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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 ofC∞_{-functions on} theC∞_{-manifold, the cohomology algebra of Poisson structure is} iso-morphic to an algebra of vertical cohomologies of the foliation corre-sponding to the Poisson structure.

§ 0. Introduction

0.1. _{Let M be a finite-dimensional C}∞_{-manifold. We use the following} notation: ΩK_{(M), k = 1,2, . . ., is the C}∞_{(M)-module of differential} k-form on M; Vk_{(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 manifoldM; Vk_{(M, S), k= 1,2, . . ., is a submodule ofV}k_{(M) consisting}

of the fields tangent to the leaves of the foliationS; Ωk_{(M, S),k= 1,2, . . .,}

is theC∞_{(M)-module of homomorphisms from the moduleV}k_{(M, S) into}

C∞_{(M); Ω}k

s(M), k = 1,2, . . ., is a submodule of Ωk(M) consisting of

k-forms vanishing onVk_{(M, S). Also, we put}

Ω0(M) =V0(M) = Ω0(M, S) =V0(M, S) =C∞(M);

Ω∗(M) = ∞⊕

k=0Ω

k

(M); V∗(M) = ∞⊕

k=0V

k

(M);

Ω∗(M, S) = ∞⊕

k=0Ω

k_{(M, S); V}∗_{(M, S) =} ∞_⊕

k=0V

k_{(M, S); Ω}∗

s(M) =

∞

⊕

k=0Ω

∗

s(M),

where Ω0

s(M) is a subalgebra of C∞(M) consisting of functions constant

along the leaves of the foliationS.

1991Mathematics Subject Classification. 53B50, 70H05.

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

513

0.2. _{The exterior derivation d: Ω}k_{(M)→ Ω}k+1_{(M) carries Ω}k_{s(M) into} Ωk+1

s (M) and thus induces a differential de: Ω

k_(M)/Ωk

s(M)→ Ω

k+1_(M)/

Ωk+1

s (M).

A cohomology of the complex (Ω∗_(M)/Ω∗

s(M),d) is called a relative coho-e

mology of the foliated manifold (M, S). It is a generalization of cohomology of the family of manifold defined in [1]. We denote thepth cohomology space byHp_{(M, S), and the cohomology algebra} ∞_⊕

k=0H

p_{(M, S) byH}∗_{(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 bydsthe operator of exterior derivation

on Ωk_{(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 homomorphismRe: (Ω∗_(M)/Ω∗

s(M),d)e →(Ω∗(M, S), ds) is not an

isomorphism.

If we denote thepth cohomology space of the complex (Ω∗_{(M, S), d}

s) by

Hp

s(M) and the cohomology algebra

∞

⊕

k=0H

p

s(M) byHs∗(M), we can say that,

in general, the algebras H∗_{(M, S) andH}∗

s(M) are not isomorphic though

we have the natural homomorphism [R] :H∗_{(M, S)→H}∗

s(M) induced byR.e

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 mapπinduces the endomorphismπ∗ _{of the algebra Ω}∗_{(M) defined as} (π∗_w)(v

1, . . . , vk) = w(πv1, . . . , πvk). 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), dv) the algebra of vertical cohomologies of the foliation Si and

denote it byH∗

v(M) (see [2]).

0.5. _{IfM is a Riemannian manifold, we can define the reverse map of R}e as follows: (R−1_w)(v

1, . . . , vk) =w(πv1, . . . , πvk), and Re−1(w) = [R−1w].

So, the complexes (Ω∗_(M)/Ω∗

s(M),d), (Ωe ∗(M, S), ds), and (Ω∗v(M), ds) are

isomorphic.

For the foliated Riemannian manifold (M, S), three cohomology algebras H∗_{(M, S),H}∗

s(M), andHv∗(M) are isomorphic.

0.6. _{The definition of the complexes (Ω}∗_(M)/Ω∗

s(M),d),e

€

Ωk_{(M, S), d} s



and€Ω∗

v(M), dvcan be generalized as follows: LetLbe aC∞(M

)-submo-dule of V′_{(M), and also be a Lie subalgebra ofV}′_{(M). Let us denote by} Ω∗

L(M) a subalgebra of the exterior algebra Ω∗(M) consisting of the forms

we denote by Ω∗_{(M, L) the algebra of C}∞_{(M)-multilinear antisymmetric} maps fromLk _intoC∞_(M).

The definition of derivations dL : Ω∗(M, L) → Ω∗(M, L) and de :

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

coho-mologies 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 submoduleLisV′_{(M, S).} If there is some projectorπ: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),d),e

€

Ω∗(M, L), dL



, and €Ω∗v(M), dv



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

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

In §1 we introduce the notion of Poisson algebra and define its coho-mologies. 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. _{LetF be a real or complex vector space. For each positive integer k} we denote byAk_{(F) the space of multilinear antisymmetric maps fromF}k

intoF. Also we putA0_{(F) =F andA}∗_{(F) =} ∞_⊕

k=0A

k_(F).

