Computations of Nambu-Poisson Cohomologies:

42(2006), 323–359

Computations of Nambu-Poisson Cohomologies:

Case of Nambu-Poisson Tensors of Order 3 on R

By

NobutadaNakanishi^∗

Abstract

We compute Nambu-Poisson cohomology for Nambu-Poisson tensors of order three which are defined onR⁴. In particular, we prove that Nambu-Poisson cohomol- ogy of exact Nambu-Poisson tensors is equivalent to relative cohomology.

§1. Introduction

A Nambu-Poisson structure was given by L. Takhtajan [14] in 1994 in order to extend Nambu mechanics defined onR³to Nambu-Poisson mechanics defined on an n-dimensional manifold, n ≥ 3. One of the main objects of Nambu- Poisson geometry is to study Nambu-Poisson cohomology and its related topics.

The notion of Nambu-Poisson cohomology was first introduced by R. Ib´a˜nez et al. [7], and it is an extension of Poisson cohomology (or Lichnerowicz-Poisson cohomology) on a Poisson manifold. Let (M, η) be anm-dimensional Nambu- Poisson manifold. (See Definition 2.1 for the precise definition.) Whenever we mention a Nambu-Poisson manifold, m is assumed to bem ≥ 3. Then a Nambu-Poisson tensorη defines the so-calledcharacteristic foliation, which is, in general, a singular foliation onM. In case thatηis a Nambu-Poisson tensor, then the set of Hamiltonian vector fields becomes a Lie subalgebra of χ(M),

Communicated by K. Saito. Received February 2004. Revised July 29, 2004, December 9, 2004.

2000 Mathematics Subject Classification(s): 53D, 17.

Key words: Nambu-Poisson tensor, Nambu-Poisson cohomology, relative cohomology.

∗Department of Mathematics, Gifu Keizai University, 5-50 Kitagata, Ogaki-city, Gifu, 503-8550, Japan.

e-mail: nakanisi@gifu-keizai.ac.jp

the Lie algebra of all vector fields on M. This Lie subalgebra will be denoted byH.

Let Ω^k(M) be the space ofk-forms onM,and let the order ofηben. (i.e.

η∈Γ(ΛⁿT M),whereΓ(ΛⁿT M) is the space of cross-sectionsM −→ΛⁿT M.) Herem≥n≥3,andn≥k. We define a mapping

_k : Ω^k(M)−→Γ(Λ^n−kT M)

byk(α) = i(α)η forα∈Ω^k(M). If k=n−1, Ωⁿ⁻¹(M) has a structure of Leibniz algebra, which is defined by

{α, β}=L_n−1(α)β+ (−1)ⁿn(dα)β, α, β∈Ωⁿ⁻¹(M),

whereLstands for the Lie derivative. The image of_n−1,which is denoted byg, becomes a Lie subalgebra of χ(M). (See Proposition 3.1 and its explanation.) It is clear thatHis contained ing. Nambu-Poisson cohomology is a cohomology group of a Lie algebraghavingC^∞(M,R) as its representation space, which is also called Chevalley-Eilenberg cohomology of g. It will be denoted byH_{N P}^∗ . It is easy to see that H_{N P}⁰ is equal to the space of invariant functions of g.

MoreoverH_{N P}¹ is deeply related to themodular class of (M, η) [7]. It will be expected that other cohomologiesH_{N P}^∗ have also some geometric meanings.

Ifηdoes not vanish anywhere onM, it is said to beregular. Then R. Ib´a˜nez et al. computed Nambu-Poisson cohomology of a regular Nambu-Poisson man- ifold (M, η) [7]. If η has some singularities, it is quite difficult to compute its Nambu-Poisson cohomology. As an example of a singular Nambu-Poisson man- ifold, they also considered (R³, η= (x²+y²+z²)_∂x^∂ ∧_∂y^∂ ∧_∂z^∂ ),and they proved that the first Nambu-Poisson cohomology groupH_{N P}¹ (R³, η) is isomorphic toR. On the other hand, P. Monnier [9] computed Nambu-Poisson cohomology for germs at 0 of n-vectors η = f_∂x^∂

1 ∧ · · · ∧ _∂x^∂_n on Kⁿ(K =R or C), with the assumption thatf is a quasihomogeneous polynomial of finite codimension.

His results contain the result of R. Ib´a˜nezet al., (at least in the formal case).

