Lobachevskii Journal of Mathematics Vol. 3, 1999, 201–207 \begin{matrix}c \\ ◯ \end{matrix}Kentaro Mikami

Kentaro Mikami

NAMBU-POISSON STRUCTURES AND THEIR FOLIATIONS

(submitted by B.N.Shapukov)

Abstract. Nambu-Poisson bracket is a natural generalization of Poisson bracket. A very distinguished property is its decomposability. This is investigated from the second order term of the fundamental identity (see [2] or [5]). In this paper, we shall study the first order term of the fundamental identity and get a relation with the Schouten-Nijenhuis bracket. And also we shall show that for a given Poisson structure, the top power of it gives a Nambu-Poisson structure. We shall characterize the Godbillon-Vey class of the foliation defined from a regular Nambu-Poisson tensor.

1. Introduction

We recall the definition of Nambu-Poisson bracket and fix notation we use in this paper. Let M be an n-dimensional C^{∞}-manifold.

Definition 1.1. An order p Nambu-Poisson bracket on M is a p-fold skewsymmetric R-multilinear operation on C^{∞}(M), i.e.,

\{… \} : \underbrace{C^{∞}(M) × ⋯ × C^{∞}(M)}_{p} ⟶ C^{∞}(M)

provided with 1. Leibniz rule for each argument,

2. the fundamental identity (or generalized Jacobi identity), i.e.,

{F, {G}} = ∑_{ℓ=1}^{p}{g_{1}, …, {F, g_{ℓ}}, …, g_{p}}

Partially supported by Grand-in-Aid for Scientific Research, The Priority Area of the infinite dimensional integrability systems (No. 08211207) and the Fundamental Research (C) (No. 08640084,10640057), The ministry of Education, Science and Culture, Japan..

1991 Mathematical Subject Classification. 58F05.. Key words and phrases. Nambu-Poisson structure/bracket/tensor, foliation,

Godbillon-Vey class..

where F = (f_{1}, …, f_{p−1}) ∈ C^{∞}(M)^{p−1}, G = (g_{1}, …, g_{p}) ∈ C^{∞}(M)^{p}.

An order 2 Nambu-Poisson bracket is a Poisson bracket. Like as Poisson brackets, each order p Nambu-Poisson bracket is defined by the unique pmultivector field π as

{f_{1}, …, f_{p}} = ⟨π, df_{1} ∧ ⋯ ∧ df_{p}⟩ .

For a p-fold bracket of functions defined from a p-multivector field π, let

Jac(F; G) := {F, {G}} − ∑_{ℓ=1}^{p}{g_{1}, …, {F, g_{ℓ}}, …, g_{p}}

where F ∈ C^{∞}(M)^{p−1} and G ∈ C^{∞}(M)^{p}. Namely, Jac = 0 if and only if the pfold bracket {⋯ } satisfies the fundamental identity. Analogy of Hamiltonian vector fields, we can consider the vector fields H_{(f_{1},... ,f_{p−1})} := {f_{1}, …, f_{p−1}, ·} for each f_{j} ∈ C^{∞}(M). Then the fundamental identity shows that

[H_{F}, H_{G}] = ∑_{j=1}^{p−1} H_{(g_{1}...{F,g_{j}}...g_{p−1})}

where G = (g_{1} … g_{p−1}). Thus we see that the characteristic distribution defined by H_{(f_{1},... ,f_{p−1})} is involutive.

The main result in this paper is the following theorem which is a part of Theorem 3.3.

Theorem 1.1. Each nowhere vanishing order p Nambu-Poisson structure gives a transversely oriented foliation of rank p and whose Godbillon-Vey class is given by [θ ∧ (dθ)^{n−p}] ∈ HDR^{2n−2p+1}(M; R), where 1-form θ := (−1)^{n}[π, ⋆(π)]_{SN} vol_{M}, ⋆(π) is defined by (π ∧ ⋆(π)) vol_{M} = 1 for some volume form vol_{M} of M.