1.2. _{There is a natural structure of the Lie subalgebra onA}′_{(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∈Am_{(F) andβ∈A}n_{(F) is defined as follows (see [4]):}

[α, β](v1, . . . , vm+n−1) =

= 1 m!n!

X

s

sgn(s)€(−1)mn+n_α(β(v

s(1), . . . , vs(n)), vs(n+1), . . .

. . . , vs(m+n−1)+ (−1)mβ€α(vs(1), . . . , vs(n)), vs(n+1), . . . , vs(m+n−1));

also, forv, w∈A0_{(F) =Fwe put [α, v](v}

1,. . ., vm−1) = [v, α](v1, . . . , vm−1) =

1.3. _{It is easy to check that the bracket as defined above satisfies the} axioms of the Lie superalgebra: Forα∈Am_{(F),β ∈A}n_{(F) andγ∈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 onF is an elementµ∈A2_{(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 µe :}

A∗_(F)→A∗_{(F), µ(α) = [µ, α]. It is clear that ife α∈A}k_{(F), then}

e µ(α)∈

Ak+1_{(F). Moreover, the property (b) in 1.3 implies}

e

µ2 = 0, i.e., µe 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 inF (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 ∈ Am_{(F), β ∈}

An_{(F), anda∈A}0_{(F) =F we have}

(αβ)(v1, . . . , vm+n) =

= 1 m!n!

X

s

sgn(s)α(vs(1), . . . , vs(m))β(vs(m+1), . . . , vs(m+n)),

and (aα)(v1, . . . , vm) = (αa)(v1, . . . , vm) =a·α(v1, . . . , vm).

1.7. Definition. _{For every positive integer k we denote by Der}k_(F)

the subspace of such elements α in Ak_{(F) that α(a, a}

1, a2, . . . , ak) =

aα(a1, . . . , ak) +a1α(a, a1, a2, . . . , ak) for every system{a, a1, . . . , ak} ⊂F.

Also, we put Der0(F) =F and Der∗(F) = ∞⊕

k=0Der

k

(F).

We call elements of the space Derk_{(F) k-derivations of the algebra F,}

1.8. _{It is easy to check that the subspace Der}∗_{(F) inA}∗_{(F) is closed} un-der the operation of exterior multiplication defined in 1.6 as well as unun-der the bracket defined in 1.2. In other words, Der∗(F) is an anticommutative algebra and a Lie superalgebra. Moreover, these two structures are con-nected by the following property: For α ∈ Derm(F), β ∈ Dern(F), and γ∈Der∗(F) we have (c) [α, βγ] = [α, β]·γ+ (−1)mn+n_{β·[α, γ].}

1.9. _{Fork= 0,1,2, . . . let∧}k_Der′_{(F) be the subspace of Der}k_{(F) which} consists of elements of the formav1, . . . , vk, wherea∈Fand{v1, . . . , vk} ⊂

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, αm, β1, . . . , βn] =

=X

i,j

(−1)m+i+j−1_[α

i, βj]α1· · ·αbi· · ·αmβ1· · ·βbj· · ·βn,

where {α1, . . . , αm, β1, . . . , βn} ⊂ Der′(F), and [αi, βj] is the commutator

ofαi andβj.

1.10. _{A Poisson structure on the commutative algebraF is an involutive} element (see 1.4) P ∈ Der2(F). The pair (F, P) is said to be a Poisson algebra.

As mentioned in 1.5, an involutive element P ∈ Der2(F) defines the operator with a vanishing squarePe: 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∈ Derm(F) and β ∈Dern(F) we havePe(αβ) =Pe(α)β+ (−1)m_αPe(β).

Such an operator is said to be an antiderivation of degree +1.

Therefore, on the space of cohomologies defined by Pe, we can intro-duce a structure of anticommutative algebra. This cohomology algebra will be called the cohomology of Poisson structure (F, P). We denote by Hk_{(F, P) thekth cohomology space, and byH}∗_{(F, P) the comology algebra}

∞

⊕

k=0H

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_orC_.

The space ofk-linear antisymmetric homomorphisms ofF-modules from (Der′(F))k_{) intoFis denoted byA}k_(Der′_{(F), F),k= 1,2, . . .. It is assumed} thatA0(Der′_{(F), F) =F.}

There is a classical operator of derivation on the exterior algebra A∗_(Der′_{(F), F) =} ∞_⊕

k=0A

2.2. _{LetP be a Poisson structure on the algebraF. For each k∈} N_{, P}

defines the homomorphismPk _:Ak_(Der′_{(F), F)→Der}k

(F) as follows: fora∈A0_(Der′_{(F), F) =F we putP}0_{(a) =a;}

for elements of the formda∈A′_(Der′_{(F), F), wherea∈Fand (da)(X) =} X(a), we put p′_{(da)(b) =P(a, b),b∈F;}

next, forw∈Ak_(Der′_{(F), F),k= 1,2, . . ., we put (P}k_w)(a

1, . . . , ak) =

(−1)k_w(P′_(da

1), . . . , P′(dak)) with every system{a1, . . . , ak} ⊂F.