As the next step, it is natural to consider the case that the order of a Nambu-Poisson tensor η is smaller than the dimension of a space on whichη is defined. In the present paper, along this concept, we will compute Nambu- Poisson cohomology for the following three cases.

(a) Exact Nambu-Poisson tensors η of order 3 defined on R⁴(x, y, z, u). A Nambu-Poisson tensor η is called exact if there is a functionf such that i(η)Ω =df for Ω =dx∧dy∧dz∧du.

(b) Linear Nambu-Poisson tensors of order 3 defined onR⁴(x, y, z, u).

(c) A quadratic Nambu-Poisson tensorη = (x²+y²+z²+u²)_∂x^∂ ∧_∂y^∂ ∧_∂z^∂ of order 3 defined onR⁴(x, y, z, u).

The computation for the case (a) naturally leads us to the notion ofrelative cohomologywhich was studied by C. A. Roche [13]. In this case, we know that H_{N P}^k =H_rel^k for 0≤k≤2. In computing Nambu-Poisson cohomology of the case (b), we will use the classification theorem of linear Nambu-Poisson tensors which was proved by J-P. Dufour and N. T. Zung [3]. A part of this case is also discussed in (a). In treating the case (c), we will take advantage of the results of P. Monnier [9].

Here we computed Nambu-Poisson cohomology only for the case (R⁴, η), where the order ofη is three. But it is not so difficult to extend all the results we have obtained here to more general situations. In fact let us consider a Nambu-Poisson manifold (Rⁿ, η),where the order ofηisn. We can easily see that if n−n > 1, then spaces of cohomologies are, in general, greater than those of the case n−n = 1. This is because that the space of g-invariant functions becomes greater ifn−n>1.

The author would like to express his deep thanks to Professors T.Fukuda, H. Sato, Y. Agaoka and G. Ishikawa for helpful and stimulating discussions.

He is also grateful to P. Monnier for valuable communications.

§2. Reviews of Nambu-Poisson Manifolds

We will review some useful results of geometry of Nambu-Poisson man- ifolds. Details are referred to [7],[10] and [14]. Let M be an m-dimensional C^∞-manifold, andF its algebra of real valuedC^∞-functions onM. We denote by Γ(ΛⁿT M) the space of global cross-sections η : M −→ ΛⁿT M. Then for eachη∈Γ(ΛⁿT M),there corresponds the bracket defined by

{f₁, ..., f_n}=η(df₁, ..., df_n), f₁, ..., f_n ∈ F.

This bracket operation is ann-linear skew-symmetric map fromFⁿ to F which satisfies the Leibniz rule:

{f₁, ..., f_n−1, g₁·g₂}={f₁, ..., f_n−1, g₁} ·g₂+g₁· {f₁, ..., f_n−1, g₂}, for allf1, ..., fn−1, g1, g2∈ F.

LetA=

f_i1∧ · · · ∧f_in−1, f_ij ∈ F. Since the bracket operation satisfies the Leibniz rule, we can define a vector field X_A corresponding to A by the following equation:

XA(g) =

{fi1, ..., fin−1, g}, g∈ F.

Such a vector field is called a Hamiltonian vector field. The space of Hamilto- nian vector fields is denoted byH.

Definition 2.1. η ∈ Γ(ΛⁿT M) is called a Nambu-Poisson tensor of order nif it satisfies LX_Aη= 0 for all XA ∈ H, whereL is the Lie derivative.

Then a Nambu-Poisson manifold is a pair (M, η).

Let η(p)= 0, p ∈M. Then we say that η isregular at p. Now we can state the following local structure theorem for Nambu-Poisson tensors [5],[10].

Theorem 2.1. Let η ∈ Γ(ΛⁿT M), n ≥ 3. If η is a Nambu-Poisson tensor of order n,then for any regular point p,there exists a coordinate neigh- borhood U with local coordinates (x₁, ..., x_n, x_n+1, ..., x_m) around p such that

η= ∂

∂x₁ ∧ · · · ∧ ∂

∂x_n on U,and vice versa.

Let (M, η) be a Nambu-Poisson manifold with volume form Ω, and m ≥ n≥3. Put ω =i(η)Ω, where the right hand side is the interior product ofη and Ω. Hence ω is an (m−n)-form. The following theorem gives a necessary and sufficient condition for η to be a Nambu-Poisson tensor. For the proof, see [11].

Theorem 2.2. Let η∈Γ(ΛⁿT M). Thenη is a Nambu-Poisson tensor if and only ifηsatisfies the following two conditions around each regular point:

(a) ω is(locally)decomposable,and

(b) there exists a locally defined1-formθ such thatdω=θ∧ω.

§3. Nambu-Poisson Cohomology

Let (M, η) be a Nambu-Poisson manifold of ordernand letkbe an inte- ger with k ≤n. Denote by Ω^k(M) the space of k-forms on M. If Λ^k(T^∗M) (respectively, Λ^n−k(T M)) denotes the vector bundle of the k-forms (respec- tively, (n−k)-vectors) then η induces a homomorphism of vector bundles k:Λ^k(T^∗M)→Λ^n−k(T M) by defining

k(β) =i(β)η(x)

forβ ∈Λ^k(T_x^∗M) andx∈M, wherei(β) is the contraction by β. Denote also by_k the homomorphism of F-modules from the space Ω^k(M) into the space Γ(Λ^n−kT M) given by

_k(α)(x) =_k(α(x)) for allα∈Ω^k(M) andx∈M.

Next we define a Leibniz algebra structure on Ωⁿ⁻¹(M). The Leibniz algebra on Ωⁿ⁻¹(M) attached to M is the bracket of (n−1)-forms {,} : Ωⁿ⁻¹(M)×Ωⁿ⁻¹(M)→Ωⁿ⁻¹(M) defined by

{α, β}=L_n−1(α)β+ (−1)ⁿ_n(dα)β for allα, β∈Ωⁿ⁻¹(M). In particular, we have that

n−1({α, β}) = [n−1(α), n−1(β)]

for allα, β∈Ωⁿ⁻¹(M).

Using Theorem 2.1, the following proposition was proved by R. Ib´a˜nez et al. [7].

Proposition 3.1. Let (M, η) be an m-dimensional Nambu-Poisson manifold of ordern,withn≥3. Then the center of the algebra(Ωⁿ⁻¹(M),{,}) is theF-module

kern−1={α∈Ωⁿ⁻¹(M) | n−1(α) = 0}.

By the above proposition, we know that Ωⁿ⁻¹(M)/ker_n−1 is isomorphic to a Lie subalgebra ofχ(M). This Lie algebra is often denoted byg. AndF is a (Ωⁿ⁻¹(M)/kern−1)-module relative to the representation:

Ωⁿ⁻¹(M)/ker_n−1× F −→ F, ([α], f)→[α]f = (_n−1(α))(f).

According to [7], one can definethe skew symmetric-cochain complex

C^∗(Ωⁿ⁻¹(M)/kern−1;F) =

k

C^k(Ωⁿ⁻¹(M)/kern−1;F), ∂

where the space of thek-cochainsC^k(Ωⁿ⁻¹(M)/kern−1;F) consists of skew- symmetricF-linear mappings

c^k : (Ωⁿ⁻¹(M)/kern−1)× · · · ×(Ωⁿ⁻¹(M)/kern−1)→ F

and the coboundary operator∂is given by

∂c^k([α0], ...,[αk]) = k i=0

(−1)ⁱ(n−1(αi))(c^k([α0], ...,[αi], ...,[αk]))

+

0≤i<j≤k

(−1)^i+jc^k([{αi, αj}],[α0], ...,[αi], ...,[αj], ...,[αk])

for allc^k ∈C^k(Ωⁿ⁻¹(M)/ker_n−1;F), and [α₀], ...,[α_k]∈Ωⁿ⁻¹(M)/ker_n−1. Then we have ∂◦∂ = 0. The cohomology of this complex is called Nambu- Poisson cohomologyand denoted byH_{N P}^∗ (M, η).

Remark3.1. Since a Nambu-Poisson tensor η satisfies [η, η] = 0 (Schouten bracket), we can define three cohomology spaces H_η⁰(M), H_η¹(M) andH_η²(M) as in the case of usual Poisson manifolds. We see that these three spaces appear as parts of Nambu-Poisson cohomology spaces. (See [9].)

The first attempt at the computation of singular Nambu-Poisson coho- mology was carried out by R. Ib´a˜nezet al. In [7], they considered a Nambu- Poisson manifold {R³, η = (x²+y²+z²)_∂x^∂ ∧ _∂y^∂ ∧_∂z^∂ }. They obtained that H_{N P}¹ (R³, η)∼=R.

In [9], P. Monnier studied Nambu-Poisson cohomology from slightly more general viewpoint, which includes the case of R. Ib´a˜nezet al. [7]. That is to say, he computed Nambu-Poisson cohomology of Nambu-Poisson manifolds of the form (Rⁿ, η=f_∂x^∂

1 ∧ · · · ∧_∂x^∂_n),wheref is a quasihomogeneouspolynomial of finite codimension. Using his results, we compute Nambu-Poisson cohomology of (R⁴, η= (x²+y²+z²+u²)_∂x^∂ ∧_∂y^∂ ∧_∂z^∂ ) in the last section.

§4. Computation of Nambu-Poisson Cohomology: Exact Case

§4.1. Notation and general remarks

LetF be the space ofC^∞-functions onR⁴. Throughout this section, we suppose thatF fsatisfiesf(0) = 0,and is of finite codimension, which means that F/f (f is the ideal spanned by fx, fy, fz, fu) is a finite dimensional vector space. Here we simply write, for example,fx for ^∂f_∂x.

Letη be a Nambu-Poisson tensor of order 3 onR⁴(x, y, z, u). η is said to beexactifηsatisfiesi(η)Ω =df,where Ω =dx∧dy∧dz∧du. Thenηis written as follows.

η=−f_x ∂

∂y∧ ∂

∂z∧ ∂

∂u+f_y ∂

∂x∧ ∂

∂z∧ ∂

∂u−f_z ∂

∂x∧ ∂

∂y∧ ∂

∂u+f_u ∂

∂x∧ ∂

∂y∧ ∂

∂z.

A Lie subalgebrag=₂(Ω²(R⁴)) ofχ(R⁴) is spanned over F by the following six vector fields.

X1=fx ∂

∂y −fy ∂

∂x, X2=fx ∂

∂z−fz ∂

∂x, X3=fx ∂

∂u−fu ∂

∂x, X₄=f_y∂z^∂ −f_z∂y^∂ , X₅=f_y∂u^∂ −f_u∂y^∂ , X₆=f_z∂u^∂ −f_u∂z^∂ . It is easy to see thatΛ⁴g= 0. HenceH_{N P}^k = 0,fork≥4.

§4.2. Relative cohomology

In this subsection, we show that Nambu-Poisson cohomology of exact Nambu-Poisson structure is equivalent torelative cohomologywhich was studied by C. A. Roche [13].

In the first half of this subsection, all objects are considered onR^s. And we simply write Ω^k for Ω^k(R^s). Suppose that C^∞(R^s)f satisfies f(0) = 0 and is of finite codimension. That is to say, an ideal generated by coefficients ofdf is of finite codimension inC^∞(R^s).

First note thatdf∧Ω^k is compatible with the exterior differentiald: i.e., d(df∧Ω^k−1)⊂df∧Ω^k. Hence the linear mapping

drel : Ω^k/df∧Ω^k−1−→Ω^k+1/df∧Ω^k is well-defined.

Definition 4.1. The following sequence defined onR^sis called relative complex off.

0−→Ω⁰−→^drel Ω¹/df ∧Ω⁰−→^drel Ω²/df∧Ω¹−→ · · ·^drel −→^drel Ω^s/df∧Ω^s−1−→0.

The cohomology of complex defined above is called relative cohomlogy of f,and is denoted byH_rel^∗ (f) orH_rel^∗ . In the above sequence, if we putI ·Ω^kinto Ω^k, then we have flat relative cohomology H_∞^k_rel, where I denotes the space of flat functions ofF at the origin. Moreover if we consider formal differential k-forms instead of Ω^k, we have formal relative cohomology ˆH_rel^k .

To state the structure ofH_∞^k_rel it is convenient to introduce the following notations: For a positive small number c,

b^k₊= dimH^k(X_+c,R), b^k₋ = dimH^k(X_−c,R),

m^∞(1) = the space of flat functions at the origin of 1-variable, m^∞_± ={h∈m^∞(1) | h(R^∓) = 0},

X_±c =f⁻¹(±c)∩B, whereB is a small ball centered at the origin.

Then C. A. Roche [13] proved the following theorems. All objects are considered onR^s.

