Lobachevskii Journal of Mathematics Vol.4, 1999, 5–11 \begin{matrix}c \\ ◯ \end{matrix}N. Nakanishi

N. Nakanishi A SURVEY OF NAMBU-POISSON GEOMETRY

(submitted by B.N.Shapukov)

Abstract. We survey geometry of Nambu-Poisson manifolds. First we recall Nambu’s work which is the origin of Nambu-Poisson geometry. Next adopting Takhtajan’s geometric formulation and applying the local structure theorem of Nambu-Poisson tensors, we study the projectability of left invariant Nambu-Poisson tensors on Lie groups to certain homogeneous spaces.

1. Introduction

In 1973 Nambu [14] proposed a generalization of classical Hamiltonian mechanics taking the Liouville theorem as a guiding principle. After Nambu’s work, some papers appeared to analyze his proposal. For example Nambu mechanics was treated as degenerate Hamiltonian systems with some constraints (cf. [1],[11]). Moreover in [6] and [7], I.Cohen and A.J.Kálnay studied Nambu canonical transformations. But all of these papers were rather written from physical aspects, and they used the simplest Nambu-Poisson tensor: η = \frac{∂}{∂x_{1}} ∧ ⋯ ∧ \frac{∂}{∂x_{n}} on R^{n}, which we shall call the standard Nambu-Poisson tensor.

In 1994 L.Takhtajan [15] made a geometric formulation of Nambu-Poisson bracket, and he suggested that the bracket has to satisfy the following three conditions: skew-symmetricity, the Leibniz rule and the fundamental identity.

R.Chatterjee and L.Takhtajan [3] conjectured that any Nambu-Poisson tensor, which defines Nambu-Poisson bracket is decomposable. This conjecture was affirmatively solved in [9],[12]. In [2], it was shown that some physical systems (the SU(n)-isotropic harmonic operator and the SO(4)- Kepler systems) admit a Nambu-Poisson structure. Other examples are also discussed in [15].

The author wishes to express his gratitude to Professor B.Komrakov for wonderful hospitality at Kazan. Throughout our two weeks’ stay at Kazan, everyday was a succession of magnificent experiences. The author also wishes to thank Professor T.Morimoto for his arrangements for our first visit to Russia.

We shall enumerate here some further researches on Nambu-Poisson manifolds. J-P.Dufour and N.T.Zung [8] classified linear Nambu-Poisson strucutres and solved the linearizability problem of Nambu-Poisson tensors. I.Vaisman [16] proposed a definition of Nambu-Lie groups, which is an extension of the notion of Poisson-Lie groups. Since direct product of two Nambu-Poisson manifolds is no longer a Nambu-Poisson manifold, the definition of Nambu-Lie structure must be slightly changed. J.Grabowski and G.Marmo [10] discuss relations of linear Nambu-Poisson structures to Filippov algebras and define Filippov algebroids. They also prove results describing multiplicative Nambu-Poisson structures on Lie groups.

The aim of this paper is to survey geometry of Nambu-Poisson manifolds. The plan of this paper is as follows: In Section 2, Nambu’s original work is introduced. In Section 3, we define the notion of a Nambu-Poisson manifold according to Takhtajan [15]. A Nambu-Poisson manifold is a pair of a C^{∞}-manifold and a Nambu-Poisson tensor on it. We prove the local structure theorem of Nambu-Poisson tensors. Using this theorem, we can prove some fundamental properties about a Nambu-Poisson manifold. Moreover we rewrite Nambu-Poisson tensors in differential forms. This will enable us to judge somewhat easily whether given tensors are Nambu-Poisson tensors or not. In Section 4, we consider left invariant Nambu-Poisson tensors on Lie groups, and we shall study under what conditions they can be projected down to suitable homogeneous spaces [13].