2.3. _{Let us note some interesting properties ofP}k_{,k= 0,1, . . .: The map}

P∗ ₌ ∞_⊕

k=0P

∗ _{: A}∗_(Der′_{(F), F) → Der}′_{(F) is a homomorphism of exterior}

algebras;

Theorem. _{The composition map P}′_{◦d:F →Der}′_{(F)is a}

homomor-phism of Lie algebras.

Proof. We must prove the identityP′_{(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 forP.

2.4. Theorem. _{The map P}∗ _{is a homomorphism from the complex} (A∗_(Der′_{(F), F), d) into the complex (Der}∗_{(F),Pe), where d is the}

classi-cal derivation and Pe is defined in 1.5 and1.11.

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}

Pn+1_(dw)(a

1, . . . , an+1) = (−1)n+1dw(P′(da1), . . . , P′(dan+1)) =

= (−1)n+1 X

i

(−1)i−1_(P′_(da

i))w€P′(da1), . . . ,Pd′(dai), . . . , P′(dan+1)+

+X

i<j

(−1)i+j_w€_[P′_(da

i), P′(daj)], . . . ,Pd′(dai), . . . ,Pd′(daj), . . .)

‘ =

= (−1)n+1 X

i

(−1)i−1P€aiw

€

P′(da1), . . . ,Pd′(dai), . . . , P′(dan+1)+

+X

i<j

(−1)i+j_w€_[P′_(da

i), P′(daj)], . . . ,Pd′(dai), . . . ,Pd′(daj), . . .‘.

On the other hand,

[P, Pn_(w)](a

1, . . . , an+1=

X

i

(−1)i−1_P€_Pn_(w)(a

1, . . . ,abi, . . . , an+1), ai+

+X

i<j

(−1)i+j−3_(Pn_(w))€_P(a

iaj), . . . ,abi, . . . ,abj, . . .

= (−1)n+1 X

i

(−1)i−1_P€_a

iw€P′(da1), . . . ,Pd′(dai), . . . , P′(dan+1)+

=X

i<j

(−1)i+j_w€_P′_(dP(a

iaj)), . . . ,Pd′(dai), . . . ,Pd′(daj), . . .).

As in the proof of Theorem 2.3, we obtain [P′_(da

i), P′(daj)] =P′(dP(ai, aj)),

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 algebraF into the cohomol-ogy of Poisson structureH∗_{(F, P).}

2.5. _{Further, we consider the case withF =C}∞_{(M), whereM is a} finite-dimensionalC∞_{-manifold. Then Der}k

(F) is the space of contravariant anti-symmetric tensor fields of degreekon the manifoldM andAk_(Der′_{(F), F)} is the space of differential k-forms on M. These spaces are denoted by Vk_{(M) and Ω}k_{(M) , respectively.}

The Poisson structureP onM is a contravariant antisymmetric involu-tive tensor field of degree 2.

P induces a homomorphism P : T∗_{(M) → T(M) from the cotangent} bundle ofM into the tangent bundle ofM. Letβ(P∗

x(α)) = (α∧β)(Px) for

x∈M andα, β∈T∗

x(M).

The set of subspaces {Image(Px ⊂ Tx(M) : x ∈ M} is an integrable

distribution (see [5]). Integral manifolds are called symplectic leaves of the Poisson structureP (see [5], [6]).

Thus we have a foliationFpwith different-dimensional leaves induced by

the Poisson structureP. Now we use the generalization of the cohomology from in 0.6, associated with a submodule on the Lie subalgebraL⊂V′_(M). LetLbe the set of vector fields on the manifoldM, tangent to the leaves of the foliationFp.

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}k

L(M) if

and only ifw(P′_(dϕ

1), . . . , P′(dϕk)) = 0 for every system {ϕ1, . . . , ϕk} ⊂

C∞_{(M); this is the same asP}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),d)e

(relative cohomologies)is isomorphic to the cohomology algebra of the com-plex (ImP∗_,Pe_).

2.6. _{The homomorphism of bundlesP : T}∗_{(M)→T(M) induces} homo-morphisms of the associated bundles ∧k_{P : ∧}k_T∗_{(M) → ∧}k_{T(M), k =}

1,2, . . .. We denote by Vk_{(M, P) the subspace of V}k_{(M) consisting of}

such elementsv that vx∈Image (∧kP

∗

V∗_{(M, P) =} ∞_⊕

k=0V

k_{(M, P) is invariant under the action of th e operatorPe}

(see [7]). Hence we have a complex (V∗_{(M, P),P}e_{) and the corresponding} cohomology algebra denoted byh∗_{(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 homomorphismPk

L: Ωk(M, L)→Vk(M, P) for each

k = 0,1, . . . , analogously to the homomorphisms Pk _{: 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: Pk

L(w) = 0 ⇒ (P k

L(w))(ϕ1, . . . , ϕk) = 0 for

every {ϕ1, . . . , ϕk} ⊂ C∞(M) ⇒ w(PL′(dϕ1), . . . , PL′(dϕk)) = 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 identityw= 0.

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2_{(1995), No. 4, 347–359.}

(Received 01.08.1996)

Author’s address: