Proceedings Ist International Meeting on Geometry and Topology Braga (Portugal)

Public. Centro de Matemática da Universidade do Minho p. 119–130, 1998

Some Generalizations of Poisson and Jacobi Structures

\underline{Raúl Ibáñez}, Manuel de León, Juan C. Marrero, David Martín de Diego and Edith Padrón^{1}

1 Introduction

More than twenty years ago, Nambu [35] proposed a generalization of Hamiltonian mechanics by considering brackets involving three or more functions. The main motivation of Nambu to propose such a generalization was to study other forms of dynamics enforcing the Liouville theorem as in the case of statistical mechanics. The first example of a dynamical system which admits a Nambu formulation was given by Nambu himself and it is the Euler equations for the angular momentum of a rigid body. Another examples can be seen in [7, 18, 20, 41, 42]. Immediatly after the publication of Nambu’s paper, many authors have investigated its relation with the hamiltonian theory of constraints (see for instance [6, C, 11, 26, 33, 17]).

The subject was almost dormant for many years till the appearence of Takhtajan’s paper [42]. Indeed, Takhtajan gave a geometric approach to Nambu brackets by establishing the so-called Fundamental identity, which is a generalization of the Jacobi identity. Therefore, the notion of Nambu-Poisson manifold was introduced. It is remarkable that the fundamental identity was previously considered by Filippov [15] (see [31]) in a pure algebraic context.

On the other hand, Azcárraga et al [3, 4] have introduced a different generalization of the Jacobi identity, the so-called generalized Jacobi identity, which leads to the notion of generalized Poisson manifold.

120 Raúl Ibáñez et al

The brackets considered above are derivations on each argument. However, one can weak this assumption in the same way that Poisson brackets can be extended to Jacobi brackets. Proceeding in this direction, one obtains the socalled Nambu-Jacobi and generalized Jacobi manifolds (see [30, B, 24]).

1This work has been partially supported through grants DGICYT (Spain) Projects PB94-0106, PB94-0633-C02-02, UPV 127.310-EA043/97, Consejería de Educación y Cultura de la Comunidad de Madrid, Universidad de La Laguna, and UNED (Spain).

The aim of the present paper is to present these extensions of Poisson and Jacobi brackets, and show the main examples. The dynamics and the geometry of multibrackets have been studied in several recent papers. A huge list of references is included at the end.

2 Nambu-Poisson and generalized Poisson manifolds

Let M be a differentiable manifold of dimension m. We will use the following notations:

• C^{∞}(M, ℝ) is the algebra of C^{∞} real-valued functions on M.

• 𝔛(M) is the C^{∞}(M, ℝ)-module of vector fields on M.

• V^{k}(M) is the space of k-vectors on M.

• Ω^{k}(M) is the space of k-forms on M.

An almost Poisson bracket of order n [21] on M is a n-linear mapping

{ , …, } : C^{∞}(M, ℝ) × …^{(n} … × C^{∞}(M, ℝ) ⟶ C^{∞}(M, ℝ)

satisfying the following properties:

(1) (Skew-symmetry)

{f_{1}, …, f_{n}} = ε_{σ}{f_{σ(1)}, …, f_{σ(n)}} ,

for all f_{1}, …, f_{n} ∈ C^{∞}(M, ℝ) and σ ∈ 𝔖_{n}, where 𝔖_{n} is the group of the permutations of order n and ε_{σ} is the signature of the permutation σ.

(2) (Leibniz rule)

{f_{1}g_{1}, …, f_{n}} = f_{1}{g_{1}, …, f_{n}} + g_{1}{f_{1}, …, f_{n}} ,

for all f_{1}, …, f_{n}, g_{1} ∈ C^{∞}(M, ℝ).

An alternative way to define an n-bracket of functions is to consider the skewsymmetric tensor Λ of type (n, 0) defined by

Λ(df_{1}, …, df_{n}) = {f_{1}, …, f_{n}} ,

for all f_{1}, …, f_{n} ∈ C^{∞}(M, ℝ). We call Λ an almost or pseudo Poisson n-tensor, and (M, Λ) a generalized almost Poisson manifold. An almost Poisson n-tensor Λ induces a linear mapping

# : Ω^{n−1}(M) ⟶ 𝔛(M)

Some Generalizations of Poisson and Jacobi Structures 121

by defining

⟨#(α_{1} ∧ … ∧ α_{n−1}), β⟩ = Λ(α_{1}, …, α_{n−1}, β) ,

for all α_{1}, …, α_{n−1}, β ∈ Ω^{1}(M). Therefore, given n − 1 functions f_{1}, …, f_{n−1}, we have a vector field

X_{f_{1}...f_{n−1}} = #(df_{1} ∧ ⋯ ∧ df_{n−1})

which is called the Hamiltonian vector field associated with the Hamiltonian functions f_{1}, …, f_{n−1}.

2.1 Nambu-Poisson manifolds

A generalized almost Poisson manifold (M, Λ) of order n is called Nambu-Poisson if the following fundamental identity is satisfied [21, 41, 42]:

{f_{1}, …, f_{n−1}, {g_{1}, …, g_{n}}} = ∑_{i=1}^{n} {g_{1}, …, {f_{1}, …, f_{n−1}, g_{i}}, …, g_{n}} , (2.1)

for all functions f_{1}, …, f_{n−1}, g_{1}, …, g_{n} on M. This is equivalent to say that the Hamiltonian vector fields X_{f_{1}...f_{n−1}} are infinitesimal automorphisms of Λ, that is, L_{X_{f_{1}...f_{n−1}}}Λ = 0.

For n = 2 the fundamental identity is the well-known Jacobi identity, so (M, Λ) is a Poisson manifold. Symplectic manifolds are the main examples of Poisson manifolds.

Example: [17]

The constant Nambu-Poisson tensors of order n on ℝ^{n+s} are just the decomposable ones.

Example: (Volume forms) [17, 21]

Let M be an oriented n-dimensional manifold and ν a volume form. Given n functions f_{1}, …, f_{n} on M, we define a Nambu-Poisson bracket by the formula

{f_{1}, …, f_{n}} ν = df_{1} ∧ ⋯ ∧ df_{n} .

If we denote by Λ_{ν} the associated Nambu-Poisson tensor, then Λ_{ν} ≠ 0 at every point of M. Moreover, each Nambu-Poisson tensor Λ ≠ 0 of order n over an n-dimensional manifold M comes from a volume form.

If we take M = ℝ^{n}, and ν is the standard volume form ν = dx^{1} ∧ ⋯ ∧ dx^{n}, we recover the example originally discussed by Nambu [35]:

Λ_{ν} = \frac{∂}{∂x^{1}} ∧ ⋯ ∧ \frac{∂}{∂x^{n}} .

In some sense, the Nambu-Poisson structure associated to a volume form (for n ≥ 3) can be considered as the analogous to the symplectic structure in Poisson geometry. Indeed, in this case the associated morphism # : Ω^{n−1}(M) ⟶ 𝔛(M) is an isomorphism. Moreover, there is a generalized foliation whose leaves are points or volume manifolds.

Example:(Completely integrable systems) [19, 22]

Let (M, ω, H) be a completely integrable Hamiltonian system, that is, M is a 2n-dimensional symplectic manifold with symplectic form ω such that there exist f_{1}, …, f_{n} independent functions pairwise in involution, that is, {f_{i}, f_{j}} = 0 for i, j = 1, …, n. Then the associated Hamiltonian vector fields X_{i} = X_{f_{i}} commute, that is, [X_{i}, X_{j}] = 0. Therefore, Λ = X_{1} ∧ … ∧ X_{n} is a Nambu-Poisson tensor on M of order n.

More generally, let M be a differentiable manifold of dimension m and X_{1}, …, X_{n} independent vector fields that commute, that is, [X_{i}, X_{j}] = 0, for i, j = 1, …, n. Then the n-tensor Λ = X_{1} ∧ ⋯ ∧ X_{n} is a Nambu-Poisson structure of order n. Notice that the simultaneous flow-box theorem says that for each p ∈ M there exists a local coordinate system (x^{1}, …, x^{m}) on a neighborhood U

of p such that X_{1} = \frac{∂}{∂x^{1}} , ⋯, X_{n} = \frac{∂}{∂x^{n}}; and therefore, Λ = \frac{∂}{∂x^{1}} ∧ ⋯ ∧ \frac{∂}{∂x^{n}}.

2.2 Generalized Poisson manifolds

A generalized almost Poisson manifold (M, Λ) of order n (even) is called a generalized Poisson manifold if the generalized Jacobi identity is satisfied [3, 4, 21]:

Alt ({f_{1}, …, f_{n−1}, {g_{1}, …, g_{n}}}) = 0 , (2.2)

for all functions f_{1}, …, f_{n−1}, g_{1}, …, g_{n} on M. This is equivalent to the following property: [Λ, Λ] = 0, where [ , ] is the Schouten-Nijenhuis bracket. Again, for n = 2 we recover Poisson manifolds.

Example:(Nambu-Poisson manifolds) [2, 21] Every Nambu-Poisson manifold (of even order) is generalized Poisson.

Example:(Compatible Poisson structures) [22, 23]

Let M be a differentiable manifold and Λ_{1}, Λ_{2} two compatible Poisson structures, that is, [Λ_{1}, Λ_{2}] = 0, then Λ_{1} ∧ Λ_{2} is a generalized Poisson structure of order 4.

Example:(Hyperkähler manifolds)[22, 23]

Let M be a hyperkähler manifold, that is, M is a differentiable manifold of dimension 4n with a Riemannian metric g and three complex structures J_{1}, J_{2}, J_{3} compatible with g and such that: i) the complex structures satisfy J_{3} = J_{1}J_{2} = −J_{2}J_{1}. ii) the Kähler forms ω_{i} defined by ω_{i}(X, Y) = g(X, J_{i}Y), for X, Y ∈ 𝔛(M)

are closed.

If Λ_{i} is the Poisson structure associated with the Kähler form ω_{i}, we have that Λ_{1}, Λ_{2} and Λ_{3} are compatible, i.e., [Λ_{i}, Λ_{j}] = 0, for all i, j, and consequently, the 4-vector

Λ = Λ_{1} ∧ Λ_{1} + Λ_{2} ∧ Λ_{2} + Λ_{3} ∧ Λ_{3},

is a generalized Poisson structure of order 4 on M.

Example:(Generalized Lie-Poisson structures)[4, 5, 24]

Let 𝔤 be a real vector space of dimension m. A skew-symmetric Lie 2n-bracket on 𝔤 is a 2n-linear skew-symmetric mapping

[ , …, ] : 𝔤 × …^{(2n} … × 𝔤 ⟶ 𝔤

satisfying the following condition:

Alt[a_{1}, …, a_{2n−1}, [b_{1}, …, b_{2n}]] = 0, (2.3)

for all a_{1}, …, a_{2n−1}, b_{1}, …, b_{2n} ∈ 𝔤. A vector space endowed with such a bracket is called a generalized Lie algebra of order 2n .

Now, on the dual space 𝔤^{∗}, we define the generalized almost Poisson bracket of order 2n

{, …, } : C^{∞}(𝔤^{∗}, ℝ) × … × C^{∞}(𝔤^{∗}, ℝ) ⟶ C^{∞}(𝔤^{∗}, ℝ)

by putting {f_{1}, …, f_{2n}}_{α} = α([(df_{1})_{α}, …, (df_{2n})_{α}]) (2.4)

for all f_{1}, …, f_{2n} ∈ C^{∞}(𝔤^{∗}, ℝ) and α ∈ 𝔤^{∗}. Note that (df_{i})_{α} ∈ T_{α}^{∗}𝔤^{∗} ≅ (𝔤^{∗})^{∗} ≅ 𝔤. Let Λ̄ be the 2n-vector associated with the bracket defined in (2.4). Then, (𝔤^{∗}, Λ̄) is a generalized Poisson manifold of order 2n.

3 Nambu-Jacobi and Generalized Jacobi manifolds

A generalized almost or pseudo Jacobi bracket of order n [24] on M is an n-linear mapping

{, …, } : C^{∞}(M, ℝ) × …^{(n} … × C^{∞}(M, ℝ) ⟶ C^{∞}(M, ℝ)

satisfying the following properties:

(i) Skew-symmetry:

{f_{1}, …, f_{n}} = ε_{σ}{f_{σ(1)}, …, f_{σ(n)}}

for all f_{1}, …, f_{n} ∈ C^{∞}(M, ℝ) and σ ∈ 𝔖_{n}.

(ii) {, …, } is a first order linear differential operator on M with respect to each argument, i.e.,

{f_{1}g_{1}, f_{2}, …, f_{n}} = f_{1}{g_{1}, f_{2}, …, f_{n}} + g_{1}{f_{1}, …, f_{n}} − f_{1}g_{1}{1, f_{2}, …, f_{n}}

for all f_{1}, g_{1}, f_{2}, …, f_{n} ∈ C^{∞}(M, ℝ).

If {, …, } is a generalized almost Jacobi bracket of order n then we can define an n-vector Λ and an (n − 1)-vector □ as follows

\begin{aligned}□(df_{1}, …, df_{n−1}) & = & \{1, f_{1}, …, f_{n−1}\} \\ Λ(df_{1}, …, df_{n}) & = & \{f_{1}, …, f_{n}\} + ∑_{i=1}^{n}(−1)^{i}f_{i}□(df_{1}, …, d̂f_{i}, …, df_{n}) \end{aligned}

for all f_{1}, …, f_{n} ∈ C^{∞}(M, ℝ). Conversely, any pair (Λ, □) ∈ V^{n}(M) ⊕ V^{n−1}(M) defines a vgeneralized almost Jacobi bracket of order n given by

{f_{1}, …, f_{n}} = Λ(df_{1}, …, df_{n}) + ∑_{i=1}^{n}(−1)^{i−1}f_{i}□(df_{1}, …, d̂f_{i}, …, df_{n}).

Thus, the pair (Λ, □) ∈ V^{n}(M) ⊕ V^{n−1}(M) is called a generalized almost Jacobi structure of order n and (M, Λ, □) is a generalized almost Jacobi manifold. If □ = 0, then we obtain a generalized almost Poisson structure of order n.

Given n − 1 functions f_{1}, ⋯, f_{n−1} ∈ C^{∞}(M, ℝ), the associated Hamiltonian vector field X_{f_{1},...,f_{n−1}} is defined by

X_{f_{1}...f_{n−1}} = X_{f_{1}...f_{n−1}}^{Λ} + ∑_{i=1}^{n−1}(−1)^{i−1}f_{i}X_{f_{1}...f̂_{i}...f_{n−1}}^{□} (3.1)

where we denote by X_{f_{1}...f_{n−1}}^{Λ} (respectively, X_{f_{1}...f̂_{i}...f_{n−1}}^{□}) the Hamiltonian vector field on (M, Λ) (respectively, (M, □)) associated with the functions f_{1}, …, f_{n−1} (respectively, f_{1}, …, f_{i−1}, f_{i+1}, …, f_{n−1}).

3.1 Nambu-Jacobi manifolds

A generalized almost Jacobi manifold (M, Λ, □) of order n is called Nambu-Jacobi [24, 30] if the fundamental identity (2.1) is satisfied. It can be proved [30] that in this case, Λ (resp. □) is a Nambu-Poisson structure of order n (resp. n − 1) and for x ∈ M satisfying Λ(x) ≠ 0, there exists θ ∈ T_{x}^{∗}M such that i_{θ}Λ(x) = □(x).

For n = 2 we recover Jacobi manifolds, which main examples are provided by contact and locally conformal symplectic manifolds.

Example: [24, 30]

Let (M, Λ) be a Nambu-Poisson manifold of order n, n > 2. If f ∈ C^{∞}(M, ℝ) then i_{df}Λ is also a Nambu-Poisson structure on M. Furthermore, it can be proved that the pair (Λ, i_{df}Λ) is a Nambu-Jacobi structure of order n. Since a closed 1-form θ is locally exact, the above result can be extended to (Λ, i_{θ}Λ).

Next, let M be an oriented manifold of dimension n, and choose a volume form ν on M. Denote by Λ_{ν} the corresponding Nambu-Poisson tensor on M. If θ is a 1-form on M, then the generalized almost Jacobi manifold (M, Λ_{ν}, i_{θ}Λ_{ν}) is a Nambu-Jacobi manifold if and only if θ is closed.

Example:[24, 30]

Let (M, Λ) be a Nambu-Poisson manifold of order n, n > 2 (respectively, a symplectic manifold of dimension 2). It can be proved that the pair (0, Λ) is a Nambu-Jacobi structure of order n + 1 (respectively, of order 3) on M.

3.2 Generalized Jacobi manifolds

A generalized almost Jacobi manifold (M, Λ, □) of order n (even) is called generalized Jacobi [24, B] if the generalized Jacobi identity (2.2) is satisfied. This is equivalent to the properties:

[Λ, Λ] = 2(2s − 1)Λ ∧ □  and  [Λ, □] = 0.

Example: [24] Every Nambu-Jacobi manifold (of even order) is a generalized Jacobi manifold.

Example: [24]

Let (M, Λ, □) be a generalized almost Jacobi manifold of even order n. For each p ∈ ℝ − {0}, we define on M × ℝ the n-vector

{ \atop (Λ, □)}_{p} = e^{−pt}Λ + e^{−pt} \frac{∂}{∂t} ∧ □ (3.2)

where t is the usual coordinate on ℝ. Then, (Λ, \frac{p}{n − 1}□) is a generalized Jacobi

structure on M if and only if { \atop (Λ, □)}_{p} is a generalized Poisson structure on M ×ℝ.

Example:(Compatible Jacobi structures) [24]

Two Jacobi structures (Λ_{1}, □_{1}) and (Λ_{2}, □_{2}) on a manifold M are said to be compatible if (Λ_{1} + Λ_{2}, □_{1} + □_{2}) is a Jacobi structure on M, that is, [Λ_{1}, Λ_{2}] = □_{1} ∧ Λ_{2} + □_{2} ∧ Λ_{1} and L_{□_{1}}Λ_{2} + L_{□_{2}}Λ_{1} = 0.

If we put

Λ = Λ_{1} ∧ Λ_{2}, □ = □_{1} ∧ Λ_{2} + □_{2} ∧ Λ_{1} (3.3) then (M, Λ, \frac{2}{3}□) is a generalized Jacobi manifold of order 4.

Example:(3-Sasakian manifolds)[24]

A 3-Sasakian manifold is a (4n + 3)-dimensional Riemannian manifold (M, g) that admits three Sasakian structures (φ_{i}, □_{i}, η_{i}, g)_{i=1,2,3} whose vector fields □_{1}, □_{2}, □_{3} are mutually orthogonal and satisfy [□_{σ(1)}, □_{σ(2)}] = 2ε_{σ}□_{σ(3)}, for all σ ∈ 𝔖_{3} (see [27]).

Suppose that (Λ_{i}, □_{i}) is the associated Jacobi structure with the contact 1form η_{i}, i = 1, 2, 3. Then, the structures (Λ_{1}, □_{1}), (Λ_{2}, □_{2}) and (Λ_{3}, □_{3}) are compatible and if Λ and □ are the 4-vector and the 3-vector on M given by

Λ = ∑_{i=1}^{3} Λ_{i} ∧ Λ_{i}, □ = ∑_{i=1}^{3} □_{i} ∧ Λ_{i},

we have that (Λ, \frac{2}{3}□) is a generalized Jacobi structure on M of order 4.

Finally, if M is a 3-Sasakian manifold, then M ×ℝ is an hyperkähler manifold. Moreover, if \overline{Λ} is the generalized Poisson structure on M × ℝ, then \overline{Λ} = { \atop (Λ, □)}_{2}.

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Raúl Ibáñez Departamento de Matemáticas Facultad de Ciencias Universidad del Pais Vasco Apartado 644, 48080 Bilbao Spain

Email: mtpibtor@lg.ehu.es

Manuel de León Instituto de Matemáticas y Física Fundamental Consejo Superior de Investigaciones Científicas Serrano 123, 28006 Madrid Spain

Email: mdeleon@fresno.csic.es

Juan C. Marrero, E. Padrón Departamento de Matemática Fundamental Facultad de Matemáticas Universidad de la Laguna La Laguna, Tenerife Canary Islands, Spain

Email: jcmarrer@ull.es, mepadron@ull.es

David Martín de Diego Depart. de Economía Aplicada Cuantitativa Facultad de Ciencias Económicas y Empresariales UNED 28040 Madrid Spain

Email: dmartin@sr.uned.es