GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 6, 1998, 513-520

SEVERAL COHOMOLOGY ALGEBRAS CONNECTED WITH THE POISSON STRUCTURE

Z. GIUNASHVILI

Abstract. The structure of a Lie superalgebra is defined on the space of multiderivations of a commutative algebra. This structure is used to define some cohomology algebra of Poisson structure. It is shown that when a commutative algebra is an algebra of C^{∞}-functions on the C^{∞}-manifold, the cohomology algebra of Poisson structure is isomorphic to an algebra of vertical cohomologies of the foliation corresponding to the Poisson structure.

§ 0. Introduction

0.1. Let M be a finite-dimensional C^{∞}-manifold. We use the following notation: Ω^{K}(M), k = 1, 2, …, is the C^{∞}(M)-module of differential kform on M; V^{k}(M), k = 1, 2, …, is the C^{∞}(M)-module of contravariant antisymmetric tensor fields of degree k on M; S is some foliation on the manifold M; V^{k}(M, S), k = 1, 2, …, is a submodule of V^{k}(M) consisting of the fields tangent to the leaves of the foliation S; Ω^{k}(M, S), k = 1, 2, …, is the C^{∞}(M)-module of homomorphisms from the module V^{k}(M, S) into C^{∞}(M); Ω_{s}^{k}(M), k = 1, 2, …, is a submodule of Ω^{k}(M) consisting of kforms vanishing on V^{k}(M, S). Also, we put

Ω^{0}(M) = V^{0}(M) = Ω^{0}(M, S) = V^{0}(M, S) = C^{∞}(M);

Ω^{∗}(M) = ⊕_{k=0}^{∞}Ω^{k}(M); V^{∗}(M) = ⊕_{k=0}^{∞}V^{k}(M);

Ω^{∗}(M, S)= ⊕_{k=0}^{∞}Ω^{k}(M, S); V^{∗}(M, S)= ⊕_{k=0}^{∞}V^{k}(M, S); Ω_{s}^{∗}(M)= ⊕_{k=0}^{∞}Ω_{s}^{∗}(M),

where Ω_{s}^{0}(M) is a subalgebra of C^{∞}(M) consisting of functions constant along the leaves of the foliation S.

513 0.2. The exterior derivation d : Ω^{k}(M) → Ω^{k+1}(M) carries Ω_{s}^{k}(M) into Ω_{s}^{k+1}(M) and thus induces a differential d̃: Ω^{k}(M)/Ω_{s}^{k}(M) → Ω^{k+1}(M)/ Ω_{s}^{k+1}(M).

1991 Mathematics Subject Classification. 53B50, 70H05.

Key words and phrases. Lie superalgebra, Poisson structure, vertical cohomology, cohomology of Poisson structure.

1072-947X/98/1100-0513$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

A cohomology of the complex (Ω^{∗}(M)/Ω_{s}^{∗}(M), d̃) is called a relative cohomology of the foliated manifold (M, S). It is a generalization of cohomology of the family of manifold defined in [1]. We denote the pth cohomology space by H^{p}(M, S), and the cohomology algebra ⊕_{k=0}^{∞}H^{p}(M, S) by H^{∗}(M, S).

0.3. Let R : Ω^{∗}(M) → Ω^{∗}(M, S) be a restriction map. It is clear that Kernel (R) = Ω_{s}^{∗}(M). If we denote by d_{s} the operator of exterior derivation on Ω^{k}(M, S), then it can be said that the map R is a homomorphism of the complex (Ω^{∗}(M), d) into the complex (Ω^{∗}(M, S), d_{s}). In general, the homomorphism R is not an epimorphism, and therefore, in general, the induced homomorphism R̃ : (Ω^{∗}(M)/Ω_{s}^{∗}(M), d̃) → (Ω^{∗}(M, S), d_{s}) is not an isomorphism.

If we denote the pth cohomology space of the complex (Ω^{∗}(M, S), d_{s}) by H_{s}^{p}(M) and the cohomology algebra ⊕_{k=0}^{∞}H_{s}^{p}(M) by H_{s}^{∗}(M), we can say that,

in general, the algebras H^{∗}(M, S) and H_{s}^{∗}(M) are not isomorphic though we have the natural homomorphism [R]:H^{∗}(M, S)→H_{s}^{∗}(M) induced by R̃.

0.4. In the case where the manifold M is provided with a Riemannian metric, we have the map of orthogonal projection π : V′(M) → V′(M, S). The map π induces the endomorphism π^{∗} of the algebra Ω^{∗}(M) defined as (π^{∗}w)(v_{1}, …, v_{k}) = w(πv_{1}, …, πv_{k}). It is clear that π^{∗} is the projection π^{∗} ∘ π^{∗} = π^{∗}. We denote the subalgebra Image(π^{∗}) by Ω_{v}^{∗}(M) and call its elements vertical differential forms on the foliated manifold (M, S) (see [2]).

It is easy to check that the operator π^{∗} ∘ d ≡ d_{v} : Ω_{v}^{∗}(M) → Ω_{v}^{∗}(M) is a coboundary operator, and we call the cohomology algebra of the complex (Ω_{v}^{∗}(M), d_{v}) the algebra of vertical cohomologies of the foliation S_{i} and denote it by H_{v}^{∗}(M) (see [2]).

0.5. If M is a Riemannian manifold, we can define the reverse map of R̃ as follows: (R^{−1}w)(v_{1}, …, v_{k}) = w(πv_{1}, …, πv_{k}), and R̃^{−1}(w) = [R^{−1}w]. So, the complexes (Ω^{∗}(M)/Ω_{s}^{∗}(M), d̃), (Ω^{∗}(M, S), d_{s}), and (Ω_{v}^{∗}(M), d_{s}) are isomorphic.

For the foliated Riemannian manifold (M, S), three cohomology algebras H^{∗}(M, S), H_{s}^{∗}(M), and H_{v}^{∗}(M) are isomorphic.

0.6. The definition of the complexes (Ω^{∗}(M)/Ω_{s}^{∗}(M), d̃), (Ω^{k}(M, S), d_{s}) and (Ω_{v}^{∗}(M), d_{v}) can be generalized as follows: Let L be a C^{∞}(M)-submo-dule of V′(M), and also be a Lie subalgebra of V′(M). Let us denote by Ω_{L}^{∗}(M) a subalgebra of the exterior algebra Ω^{∗}(M) consisting of the forms w such that w(u_{1}, …, u_{n}) = 0 for every system {u_{1}, …, u_{n}} ⊂ L. Further, we denote by Ω^{∗}(M, L) the algebra of C^{∞}(M)-multilinear antisymmetric maps from L^{k} into C^{∞}(M).

The definition of derivations d_{L} : Ω^{∗}(M, L) → Ω^{∗}(M, L) and d̃ : Ω^{∗}(M)/Ω_{L}^{∗}(M) → Ω^{∗}(M)/Ω_{L}^{∗}(M) is clear.

Indeed, the cohomologies of the complex (Ω^{∗}(M, L), d_{L}) are the cohomologies of the Lie algebra L, with coefficients in C^{∞}(M), denoted by H^{∗}(L, C^{∞}(M)) (see [3]).

In the cases considered in 0.1 – 0.5, the submodule L is V′(M, S).

If there is some projector π : V′(M) → V′(M) with Image (π) = L, then the algebra of vertical cohomologies can be defined as in 0.4. The proof of the fact that the cohomologies of the complexes

(Ω^{∗}(M)/Ω_{L}^{∗}(M), d̃), (Ω^{∗}(M, L), d_{L}), and (Ω_{v}^{∗}(M), d_{v})

are isomorphic is analogous to the proof of the thereom in 0.5.

We use the above-described generalization in §2 in considering a cohomology of the Poisson structure.

In §1 we introduce the notion of Poisson algebra and define its cohomologies. We also describe here some algebraic constructions which help us to arrange a connection between the cohomologies defined in §0 and the cohomologies of the Poisson structure.

§ 1. Lie Superalgebra Structure on the Space of Multiderivations of a Commutative Algebra. The Poisson Algebra

1.1. Let F be a real or complex vector space. For each positive integer k we denote by A^{k}(F) the space of multilinear antisymmetric maps from F^{k} into F. Also we put A^{0}(F) = F and A^{∗}(F) = ⊕_{k=0}^{∞}A^{k}(F).

1.2. There is a natural structure of the Lie subalgebra on A′(F) defined by the commutator. It might be defined as a structure of the Lie subalgebra on A^{∗}(F). The supercommutator [α, β] ∈ A^{m+n−1}(F) of two elements α ∈ A^{m}(F) and β ∈ A^{n}(F) is defined as follows (see [4]):

[α, β](v_{1}, …, v_{m+n−1}) =

= \frac{1}{m! n!} ∑_{s} sgn(s)((−1)^{mn+n}α(β(v_{s(1)}, …, v_{s(n)}), v_{s(n+1)}, …

…, v_{s(m+n−1)}) + (−1)^{m}β(α(v_{s(1)}, …, v_{s(n)}), v_{s(n+1)}, …, v_{s(m+n−1)}));

also, for v, w ∈A^{0}(F)=F we put [α, v](v_{1},…, v_{m−1})=[v, α](v_{1}, …, v_{m−1}) = α(v, v_{1}, …, v_{m−1}) and [v, w] = 0.

1.3. It is easy to check that the bracket as defined above satisfies the axioms of the Lie superalgebra: For α ∈ A^{m}(F), β ∈ A^{n}(F) and γ ∈ A^{k}(F) we have (a) [α, β] = (−1)^{mn}[β, α]; (b) (−1)^{mk}[[α, β], γ]+(−1)^{mn}[[β, γ], α]+ (−1)^{nk}[[γ, α], β] = 0.

1.4. One classical notion that can be translated into the language of the bracket defined in A^{∗}(F) is the notion of “a Lie algebra structure on F ”. A structure of the Lie algebra on F is an element μ ∈ A^{2}(F) satisfying the condition [μ, μ] = 0. The latter is equivalent to the Jacobi identity

μ(μ(a, b), c) + μ(μ(b, c), d) + μ(μ(c, a), b).

We call such an element an involutive element.

1.5. An involutive element μ ∈ A^{2}(F) defines the linear operator μ̃ : A^{∗}(F) → A^{∗}(F), μ̃(α) = [μ, α]. It is clear that if α ∈ A^{k}(F), then μ̃(α) ∈ A^{k+1}(F). Moreover, the property (b) in 1.3 implies μ^{2} = 0, i.e., μ̃ is a coboundary operator and therefore defines some space of cohomologies. As a matter of fact, it is the Chevalley–Eilenberg cohomology of the Lie algebra F with coefficients in F (see [4]).

1.6. Further we shall consider only the case with F as a commutative algebra over the field of real or complex numbers.

In that case, the space A^{∗}(F) has a structure of the anticommutative (exterior) algebra defined by the classical formula: For α ∈ A^{m}(F), β ∈ A^{n}(F), and a ∈ A^{0}(F) = F we have

(αβ)(v_{1}, …, v_{m+n}) =

= \frac{1}{m! n!} ∑_{s} sgn(s)α(v_{s(1)}, …, v_{s(m)})β(v_{s(m+1)}, …, v_{s(m+n)}),

and (aα)(v_{1}, …, v_{m}) = (αa)(v_{1}, …, v_{m}) = a · α(v_{1}, …, v_{m}).

1.7. Definition. For every positive integer k we denote by Der^{k}(F) the subspace of such elements α in A^{k}(F) that α(a, a_{1}, a_{2}, …, a_{k}) = aα(a_{1}, …, a_{k}) + a_{1}α(a, a_{1}, a_{2}, …, a_{k}) for every system {a, a_{1}, …, a_{k}} ⊂ F.

Also, we put Der^{0}(F) = F and Der^{∗}(F) = ⊕_{k=0}^{∞} Der^{k}(F).

We call elements of the space Der^{k}(F) k-derivations of the algebra F, and elements of Der^{∗}(F) multiderivations.

1.8. It is easy to check that the subspace Der^{∗}(F) in A^{∗}(F) is closed under the operation of exterior multiplication defined in 1.6 as well as under the bracket defined in 1.2. In other words, Der^{∗}(F) is an anticommutative algebra and a Lie superalgebra. Moreover, these two structures are connected by the following property: For α ∈ Der^{m}(F), β ∈ Der^{n}(F), and γ ∈ Der^{∗}(F) we have (c) [α, βγ] = [α, β] · γ + (−1)^{mn+n}β · [α, γ].

1.9. For k = 0, 1, 2, … let ∧^{k} Der′(F) be the subspace of Der^{k}(F) which consists of elements of the form av_{1}, …, v_{k}, where a ∈ F and {v_{1}, …, v_{k}} ⊂ Der′(F). The subalgebra ∧^{∗} Der′(F) = ⊕_{k=0}^{∞} ∧^{k} Der′(F) in Der^{∗}(F) is closed

under the bracket [ , ] which has a more explicit form on the elements of the algebra

∧^{∗} Der′(F) : [α_{1}, α_{m}, β_{1}, …, β_{n}] =

= ∑_{i,j}(−1)^{m+i+j−1}[α_{i}, β_{j}]α_{1} ⋯ α̂_{i} ⋯ α_{m}β_{1} ⋯ β̂_{j} ⋯ β_{n},

where {α_{1}, …, α_{m}, β_{1}, …, β_{n}} ⊂ Der′(F), and [α_{i}, β_{j}] is the commutator of α_{i} and β_{j}.

1.10. A Poisson structure on the commutative algebra F is an involutive element (see 1.4) P ∈ Der^{2}(F). The pair (F, P) is said to be a Poisson algebra.

As mentioned in 1.5, an involutive element P ∈ Der^{2}(F) defines the operator with a vanishing square P̃ : Der^{∗}(F) → Der^{∗}(F). By virtue of the property (c) of the bracket in Der^{∗}(F) (see 1.8) it is easy to check that for α ∈ Der^{m}(F) and β ∈ Der^{n}(F) we have P̃(αβ) = P̃(α)β + (−1)^{m}αP̃(β). Such an operator is said to be an antiderivation of degree +1.

Therefore, on the space of cohomologies defined by P̃, we can introduce a structure of anticommutative algebra. This cohomology algebra will be called the cohomology of Poisson structure (F, P). We denote by H^{k}(F, P) the kth cohomology space, and by H^{∗}(F, P) the comology algebra

⊕_{k=0}^{∞}H^{k}(F, P).

§ 2. Various Cohomology Algebras of a Manifold with Poisson Structure and Their Interconnections

2.1. As in Section 1, F is a commutative algebra over ℝ or ℂ.

The space of k-linear antisymmetric homomorphisms of F -modules from (Der′(F))^{k}) into F is denoted by A^{k}(Der′(F), F), k = 1, 2, . . .. It is assumed that A^{0}(Der′(F), F) = F.

There is a classical operator of derivation on the exterior algebra

A^{∗}(Der′(F), F) = ⊕_{k=0}^{∞}A^{k}(Der′(F), F).