Theorem 4.1. The m^∞(1)-moduleH_∞^k_rel is isomorphic to (m^∞₊)^bk+× (m^∞₋)^bk−.

Theorem 4.2. There are the following mutual relations among three cohomologies.

H_rel^k ∼=H_∞^k_rel if 0< k < s−1

H_rel⁰ /H_∞⁰_rel∼=F(1), H_rel^s−1/H_∞^s−_rel¹ ∼= ˆH_rel^s−1∼=F(1)^µ,

where F(1) is the space of formal functions of 1-variable, andµ =codim f. F(1)^µ denotes the freeF(1)-module of rank µ.

In the latter half of this subsection, let us return to the case ofR⁴. We simply write Ω^k for Ω^k(R⁴).

Definition 4.2. We define the subspaceI^k of Ω^k by I^k={c∈Ω^k|c(

k

g, ...,g) = 0}, for 1≤k≤4. PutI⁰= 0.

It is clear that I⁴ = Ω⁴ sinceΛ⁴g= 0. In the rest of this subsection, we give a characterization ofI^k fork= 1,2,3.

Proposition 4.3. I^k ={c∈Ω^k|c∧df= 0}, for0≤k≤4.

Proof. In case of k = 1, put c = Adx+Bdy +Cdz +Ddu ∈ Ω¹. Then c(g) = 0 implies that fxB = fyA, fxC = fzA, fxD = fuA, fyC = fzB, fzD=fuC,andfyD=fuB. On the other hand,

c∧df= (Adx+Bdy+Cdz+Ddu)∧(fxdx+fydy+fzdz+fudu)

= (f_yA−f_xB)dx∧dy+ (f_zA−f_xC)dx∧dz+ (f_uA−f_xD)dx∧du + (fzB−fyC)dy∧dz+ (fuB−fyD)dy∧du+ (fuC−fzD)dz∧du.

Thus we have thatc(g) = 0 if and only ifc∧df= 0.

For cases ofk≥2,we can prove in the same way as the case ofk= 1.

Now let us recall G. de Rham’s division lemma [2]. We will explain this lemma in the general situation, s-dimensional Euclidean space R^s. (Our case is, of course,s= 4.)

Definition 4.3. An element ω of Ω¹ is said to possess the property of division in Ω^∗if for anyα∈Ω^p, 1≤p≤s−1,which satisfiesω∧α= 0,there existsβ ∈Ω^p−1 such thatα=ω∧β.

Definition 4.4. Letω∈Ω¹ and letI(ω) be the ideal of Ω⁰=C^∞(R^s) spanned by the coefficients ofω. Then 0 is said to be algebraically isolated zero ofω if Ω⁰/I(ω) is a finite dimensional vector space overR.

Lemma 4.4. Let ω be an element of Ω¹. If 0 is algebraically isolated zero ofω, thenω possesses the property of division.

Sincef is of finite codimension in our situation,ω =df satisfies the con- dition of Lemma 4.4. Hence by Proposition 4.3, we know that I^k =df∧Ω^k−1 for 1≤k≤3.

Recall that a k-th cochain c ∈ C^k is F-linear skew-symmetric mapping from g× · · · ×g to F. The natural inclusion ι : g → χ(R⁴) induces the surjective mappingφ: Ω^k−→C^k as the dual mapping of the natural inclusion ι. Note that kerφ=I^k for 1≤k≤3. Then it is easy to obtain the following proposition.

Proposition 4.5. C^k ∼= Ω^k/I^k ∼= Ω^k/df∧Ω^k−1, for 1 ≤ k ≤3. For k= 0, C⁰= Ω⁰=F, and fork= 4, C⁴= 0.

Now by Proposition 4.5, we have obtained the following commutative di- agram. In particular, note that drel : Ω^k/I^k → Ω^k+1/I^k+1 coincides with

∂:C^k →C^k+1 for 0≤k≤2.

0 −−−−−→Ω⁰−−−−−→^d Ω¹ −−−−−→^d Ω² −−−−−→^d Ω³ −−−−−→^d Ω⁴ −−−−−→ 0 ^π ^π ^π ^π

0 −−−−−→Ω⁰−−−−−→^drel Ω¹/I¹−−−−−→^drel Ω²/I²−−−−−→^drel Ω³/I³−−−−−→^drel Ω⁴/df∧Ω³−−−−−→ 0 

0 −−−−−→Ω⁰−−−−−→^∂ C¹ −−−−−→^∂ C² −−−−−→^∂ C³ −−−−−→ 0

Using the above commutative diagram, we can get the following theorem.