2. Second order term of the fundamental identity

As already investigated in [1], we get an interesting phenomenon in contrast with Poisson brackets. We recall them here. By direct computations, we get

Jac(E, f_{p−1}f_{p}; G) = Jac(E, f_{p−1}; G)f_{p} + f_{p−1}Jac(E, f_{p}; G)

− ∑_{ℓ=1}^{p} (\{E, f_{p}, g_{ℓ}\}\{g_{1}, …, \underbrace{f_{p−1}}_{ℓ}, …, g_{p}\})

− ∑_{ℓ=1}^{p} (\{E, f_{p−1}, g_{ℓ}\}\{g_{1}, …, \underbrace{f_{p}}_{ℓ}, …, g_{p}\})

for E ∈ C^{∞}(M)^{p−2} and G ∈ C^{∞}(M)^{p}.

201 NAMBU-POISSON STRUCTURES AND THEIR FOLIATIONS

If the p-fold bracket {… } satisfies the fundamental identity, then

− ∑_{ℓ=1}^{p} (\{E, f_{p}, g_{ℓ}\}\{g_{1}, …, \underbrace{f_{p−1}}_{ℓ}, …, g_{p}\}) −

− ∑_{ℓ=1}^{p} (\{E, f_{p−1}, g_{ℓ}\}\{g_{1}, …, \underbrace{f_{p}}_{ℓ}, …, g_{p}\})

must vanish. We put the above by Φ(E; f_{p−1}, f_{p}; G). Gautheron ([2]) adds the trivial term

{E, f_{p−1}, f_{p}}{G} + {E, f_{p}, f_{p−1}}{G}

to Φ above and gets

Φ(E; f_{p−1}, f_{p}; G) = B(E, f_{p−1}, f_{p}, G) + B(E, f_{p}, f_{p−1}, G)

with his notation B in ([2]). He shows if Φ = 0 then B is full skew-symmetric and gets B = 0 by an elegant idea. Then it turns out π is decomposable. We can say Φ is the essence of the second order term of the fundamental identity, because we used the product f_{p−1}f_{p} at the (p − 1)-th entry of Jac.

Remark 2.1. If p = 2, then Φ = 0 automatically.

For any p, we do not get any new relation even if we deal with Jac(F; g_{1}, …, g_{p−1}, g_{p}g_{p+1}).

We summarize the fact already known:

Theorem 2.1 ([2] or [5]). If π is an order p(> 2) Nambu-Poisson structure, and π is not zero at a point x, then π_{x} is decomposable and the characteristic distribution of π at x has dimension p whose basis gives decomposition of π.

3. First order term of the fundamental identity

We would like to study about the first order term of the fundamental identity of Nambu-Poisson bracket. Jac(F; G) is skew-symmetric in F and skewsymmetric in G separately. If p = 2, namely, in the ordinary Poisson case, Jac(F; G) is skew-symmetric in the all arguments. By knowing this fact, we will skew-symmetrize Jac and study what that is.

Take any local coordinate system (x^{1}, …, x^{n}). A p-vector field π is written as

π = ∑ π^{i_{1}...i_{p}} \frac{∂}{∂x^{i_{1}}} ∧ ⋯ ∧ \frac{∂}{∂x^{i_{p}}} ,

where π^{i_{1}...i_{p}} are skew-symmetric. We abbreviate

Jac(x^{i_{1}}, …, x^{i_{p−1}}; x^{i_{p}}, …, x^{i_{2p−1}})

of π by Jac(i_{1}, …, i_{p−1}; i_{p}, …, i_{2p−1}). Skew-symmetrizing them, we put the result by SJac(i_{1}, …, i_{p−1}, i_{p}, …, i_{2p−1}). We get the following result.

