and V and W in degree one. In this paper we obtain the same description for the Poisson cohomology ring of Λ(a, b), giving rise to Theorem 1.1. For all a, b ≥ 2 one has HP * (Λ(a, b)) ≃ F ≃ HH * (Λ q (a, b)) as graded commutative algebras, where HP * (Λ(a, b)) is endowed with the Poisson cup product induced by the exterior product on skew-symmetric multiderivations.
Note that Λ(a, b) is not unimodular (see for instance [11] for a recent survey on the modular class of a Poisson manifold). Indeed, there is no Poincaré duality between HP k (Λ(a, b)) and HP 2−k (Λ(a, b)). For instance HP 2 (Λ(a, b)) has dimension 1 whereas HP 0 (Λ(a, b)) has dimension a + b − 1. One cannot even expect a twisted Poincaré duality as in [12] (see the paragraph below Lemma 3.1). However, the Nakayama automorphism ν coming from the Frobenius structure of the quantum complete intersection Λ q (a, b) (see [5]) allows us to construct a Poisson module M ν such that

Poisson cohomology of truncated polynomials
The algebra Λ(a, b) has dimension ab, with basis One easily checks that the following formulas hold in Λ(a, b) for all i, j: Recall that the complex computing the Poisson cohomology is (χ k , δ k ), with χ k the space of skew-symmetric k-derivations of Λ(a, b). Interestingly, these spaces vanish for k ≥ 3, since Λ(a, b) is 2-generated. The Poisson coboundary operator δ k : χ k → χ k+1 is defined by It is easy to check that δ k (P ) belongs indeed to χ k+1 and that δ k+1 • δ k = 0. The k th Poisson cohomology space of Λ(a, b), denoted by HP k (Λ(a, b)), is the k th cohomology space of this complex. The Poisson cohomology ring is the space a, b)). Endowed with the cup product induced by the exterior product on χ * := ⊕ k χ k it becomes a graded commutative algebra. We start by describing the spaces HP k (Λ(a, b)) for k = 0, 1, 2 as it is clear that HP k (Λ(a, b)) = 0 for k ≥ 3.
Let λ = λ ij e ij ∈ Λ(a, b), and assume it satisfies {λ, X} = {λ, Y } = 0. Then equations (1) lead successively to λ ij = 0 for all i = a − 1 and all j = 0, and then to λ ij = 0 for all i = 0 and j = b − 1. Hence Any derivation d ∈ χ 1 is uniquely determined by the values of d(X) = λ ij e ij and d(Y ) = λ ′ ij e ij . Moreover, d must satisfy the relations d(X a ) = d(Y b ) = 0, that is X a−1 d(X) = Y b−1 d(Y ) = 0, since d is a derivation. From that one easily deduces that λ 0j = λ ′ i0 = 0 for all i, j. Hence the space χ 1 = Der(A, A) has basis {d ij } ∪ {d ′ ij }, where: 1. for 1 ≤ i < a and 0 ≤ j < b, the derivation d ij is defined by d ij (X) = X i Y j and d ij (Y ) = 0; Identifying the coefficients in front of X i Y j leads to (2) From the construction we get that d 2 = α 1,0 d 1,0 + β 0,1 d ′ 0,1 , so that the images of d 1,0 and d ′ 0,1 span HP 1 (Λ(a, b)). One deduces from (1) that they actually form a basis of HP 1 (Λ(a, b)).
The complex computing the Poisson cohomology is vanishing after χ 2 , so we use the Euler-Poincaré principle to compute the dimension of HP 2 (Λ(a, b)). First note that a skew-symmetric derivation f ∈ χ 2 is determined by f (X ∧Y ) = c ij e ij , with aX a−1 f (X ∧Y ) = bY b−1 f (X ∧Y ) = 0, so that c 0j = c i0 = 0 for all i, j. Thus χ 2 has dimension (a − 1)(b − 1).
Proof. We first prove that HP 2 (Λ(a, b)) has dimension 1. From the Euler-Poincaré principle we get dim(HP 2 (Λ(a, b) . Moreover one has P (XY ) = P (X)Y + P (Y )X, and it results straight from formulas (1) that one cannot have δ 1 (P )(X ∧ Y ) = XY .

(Twisted) Poisson homology
Let M be a right Poisson module over Λ(a, b) (the reader is referred to [12,Section 3.1] for the definition of a Poisson module). We denote by {−, −} M : M ⊗ C Λ(a, b) −→ M its external bracket. Then one defines a chain complex on the Λ(a, b)-modules M ⊗ Λ(a,b) Ω k , where Ω k denotes the so-called Kähler differential k-forms of Λ(a, b), as follows. The boundary operator where we have removed the expressions under the hats in the previous sums and d denotes the exterior differential. The homology of this complex is denoted by HP * (Λ(a, b), M ) and called the Poisson homology of the Poisson algebra Λ(a, b) with values in the Poisson module M . Before computing the Poisson homology spaces for a specific module, we describe the spaces Ω k . By definition Ω k is a Λ(a, b)-module generated by the wedge products of length k of the 1-differential forms dX, dY . In particular Ω k = 0 for k ≥ 3. In the remaining cases the torsion coming from the relations X a = Y b = 0 leads to the following spaces: and Their dimensions are respectively ab, (a − 1)b + b(a − 1), and (a − 1)(b − 1). As the bracket {λ, µ} belongs to the ideal generated by XY for any λ, µ ∈ Λ(a, b), one easily checks that: Lemma 3.1. HP 0 (Λ(a, b)) has dimension a + b − 1.
We may ask now if there is a twisted duality similar to the one obtained in [12]. The Poisson algebra Λ(a, b) is the semi-classical limit of the quantum complete intersection Λ q (a, b) which is the C-algebra generated by x, y with relations xy = qyx, x a = 0, y b = 0 (see [5]). Any diagonal automorphism σ of Λ q (a, b) defined by σ(x) = q α x and σ(y) = q β y gives rise to a Poisson module M σ of Λ(a, b) via a semi-classical limit process (see [12, Section 3.1] for details). As a vector space M σ is equal to Λ(a, b), and the external Poisson bracket is defined by: From these formulas one easily deduces that if α < −b+1 and β > a−1, then HP 0 (Λ(a, b), M σ ) = C. So one might be tempted to use such a Poisson module to restore a twisted Poincaré duality between HP k (Λ(a, b)) and HP 2−k (Λ(a, b), M σ ) as in [12,Theorem 3.4.2]. Unfortunately, these hypotheses on α and β also lead to HP 2 (Λ(a, b), M σ ) = 0, so that HP 2 (Λ(a, b), M σ ) is not isomorphic to HP 0 (Λ(a, b)).