Theorem 4.6. Let ηbe the exact Nambu-Poisson tensor corresponding tof ∈ F defined on R⁴, wheref is of finite codimension. Then

H_{N P}^k ∼=H_rel^k for 0≤k≤2, H_{N P}³ ∼=H_rel³ ⊕Ω⁴/df∧Ω³, H_{N P}^k = 0 for 4≤k.

To compute some examples of exact Nambu-Poisson cohomology, let us recall the results of C. A. Roche [13]. (See Theorem 4.1 and Theorem 4.2.)

Examples. Letf =x^k+y²+z²+u², k≥3. Then ifkis an odd positive integer, bothX_+candX_−care homeomorphic toD³,whereD³denotes a three dimensional ball. Hence by Theorem 4.1 and Theorem 4.2, we have

H_∞⁰_rel ∼=m^∞₊ ×m^∞₋, H_∞¹_rel= 0, H_∞²_rel= 0, H_∞³_rel= 0.

H_rel⁰ ∼=C^∞(R⁺)×C^∞(R⁻), H_rel¹ = 0, H_rel² = 0, H_rel³ ∼=F(1)^k−1. Moreover if we use Theorem 4.6, we have

H_{N P}⁰ ∼=C^∞(R⁺)×C^∞(R⁻), H_{N P}¹ = 0, H_{N P}² = 0, H_{N P}³ ∼=F(1)^k−1⊕R^k−1. On the other hand, ifkis an even positive integer, thenX_+c is homeomorphic toS³ andX₋c=φ. Hence we have

H_∞⁰_rel ∼=m^∞₊, H_∞¹_rel= 0, H_∞²_rel= 0, H_∞³_rel∼=m^∞₊. H_rel⁰ ∼=C^∞(R⁺), H_rel¹ = 0, H_rel² = 0, H_rel³ ∼= (C^∞(R⁺))^k−1. Moreover if we use Theorem 4.6, we have

H_{N P}⁰ ∼=C^∞(R⁺), H_{N P}¹ = 0, H_{N P}² = 0, H_{N P}³ ∼= (C^∞(R⁺))^k−1⊕R^k−1.

§5. Computation of Nambu-Poisson Cohomology: Linear Case

§5.1. Notation and general remarks

In this section we consider linear Nambu-Poisson tensors which are of order 3 onR⁴(x, y, z, u). By the classification theorem of linear Nambu-Poisson structures [3],[6], we know that there are the following four types of linear Nambu-Poisson tensors.

(I) η =−fx ∂

∂y∧_∂z^∂ ∧_∂u^∂ +fy ∂

∂x∧_∂z^∂ ∧_∂u^∂ −fz ∂

∂x∧_∂y^∂ ∧_∂u^∂ +fu ∂

∂x∧_∂y^∂ ∧_∂z^∂ , where f is a homogeneous quadratic function onR⁴.

(II) η=_∂x^∂ ∧_∂y^∂ ∧ {(a₁₁z+a₁₂u)_∂z^∂ + (a₂₁z+a₂₂u)_∂u^∂ }, (a_ij ∈R).

(III) ηφ=φ_∂x^∂ ∧_∂y^∂ ∧_∂z^∂ ,whereφis any linear function onR⁴.

(IV) ηψ={px+ (q−1)y−b3z−b4u}_∂y^∂ ∧_∂z^∂ ∧_∂u^∂ − {(q+ 1)x+ry+a3z+ a4u}_∂x^∂ ∧_∂z^∂ ∧_∂u^∂ ,wherep, q, r, a3, a4, b3, b4∈R. Putα=dψ+ (x+a3z+ a4u)dy−(y+b3z+b4u)dx, whereψ = ¹₂px²+qxy+¹₂ry². Then ηψ is defined byi(η_ψ)dx∧dy∧dz∧du=α.

In (IV), recall that η_ψ becomes a Nambu-Poisson tensor if and only if α∧dα = 0. Thus seven constants must satisfy a₃b₄ = a₄b₃, a₃p+b₃(q+ 1)

= 0, a3(q−1) +b3r= 0, a4p+b4(q+ 1) = 0, a4(q−1) +b4r= 0.