Theorem 3.1. For each p-vector field π, we have

[π, π]_{SN} = −(1 + (−1)^{p})\frac{p}{1 + (−1)^{p}p}SJac

Kentaro Mikami 202

as (2p − 1)-multivector, where [·, ·]_{SN} is the Schouten-Nijenhuis bracket, which is defined by

[X_{1} ∧ ⋯ X_{q}, Y_{1} ∧ ⋯ Y_{r}]_{SN}

= ∑_{i=1}^{q} ∑_{j=1}^{r}(−1)^{i+j}[X_{i}, Y_{j}] ∧ X_{1} ∧ ⋯ X̂_{i} ⋯ ∧ X_{q} ∧ Y_{1} ∧ ⋯ Ŷ_{j} ⋯ ∧ Y_{r}

for each vector field X_{1}, …, Y_{r}. Proof of Theorem 3.1: Using the local expression of π by some local coordinate, we have

[π, π]_{SN} = (1 + (−1)^{p}) ∑_{IJ} π^{I}[π^{J}, ∂_{I}]_{SN} ∧ ∂_{J}

where we use the multi-index notation, i.e., π = ∑_{I} π^{I}∂_{I} with

∂_{I} = ∂_{i_{1}} ∧ ⋯ ∧ ∂_{i_{p}} = ∂/∂x^{i_{1}} ∧ ⋯ ∧ ∂/∂x^{i_{p}}

if I = (i_{1} … i_{p}), and

[π, π]_{SN}^{K} = −(1 + (−1)^{p})p ∑_{IJ} sgn\binom{K}{IJ}π^{Iℓ} \frac{∂π^{J}}{∂x^{ℓ}}

where [π, π]_{SN}^{K} means ∂_{K}-coefficient of [π, π]_{SN}, i.e.,

[π, π]_{SN} = ∑_{K}[π, π]_{SN}^{K}∂_{K} .

On the other hand,

Jac(i_{1}, …, i_{p−1}; i_{p}, …, i_{2p−1})

= π^{i_{1}...i_{p−1}ℓ}∂_{ℓ}π^{i_{p}...i_{2p−1}} − ∑_{j=p}^{2p−1} π^{i_{p}...ℓ...i_{2p−1}}∂_{ℓ}π^{i_{1}...i_{p−1}i_{j}}
= π^{i_{1}...i_{p−1}ℓ}∂_{ℓ}π^{i_{p}...i_{2p−1}}

− ∑_{j=p}^{2p−1}(−1)^{2p−1−j}π^{i_{p}...î_{j}...i_{2p−1}ℓ}∂_{ℓ}π^{i_{1}...i_{p−1}i_{j}} ,

where means omitting the index there. Thus, skewsymmetrizing Jac, we have

SJac(K)

= ∑_{IJ} sgn\binom{K}{IJ}π^{Iℓ}∂_{ℓ}π^{J}

− ∑_{IJ} ∑_{k=p}^{2p−1}(−1)^{2p−1−k}sgn\binom{K}{IJ}π^{i_{p}...î_{k}...i_{2p−1}ℓ}∂_{ℓ}π^{i_{1}...i_{p−1}i_{k}}

= ∑_{IJ} sgn\binom{K}{IJ}π^{Iℓ}∂_{ℓ}π^{J}

− ∑_{k=p}^{2p−1}(−1)^{2p−1−k}sgn\binom{K}{J′_{k}J′_{k}}sgn\binom{J′_{k}I′_{k}}{IJ}π^{J′_{k}ℓ}∂_{ℓ}π^{I′_{k}}

where J′_{k} = (i_{p} … î_{k} … i_{2p−1}), I′_{k} = (i_{1} … i_{p−1}i_{k}),

= ∑_{IJ} sgn\binom{K}{IJ}π^{Iℓ}∂_{ℓ}π^{J} − ∑_{IJ} ∑_{k=p}^{2p−1}(−1)^{p+1}sgn\binom{K}{I′J′}π^{I′ℓ}∂_{ℓ}π^{J′}

= (1 + p(−1)^{p}) ∑_{IJ} sgn\binom{K}{IJ}π^{Iℓ}∂_{ℓ}π^{J} .