2. Nambu’s original work

In [14], Nambu proposed a possible generalization of classical Hamiltonian dynamics to a three dimensional phase space. Let (x, y, z) ≡ r⃗ be a triplet of dynamical variables which span the 3-dimensional phase space R^{3}. Next let R, S ∈ C^{∞}(R^{3}) be two functions which are called a pair of Hamiltonians to determine the equations of motion in R^{3}. More precisely, Nambu postulates the following Hamiltonian equations:

\frac{dx}{dt} = \frac{∂(R, S)}{∂(y, z)}, \frac{dy}{dt} = \frac{∂(R, S)}{∂(z, x)}, \frac{dz}{dt} = \frac{∂(R, S)}{∂(x, y)},  (2.1)

or in a vector notation,

\frac{dr⃗}{dt} = ∇R × ∇S.  (2.2)

For any function F ∈ C^{∞}(R^{3}), we have under the assumption (2.1),

\frac{dF}{dt} = \frac{∂F}{∂x} \frac{dx}{dt} + \frac{∂F}{∂y} \frac{dy}{dt} + \frac{∂F}{∂z} \frac{dz}{dt} = ∇F · (∇R × ∇S).  (2.3)

The right hand side of (2.3) is called Nambu-Poisson bracket and is denoted by {F, R, S}. This generalized Poisson bracket does not satisfy the Jacobi identity. In our terminology, this bracket is called Nambu-Poisson bracket, and the corresponding Nambu-Poisson tensor is \frac{∂}{∂x} ∧ \frac{∂}{∂y} ∧ \frac{∂}{∂z} . Both R and S are constants of motion. Note that the velocity field \frac{dr⃗}{dt} is divergenceless:

∇ · (∇R × ∇S) = 0,  (2.4)

and this equation shows that Liouville’s theorem is still maintained in Nambu’s phase space. This means that if we regard ∇R ×∇S as the Hamiltonian vector field corresponding to the Hamiltonians R and S, it leaves the standard volume element of R^{3} invariant.

Consider the motions of a rigid body around a stationary point 0. Let M⃗ = (M_{x}, M_{y}, M_{z}) be the angular momentum vector of the body, V⃗ the angular velocity vector in the body. Then the Euler equation is defined by

\frac{dM⃗}{dt} = M⃗ × V⃗ .  (2.5)

Put R = \frac{1}{2}(M_{x}^{2} + M_{y}^{2} + M_{z}^{2}). Then we have ∇R = M⃗. If we put S = \frac{1}{2}(M_{x}^{2}/I_{x} + M_{y}^{2}/I_{y} + M_{z}^{2}/I_{z}), where I = (I_{x}, I_{y}, I_{z}) denotes the inertia operator. Then V⃗ is given by ∇S. Thus the equation (2.5) is written as follows:

\frac{dM_{x}}{dt} = M_{x}M_{y}(\frac{1}{I_{z}} − \frac{1}{I_{y}}) = \frac{∂(R, S)}{∂(M_{y}, M_{z})},  (2.6)

\frac{dM_{y}}{dt} = M_{z}M_{x}(\frac{1}{I_{x}} − \frac{1}{I_{z}}) = \frac{∂(R, S)}{∂(M_{z}, M_{x})},  (2.7)
\frac{dM_{z}}{dt} = M_{x}M_{y}(\frac{1}{I_{y}} − \frac{1}{I_{x}}) = \frac{∂(R, S)}{∂(M_{x}, M_{y})}.  (2.8)

These equations are equivalent to the equation (2.1). Hence the Euler equation of the motions of a rigid body gives a typical example of real physical systems which may be written in Nambu’s scheme.

3. Nambu-Poisson manifolds

In this section we give a definition of Nambu-Poisson manifolds, which is equivalent to that of Takhtajan [15]. Let M be an m-dimensional C^{∞}- manifold, and F its algebra of real valued C^{∞} functions. We denote by Γ (Λ^{n}T M) the space of global cross-sections η : M ⟶ Λ^{n}T M. Then for each η ∈ Γ (Λ^{n}T M), there corresponds the bracket defined by

{f_{1}, ..., f_{n}} = η(df_{1}, ..., df_{n}), f_{1}, ..., f_{n} ∈ F.

Let A = ∑ f_{i_{1}} ∧ ⋯ ∧ f_{i_{n−1}}, f_{i_{j}} ∈ F. Since the bracket operation clearly satisfies the Leibniz rule, we can define a vector field X_{A} corresponding to A by the following equation:

X_{A}(g) = ∑{f_{i_{1}}, ..., f_{i_{n−1}}, g}, g ∈ F.

Such a vector field is called a Hamiltonian vector field. The space of Hamiltonian vector fields is denoted by H.

Definition 3.1. η ∈ Γ (Λ^{n}T M) is called a Nambu-Poisson tensor of order n if it satisfies L(X_{A})η = 0 for all X_{A} ∈ H, where L is the Lie derivative. Then a Nambu-Poisson manifold is a pair (M, η).