In considering type (II), since a matrix (aij) can be chosen to be in Jordan form, there are five classes with nondegenerate Jordan forms (η1∼η5) and two classes with degenerate Jordan forms (η₆∼η₇) as follows.

(i) η₁=_∂x^∂ ∧_∂y^∂ ∧ {(αz+u)_∂z^∂ + (αu)_∂u^∂ }, α= 0,

(ii) η₂=_∂x^∂ ∧_∂y^∂ ∧ {(αz)_∂z^∂ + (βu)_∂u^∂ }, α= 0, β= 0, α=β, (iii) η3=_∂x^∂ ∧_∂y^∂ ∧ {(αz−βu)_∂z^∂ + (βz+αu)_∂u^∂ }, α= 0 β = 0, (iv) η4=_∂x^∂ ∧_∂y^∂ ∧α(z_∂z^∂ +u_∂u^∂ ), α= 0,

(v) η5=_∂x^∂ ∧_∂y^∂ ∧β(u_∂z^∂ −z_∂u^∂ ), β= 0, (vi) η₆=_∂x^∂ ∧_∂y^∂ ∧(αz)_∂z^∂ , α= 0, (vii) η₇=_∂x^∂ ∧_∂y^∂ ∧u_∂z^∂ .

A linear Nambu-Poisson tensor of type (I) is one of exact Nambu-Poisson tensors. And this case was already considered in the previous section. Hence in this section we will only give the results concerningnondegenerate Nambu- Poisson tensors (i.e. f =±x²±y²±z²±u²) for type (I). And here we will mainly study the computation for type (II).

Throughout this section, we will use the following notations:

• F is the algebra of real-valuedC^∞ functions onR⁴(x, y, z, u);

• G˜is the algebra of real-valuedC^∞functions onR³(y, z, u);

• F˜ is the algebra of real-valuedC^∞ functions onR²(z, u);

• F(1) is the algebra of formal functions of one variable;

• χ(R⁴) is the Lie algebra of all vector fields onR⁴;

• Ω^k is the space ofk-forms onR⁴.

§5.2. Computation of Nambu-Poisson cohomology of type (I) In this subsection, we confine ourselves to nondegenerate linear Nambu- Poisson tensors of type (I). This means that f = ±x²±y²±z²±u² and it is clear thatf is of finite codimension. We get the following results by using Theorem 4.1 of C. A. Roche [13]. We use the same notations as those of the previous section. Let η be a linear Nambu-Poisson tensor of type (I) defined byi(η)Ω =df. Then we get the following flat relative cohomology. In Table 1, Dⁱ denotes ani-dimensional ball.

Table 1. Flat Relative Cohomology

f X_+c X_−c H⁰_∞rel H_∞rel¹ H²_∞rel H_∞rel³

x²+y²+z²+u² S³ φ m^∞₊ 0 0 m^∞₊

x²+y²+z²−u² S²×D¹ S⁰×D³ m^∞₊×m^∞₋×m^∞₋ 0 m^∞₊ 0 x²+y²−z²−u² S¹×D² S¹×D² m^∞₊×m^∞₋ m^∞₊×m^∞₋ 0 0 x²−y²−z²−u² S⁰×D³ S²×D¹ m^∞₊×m^∞₊×m^∞₋ 0 m^∞₋ 0

−x²−y²−z²−u² φ S³ m^∞₋ 0 0 m^∞₋

Combining the results in Table 1 with Theorem 4.2 and Theorem 4.6, we can compute cohomology of type (I). In computingH_{N P}³ ,note that Ω⁴/df∧Ω³∼= R,andµ= 1. We collect the results in the following table.

Table 2. Exact Nambu-Poisson Cohomology

f H_{N P}⁰ H_{N P}¹ H²_{N P} H_{N P}³

x²+y²+z²+u² C^∞(R⁺) 0 0 C^∞(R⁺)⊕R x²+y²+z²−u² C^∞(R⁺)×C^∞(R⁻)×C^∞(R⁻) 0 m^∞₊ F(1)⊕R x²+y²−z²−u² C^∞(R⁺)×C^∞(R⁻) m^∞₊×m^∞₋ 0 F(1)⊕R x²−y²−z²−u² C^∞(R⁺)×C^∞(R⁺)×C^∞(R⁻) 0 m^∞₋ F(1)⊕R

−x²−y²−z²−u² C^∞(R⁻) 0 0 C^∞(R⁻)⊕R

§5.3. Computation of Nambu-Poisson cohomology of type (II) In this subsection, we compute Nambu-Poisson cohomology of type (II).