Remark 3.1. If p = 2, then SJac = Jac and the equation above is wellknown. If p is odd, the relation above gives nothing. If π = X_{1} ∧ ⋯ X_{p} locally for some smooth local vector fields X_{1}, …, X_{p}, then we always have [π, π]_{SN} = 0.

From Theorem 3.1, we get

Corollary 3.1. An even-order Nambu-Poisson bracket must satisfy [π, π]_{SN} = 0.

This result is already known from the decomposability of Nambu-Poisson tensor. But, we would like to emphasize that the relation of the first order term of the fundamental identity also yields this result, using the Schouten-Nijenhuis bracket of the Nambu-Poisson tensor.

Nakanishi got a characterization of the Nambu-Poisson tensor by translating the multi-vector field to a differential form through some volume form of M in [4]. Roughly speaking, let vol_{M} be a volume form of M and η := π vol_{M} which is an (n−p)-form. Then π is a Nambu-Poisson tensor, namely π satisfies the fundamental identity if and only if dη = θ ∧ η holds locally for some local 1-form θ when π ≠ 0 and π is locally decomposable.

Mikami studied about the Godbillon-Vey class of symplectic foliations of regular Poisson structures in [3]. If π_{0} is a regular Poisson structure with rank 2m, then π_{0}^{m} is decomposable and we can find a global 1-form θ = [π_{0}^{m}, ⋆(π_{0}^{m})]_{SN} vol_{M} where ⋆(π_{0}^{m}) is defined by (π_{0}^{m} ∧ ⋆(π_{0}^{m})) vol_{M} = 1. Thus, combining Nakanishi’s criterion of Nambu-Poisson tensor with Mikami’s result, we can state the next elementary fact which gives many examples of Nambu-Poisson structures.

Proposition 3.2. Let π_{0} be a Poisson tensor with the maximal rank, say 2m. Then π_{0}^{m} gives order 2m Nambu-Poisson bracket.

As well-known in Theorem 2.1, Nambu-Poisson tensor is characterized as local decomposability around non-vanishing point and the involutive condition among local vector fields which decompose π locally. Nakanishi states the involutive condition by forms like as Frobenius type. Here, I will show the condition by using the Schouten-Nijenhuis bracket.

Theorem 3.3. Let π be a p-vector field which is locally decomposable around each point where π does not vanish. Take a volume form vol_{M}. Take a global 1-form θ := (−1)^{n}[π, ⋆(π)]_{SN} vol_{M}, where ⋆(π) is defined by (π ∧ ⋆(π)) vol_{M} = 1. Then π is a Nambu-Poisson tensor if and only if

ψ(π) = θ π

where ψ is the generalized divergence with respect to vol_{M} (see below).

For a given nowhere vanishing Nambu-Poisson tensor π, θ defines the Godbillon-Vey characteristic class represented by the form θ ∧ (dθ)^{n−p}.

The generalized divergence ψ is defined by the usual divergence ψ(X) for each vector field X and

ψ(X_{1} ∧ ⋯ ∧ X_{k})
= (−1)^{k} (∑_{i<j}(−1)^{i+j}[X_{i}, X_{j}] ∧ X_{1} ∧ ⋯ X̂_{i} ⋯ X̂_{j} ⋯ ∧ X_{k}

+ ∑_{i=1}^{k}(−1)^{i}ψ(X_{i})X_{1} ∧ ⋯ X̂_{i} ⋯ ∧ X_{k})

where X_{1}, …, X_{k} are vector fields with k > 1 (cf. [3]).