The above definition is clearly equivalent to Fundamental Identity, which Takhtajan proposed in [15]. If n = 2, Fundamental Identity is nothing but the Jacobi identity, and we have usual Poisson manifolds.

A point p ∈ M is said to be regular if η(p) ≠ 0. Now we can state the following local structure theorem for Nambu-Poisson tensors [9],[12].

Theorem 3.2. Let η ∈ Γ (Λ^{n}T M), n ≥ 3. If η is a Nambu-Poisson tensor of order n, then for any regular point p, there exists a coordinate neighborhood U with local coordinates (x_{1}, ..., x_{n}, x_{n+1}, ..., x_{m}) around p such that

η = \frac{∂}{∂x_{1}} ∧ ⋯ ∧ \frac{∂}{∂x_{n}}

on U, and vice versa.

In Theorem 3.2, the condition n ≥ 3 is essential. If n = 2, as is wellknown, A.Weinstein [17] proved the splitting theorem for Poisson tensors. Comparing Weinstein’s splitting theorem with our theorem, we know that the local structure of Nambu-Poisson manifolds is more rigid than that of usual Poisson manifolds. Some immediate applications of Theorem 3.2 are the following.

Corollary 3.3. (1) Let η be a Nambu-Poisson tensor of order n ≥ 3. If f is a smooth function, then fη is again a Nambu-Poisson tensor.

(2) If m = n ≥ 3, then every η ∈ Γ (Λ^{n}T M) is a Nambu-Poisson tensor.

(3) For every Nambu-Poisson tensor η, its Schouten bracket satisfies that [η, η] = 0.

By the above corollary, we know that 3-tensor on R^{3} corresponding to the Nambu’s generalized Poisson bracket is a Nambu-Poisson tensor. Another criterion can be found in the next section.

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 of η and Ω. Hence ω is (m − n)-form. In the following theorem, we will characterize a Nambu-Poisson tensor by using this differential form ω. (For the proof, see [13].)

Theorem 3.4. Let η ∈ Γ (Λ^{n}T M). Then η is a Nambu-Poisson tensor if and only if η satisfies the following two conditions around each regular point:

1) ω is (locally) decomposable, and

2) there exists a locally defined 1-form θ such that dω = θ ∧ ω.

Remark. It is clear that the above criterion for Nambu-Poisson tensors does not depend on the choice of volume form. A differential form ω which satisfies the conditions in Theorem 3.4 is called a Nambu-Poisson form.

Corollary 3.5. If m = n + 1, then η is a Nambu-Poisson tensor if and only if ω ∧ dω = 0.

4. Left Invariant Nambu-Poisson tensors on Lie groups

Let G be an m-dimensional connected Lie group, m ≥ 3. First we shall determine the form of left invariant Nambu-Poisson tensors on G. Denote by 𝔤 the Lie algebra of left invariant vector fields on G. By using Theorem 3.2, we can easily obtain the following lemma.

Lemma 4.1. Let η be a (non-zero) left invariant Nambu- Poisson tensor of order n ≥ 3 on a Lie group G. Then η is globally decomposable.

By the above lemma, any left invariant Nambu-Poisson tensor η of order n can be written as a decomposable element of Λ^{n}𝔤.

We denote by < X_{1}, ..., X_{n} > the Lie subalgebra of 𝔤 generated by X_{1}, ..., X_{n} ∈ 𝔤. The following theorem states that for every Lie subalgebra 𝔥, dim 𝔥 ≥ 3, there corresponds a left invariant Nambu-Poisson tensor of order dim 𝔥. (For the proof, see [13].)

Theorem 4.2. Let G be an m-dimensional Lie group.

(i) Let 𝔥 =< X_{1}, ..., X_{n} > be an n-dimensional Lie subalgebra of 𝔤, n ≥ 3. For a basis X_{1}, ..., X_{n} of 𝔥, put η = X_{1} ∧ ⋯ ∧ X_{n}. Then η is a left invariant Nambu-Poisson tensor of order n on G.

(ii) Conversely given a left invariant Nambu-Poisson tensor η = X_{1} ∧⋯∧ X_{n} ∈ Λ^{n}𝔤 on G, then 𝔥 =< X_{1}, ..., X_{n} > is an n-dimensional Lie subalgebra of 𝔤.