Denote bygi the Lie algebra corresponding toη_i, i= 1,2, ...,7. Recall thatgi

is defined bygi =i(Ω²)ηi. Then each gi is spanned overF by several vector

fields as follows.

g1=z ∂

∂x, u ∂

∂x, z ∂

∂y, u ∂

∂y,(αz+u) ∂

∂z+αu ∂

∂u; g2=z ∂

∂x, u ∂

∂x, z ∂

∂y, u ∂

∂y, αz ∂

∂z+βu ∂

∂u; g₃=z ∂

∂x, u ∂

∂x, z ∂

∂y, u ∂

∂y,(αz−βu)∂

∂z + (βz+αu) ∂

∂u; g4=z ∂

∂x, u ∂

∂x, z ∂

∂y, u ∂

∂y, z ∂

∂z+u ∂

∂u; g5=z ∂

∂x, u ∂

∂x, z ∂

∂y, u ∂

∂y, u ∂

∂z−z ∂

∂u; g6=z ∂

∂x, z ∂

∂y, z ∂

∂z; g7=u ∂

∂x, u ∂

∂y, u ∂

∂z. As is easily seen, we know that

Λ⁴gi= 0, for 1≤i≤7.

Denote by H_{N P}^k (ηi) the k-th cohomology group corresponding to the Nambu-Poisson tensorη_i. Then for 1≤i≤7, H_{N P}^k (η_i) = 0 if 4≤k.

For 0≤k≤4, I^k⊂Ω^k is similarly defined as in the previous section (see Definition 4.2). Then we also haveC^k∼= Ω^k/I^k. First let us determine explicit forms of allI^k. They are summarized in the following lemma.

Lemma 5.1. Let A, B, C, D, E, F be elements of F. (a) In case ofη₁,

I¹={Cdz+Ddu | (αz+u)C+αuD= 0}, I²={Bdx∧dz+Cdx∧du+Ddy∧dz+Edy∧du

+F dz∧du | (αz+u)B+αuC= 0,(αz+u)D+αuE = 0}, I³={Adx∧dy∧dz+Bdx∧dy∧du+Cdx∧dz∧du

+Ddy∧dz∧du | (αz+u)A+αuB= 0}, I⁴= Ω⁴.

(b) In case ofη₂,

I¹={Cdz+Ddu | αzC+βuD},

I²={Bdx∧dz+Cdx∧du+Ddy∧dz+Edy∧du +F dz∧du | αzB+βuC= 0, αzD+βuE= 0}, I³={Adx∧dy∧dz+Bdx∧dy∧du+Cdx∧dz∧du

+Ddy∧dz∧du | αzA+βuB= 0}, I⁴= Ω⁴.

(c) In case ofη₃,

I¹={Cdz+Ddu | (αz−βu)C+ (βz+αu)D= 0}, I²={Bdx∧dz+Cdx∧du+Ddy∧dz+Edy∧du

+F dz∧du | (αz−βu)B+ (βz+αu)C= 0, (αz−βu)D+ (βz+αu)E= 0},

I³={Adx∧dy∧dz+Bdx∧dy∧du+Cdx∧dz∧du +Ddy∧dz∧du | (αz−βu)A+ (βz+αu)B = 0}, I⁴= Ω⁴.

(d) In case ofη4,

I¹={Cdz+Ddu | zC+uD= 0},

I²={Bdx∧dz+Cdx∧du+Ddy∧dz+Edy∧du +F dz∧du | zB+uC = 0, zD+uE= 0}, I³={Adx∧dy∧dz+Bdx∧dy∧du+Cdx∧dz∧du

+Ddy∧dz∧du | zA+uB= 0}, I⁴= Ω⁴.

(e) In case ofη5,

I¹={Cdz+Ddu | zD−uC = 0},

I²={Bdx∧dz+Cdx∧du+Ddy∧dz+Edy∧du +F dz∧du | uB−zC = 0, uD−zE= 0}, I³={Adx∧dy∧dz+Bdx∧dy∧du+Cdx∧dz∧du

+Ddy∧dz∧du | uA−zB= 0}, I⁴= Ω⁴.