2.2. Let P be a Poisson structure on the algebra F. For each k ∈ ℕ, P defines the homomorphism P^{k} : A^{k}(Der′(F), F) → Der^{k}(F) as follows:

for a ∈ A^{0}(Der′(F), F) = F we put P^{0}(a) = a;

for elements of the form da ∈ A′(Der′(F), F), where a ∈ F and (da)(X) = X(a), we put p′(da)(b) = P (a, b), b ∈ F;

next, for w ∈ A^{k}(Der′(F), F), k = 1, 2, …, we put (P^{k}w)(a_{1}, …, a_{k}) = (−1)^{k}w(P′(da_{1}), …, P′(da_{k})) with every system {a_{1}, …, a_{k}} ⊂ F.

2.3. Let us note some interesting properties of P^{k}, k = 0, 1, …: The map P^{∗} = ⊕_{k=0}^{∞}P^{∗} : A^{∗}(Der′(F), F) → Der′(F) is a homomorphism of exterior

algebras;

Theorem. The composition map P′ ∘ d : F → Der′(F) is a homomorphism of Lie algebras.

Proof. We must prove the identity P′(dP (a, b)) = [P′(da), P′(db)] for each a, b ∈ F. By the definitions of P′ and [ , ] we have P′(dP (a, b))(c) = (P′(da))(P′(db)c) − (P′(db))(P′(da)c) = P (a, P (b, c)) − P (b, P (a, c))). Now the identity we want to prove follows from the Jacobi identity for P.

2.4. Theorem. The map P^{∗} is a homomorphism from the complex (A^{∗}(Der′(F), F), d) into the complex (Der^{∗}(F), P̃), where d is the classical derivation and P̃ is defined in 1.5 and 1.11.

Proof. We must prove the identity P^{∗}(dw) = [P, P^{∗}(w)] for every w ∈ A^{∗}(Der′(F), F), n = 0, 1, . . .. By the definitions we have

P^{n+1}(dw)(a_{1}, …, a_{n+1}) = (−1)^{n+1}dw(P′(da_{1}), …, P′(da_{n+1})) =

= (−1)^{n+1}( ∑_{i}(−1)^{i−1}(P′(da_{i}))w(P′(da_{1}), …, P̂′(da_{i}), …, P′(da_{n+1})) +

+ ∑_{i<j}(−1)^{i+j}w([P′(da_{i}), P′(da_{j})], …, P̂′(da_{i}), …, P̂′(da_{j}), …))) =

= (−1)^{n+1}( ∑_{i}(−1)^{i−1}P (a_{i}w(P′(da_{1}), …, P̂′(da_{i}), …, P′(da_{n+1}))) +

+ ∑_{i<j}(−1)^{i+j}w([P′(da_{i}), P′(da_{j})], …, P̂′(da_{i}), …, P̂′(da_{j}), …)).

On the other hand,

[P, P^{n}(w)](a_{1}, …, a_{n+1} = ∑_{i}(−1)^{i−1}P (P^{n}(w)(a_{1}, …, â_{i}, …, a_{n+1}), a_{i}) +

+ ∑_{i<j}(−1)^{i+j−3}(P^{n}(w))(P (a_{i}a_{j}), …, â_{i}, …, â_{j}, …) =

= (−1)^{n+1}( ∑_{i}(−1)^{i−1}P (a_{i}w(P′(da_{1}), …, P̂′(da_{i}), …, P′(da_{n+1}))) +

= ∑_{i<j}(−1)^{i+j}w(P′(dP (a_{i}a_{j})), …, P̂′(da_{i}), …, P̂′(da_{j}), …)).

As in the proof of Theorem 2.3, we obtain [P′(da_{i}), P′(da_{j})]=P′(dP (a_{i}, a_{j})), which completes the proof of the theorem.

As a consequence, P^{∗} defines a homomorphism from the cohomology of the Lie algebra Der′(F) with coefficients in the algebra F into the cohomology of Poisson structure H^{∗}(F, P).