If a left invariant Nambu-Poisson tensor η has two expressions: η = X_{1} ∧ ⋯ ∧ X_{n} = Y_{1} ∧ ⋯ ∧ Y_{n}, then we know that < X_{1}, ..., X_{n} >=< Y_{1}, ..., Y_{n} >. Thus we have

Corollary 4.3. There is a one to one correspondence up to constant multiple between the set of left invariant Nambu-Poisson tensors of order n on G and the set of n-dimensional Lie subalgebras of 𝔤.

Let G be an m-dimensional connected Lie group and H an n-dimensional closed subgroup of G. Denote by 𝔤 and 𝔥 the Lie algebras of G and H respectively. Let π : G ⟶ G/H be the natural projection. The mapping ω̄ ⟶ π^{∗}ω̄ establishes a 1-1 correspondence between the G-invariant p-forms on G/H and the left invariant p-forms ω on G which satisfy

(a) i(X)ω = 0 for all X ∈ 𝔥,

(b) L(X)ω = 0 for all X ∈ 𝔥 [5].

If ω̄ is a G-invariant (m−n)-form (i.e. G-invariant volume form) on G/H, then ω = π^{∗}ω̄ is a left invariant (m − n)-form on G. Since ω is closed and decomposable, ω induces a left invariant Nambu-Poisson tensor η of order n on G by the equation i(η)Ω = ω. It is clear that η is equal to the left invariant Nambu-Poisson tensor corresponding to the Lie algebra 𝔥 of H up to constant multiple. Define 𝔥_{ω} = {X ∈ 𝔤 | i(X)ω = 0}. Then 𝔥_{ω} is a Lie subalgebra of 𝔤 and 𝔥_{ω} = 𝔥. The maximal integral submanifold H_{ω} through e is the identity component of H. Since H is closed, H_{ω} is also a closed subgroup of G.

Conversely let us give a left invariant Nambu-Poisson tensor η of order n ≥ 3. Then as we have seen in Theorem 4.2, η determines an n-dimensional Lie subalgebra 𝔥, and η also induces the left invariant (m − n)-form ω on G by i(η)Ω = ω. In the following proposition, we give a sufficient condition for ω to be projected down to the G-invariant volume form of G/H. This is essentially due to S.S.Chern [4].

Proposition 4.4. Let G be an m-dimensional connected unimodular Lie group, and η a left invariant Nambu-Poisson tensor of order n ≥ 3 on G. Then there corresponds an n-dimensional Lie subalgebra 𝔥. Denote by H the connected Lie subgroup corresponding to 𝔥. If H is closed and unimodular, then ω is projected down to the G-invariant volume form of G/H.

Another easy sufficient condition for ω to be projectable is the following. If 𝔥 is an ideal of 𝔤, then ad(X) is 𝔥-valued for X ∈ 𝔥, and we easily obtain that dω = 0. Thus we have

Proposition 4.5. Let η be a left invariant Nambu-Poisson tensor of order n on G. Suppose that 𝔥 induced by η is an ideal of 𝔤 and the connected Lie group H which corresponds to 𝔥 is a closed subgroup of G. Then ω is projected down to the G-invariant volume form of G/H.

Here let us give one example of the pair of Lie groups (G, H) such that ω can not be projected down to any G-invariant volume form of G/H. In this case, of course, H is not unimodular. Let 𝔤 = 𝔰𝔩(3, R) and let 𝔤 = 𝔞+𝔫+𝔨 be the usual Iwasawa decomposition. Take 𝔞+𝔫 as 𝔥. Then 𝔥 is not an ideal but a Lie subalgebra of 𝔤. Let A and N be the connected Lie groups corresponding to 𝔞 and 𝔫 respectively. Then A and N are closed Lie subgroups of SL(3, R), and H is diffeomorphic to A × N. Hence H is a closed subgroup of SL(3, R). Now we can find a basis < X_{1}, ..., X_{8} > of 𝔤 such that 𝔞 =< X_{1}, X_{2} > and 𝔫 =< X_{3}, X_{4}, X_{5} >. Put η = X_{1} ∧ ⋯ ∧ X_{5}. Then ω = i(η)Ω can be written as ω = ω_{6} ∧ ω_{7} ∧ ω_{8} with respect to the dual basis < ω_{1}, ..., ω_{8} > of < X_{1}, ..., X_{8} > . Then we know that i(𝔥)dω ≠ 0. Hence ω can not be projected down to any G-invariant form of G/H.

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E-mail address: nakanisi@gifu-keizai.ac.jp

Received July 27, 1999