Proof of Theorem 3.3: Let π = X_{1} ∧ ⋯ ∧ X_{p} be a local decomposition of π. We can find some local frame {X_{1}, …, X_{p}, … X_{n}} with the dual frame {θ^{1}, …, θ^{n}} such that vol_{M} = θ^{1} ∧ ⋯ ∧ θ^{n} locally. We denote the bracket relations among {X_{i}} by

[X_{i}, X_{j}] = ∑_{k=1}^{n} C_{ij}^{k}X_{k}

for i, j = 1, …, n. Then we see that ψ(X_{i}) = − ∑_{j=1}^{n} C_{ij}^{j} for each i = 1, …, n and

θ = (−1)^{p} ( ∑_{i≤p<λ} C_{iλ}^{λ}θ^{i} + ∑_{μ>p≥j} C_{μj}^{j}θ^{μ}) .

We have

θ π = ∑_{i≤p}(−1)^{i+1}θ(X_{i})X_{1} ∧ ⋯ X̂_{i} ⋯ ∧ X_{p}

= (−1)^{p} ∑_{i≤p}(−1)^{i+1} ∑_{λ>p} C_{iλ}^{λ}X_{1} ∧ ⋯ X̂_{i} ⋯ ∧ X_{p}

and

(−1)^{p}ψ(π) = (−1)^{p}ψ(X_{1} ∧ ⋯ X_{p})
= ∑_{i<j≤p}(−1)^{i+j} ∑_{k≤p} C_{ij}^{k}X_{k} ∧ X_{1} ∧ ⋯ X̂_{i} ∧ ⋯ X̂_{j} ⋯ X_{p}

+ ∑_{i<j≤p}(−1)^{i+j} ∑_{λ>p} C_{ij}^{λ}X_{λ} ∧ X_{1} ∧ ⋯ X̂_{i} ∧ ⋯ X̂_{j} ⋯ X_{p}

+ ∑_{i≤p}(−1)^{i}ψ(X_{i})X_{1} ∧ ⋯ X̂_{i} ⋯ X_{p}

= ∑_{i<j≤p}(−1)^{i+j} ∑_{λ>p} C_{ij}^{λ}X_{λ} ∧ X_{1} ∧ ⋯ X̂_{i} ∧ ⋯ X̂_{j} ⋯ X_{p}

− ∑_{i≤p}(−1)^{i} ∑_{λ>p} C_{iλ}^{λ}X_{1} ∧ ⋯ X̂_{i} ⋯ X_{p}

Thus, we have

ψ(π) − θ π = ∑_{i<j≤p≤λ}(−1)^{i+j}C_{ij}^{λ}X_{1} ∧ ⋯ X̂_{i} ⋯ ∧ X_{p} ∧ X_{λ}

and we see that ψ(π) − θ π = 0 if and only if C_{ij}^{λ} = 0 for i, j = 1, …, p and λ = p + 1, …, n, namely {X_{1}, …, X_{p}} is involutive. From the definition of Godbillon-Vey class of transversely oriented foliations, we can say that θ ∧ (dθ)^{n−p} is a closed (2(n − p) + 1)-form which is a representative of the cohomology class.

References

[1] Rupka Chatterjee and Leon Takhtajan. Aspects of classical and quantum Nambu mechanics. Lett. Math. Phys., 37(4):475–482, August 1996.

[2] Philippe Gautheron. Some remarks concerning Nambu mechanics. Lett. Math. Phys., 37(1):103–116, May 1996.

[3] Kentaro Mikami. Godbillon-Vey classes of symplectic foliations. (to appear in Pacific J. Math.)

[4] Nobutada Nakanishi. Nambu-Poisson tensors on Lie groups. (to appear in Banach Center Publications).

[5] Nobutada Nakanishi. On Nambu-Poisson manifolds. Rev. in Math. Phys., 10:499–510, 1998.

Department of Computer Science and Engineering, Akita University, Akita, 010–8502, Japan. E-mail address: mikami@math.akita-u.ac.jp

Received July 27, 1999