2.5. Further, we consider the case with F = C^{∞}(M), where M is a finitedimensional C^{∞}-manifold. Then Der^{k}(F) is the space of contravariant antisymmetric tensor fields of degree k on the manifold M and A^{k}(Der′(F), F) is the space of differential k-forms on M. These spaces are denoted by V^{k}(M) and Ω^{k}(M) , respectively.

The Poisson structure P on M is a contravariant antisymmetric involutive tensor field of degree 2.

P induces a homomorphism \overline{P} : T^{∗}(M) → T (M) from the cotangent bundle of M into the tangent bundle of M. Let β(P_{x}^{∗}(α)) = (α ∧ β)(P_{x}) for x ∈ M and α, β ∈ T_{x}^{∗}(M).

The set of subspaces \{Image(\overline{P}_{x} ⊂ T_{x}(M) : x ∈ M\} is an integrable distribution (see [5]). Integral manifolds are called symplectic leaves of the Poisson structure P (see [5], [6]).

Thus we have a foliation F_{p} with different-dimensional leaves induced by the Poisson structure P. Now we use the generalization of the cohomology from in 0.6, associated with a submodule on the Lie subalgebra L ⊂ V′(M).

Let L be the set of vector fields on the manifold M, tangent to the leaves of the foliation F_{p}.

Since L is a submodule of V′(M) generated by elements of the form P′(dφ), where φ ∈ C^{∞}(M), it is clear that w is an element of Ω_{L}^{k}(M) if and only if w(P′(dφ_{1}), …, P′(dφ_{k})) = 0 for every system {φ_{1}, …, φ_{k}} ⊂ C^{∞}(M); this is the same as P^{k}(w) = 0. So we have Ω_{L}^{∗}(M) = Kernel(P^{∗}).

The consequence of the above result can be formulated as

Theorem. The cohomology algebra of the complex (Ω^{∗}(M)/Ω_{L}^{∗}(M), d̃) (relative cohomologies) is isomorphic to the cohomology algebra of the complex (Im P^{∗}, P̃).

2.6. The homomorphism of bundles \overline{P} : T^{∗}(M) → T (M) induces homomorphisms of the associated bundles ∧^{k}\overline{P} : ∧^{k}T^{∗}(M) → ∧^{k}T (M), k = 1, 2, . . .. We denote by V^{k}(M, P) the subspace of V^{k}(M) consisting of such elements v that v_{x} ∈ Image (∧^{k}\overline{P} _{x}^{∗}) for every x ∈ M. The subalgebra V^{∗}(M, P) = ⊕_{k=0}^{∞}V^{k}(M, P) is invariant under the action of th e operator P̃

(see [7]). Hence we have a complex (V^{∗}(M, P), P̃) and the corresponding cohomology algebra denoted by h^{∗}(M, P) (see [7]).

Theorem. The cohomology of the Lie algebra L with coefficients in C^{∞}(M) (in other words, the cohomology of the complex (Ω^{∗}(M, L), d_{L}) (see 0.6)) is isomorphic to h^{∗}(M, P).

Proof. We construct a homomorphism P_{L}^{k} : Ω^{k}(M, L) → V^{k}(M, P) for each k = 0, 1, …, analogously to the homomorphisms P^{k} : Ω^{k}(M) → V^{k}(M) defined in 2.2. To prove that it is an isomorphism, it is sufficient to show that it is a monomorphism: P_{L}^{k}(w) = 0 ⇒ (P_{L}^{k}(w))(φ_{1}, …, φ_{k}) = 0 for every {φ_{1}, …, φ_{k}} ⊂ C^{∞}(M) ⇒ w(P′_{L}(dφ_{1}), …, P′_{L}(dφ_{k})) = 0. Since L is a module generated by elements of the form P′_{L}(dφ), φ ∈ C^{∞}(M), the above identity is equivalent to the identity w = 0.

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(Received 01.08.1996)

Author’s address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, Aleksidze St., Tbilisi 390093

Georgia