GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 4, 1995, 347-359

GEOMETRY OF POISSON STRUCTURES

Z. GIUNASHVILI

Abstract. The purpose of this paper is to consider certain mechanisms of the emergence of Poisson structures on a manifold. We shall also establish some properties of the bivector field that defines a Poisson structure and investigate geometrical structures on the manifold induced by such fields. Further, we shall touch upon the dualism between bivector fields and differential 2-forms.

1. Schoten Bracket: Definition and Some Properties

1.1. Let L be any Lie algebra over the field of real numbers and F be any commutative real algebra with unity. It is assumed that L acts on F and this action has the following properties:

(a) F is an L-modulus: for each (u, v, a, b) ∈ L × L × F × F we have [u, v]a = uva − vua;

(b) Leibnitz’ rule: u(a · b) = (ua) · b + a · (ub). 1.2. Let us consider the spaces: C^{k}(L, F) = {α : L×⋯×L ⟶ F | α is an antisymmetric and polylinear

form}, k ≥ 0; C^{0}(L, F) = F; C^{k}(L, F) = {0} for k < 0. The space C(L, F) = ∑_{k∈ℤ} C^{k}(L, F) is an antisymmetric graded algebra

with the operation of exterior multiplication (see [1]).

347

1991 Mathematics Subject Classification. 30C45.

Key words and phrases. Poisson structure, Schoten bracket, bivector field, degenerate Poisson bracket, cohomology of Poisson structure.

1072-947X/95/0700-0347$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

1.3. We have two endomorphisms on the space C(L, F):

(∂_{1}α)(u_{1}, …, u_{k+1}) = ∑_{i<j}(−1)^{i+j−1}α([u_{i}, u_{j}], u_{1}, …, û_{i}, …, û_{j}, …, u_{k+1}),

(∂_{2}α)(u_{1}, …, u_{k+1}) = ∑_{i=1}^{k+1}(−1)^{i−1}u_{i}α(u_{1}, …, û_{i}, …, u_{k+1}),

where α is an element of C^{k}(L, F).

The endomorphism d = ∂_{2} − ∂_{1} is the coboundary operator defining the cohomology algebra of L (see [1]).

1.4. It is easy to check that the operators ∂_{1} and ∂_{2} are antidifferentiations, i.e., for each α ∈ C^{m}(L, F) and β ∈ C(L, F) we have

∂_{1}(α ∧ β) − (∂_{1}α ∧ β + (−1)^{m}α ∧ ∂_{1}β) = 0,
∂_{2}(α ∧ β) − (∂_{2}α ∧ β + (−1)^{m}α ∧ ∂_{2}β) = 0.

Therefore the operator d is an antidifferentiation, too.

1.5. For each k ∈ ℤ the space C_{k}(L, F) = End(F)⊗(∧^{k}L), where End(F) is the algebra of endomorphisms of F and ∧^{k}L is the exterior degree of L, is a subspace of Hom(C^{k}(L, F), F): for φ ⊗ u ∈ End(F) ⊗ (∧^{k}L) and ω ∈ C^{k}(L, F), we have (φ ⊗ u)(ω) = φ(ω(u)).

The multiplication in C^{∗}(L, F) = ∑_{k∈ℤ} C_{k}(L, F) is defined by the equation (φ ⊗ u) · (ψ ⊗ v) = (φ ∘ ψ) ⊗ (u ∧ v).

1.6. Define the operators:

∂^{1} = (∂_{1})^{∗}, ∂^{2} = (∂_{2})^{∗} : Hom(C^{k}(L, F), F) ⟶ Hom(C^{k−1}(L, F), F)

(∂^{i}(φ))(α) = φ(∂_{i}(α)), i = 1, 2, φ ∈ Hom(C^{k}(L, F), F),
α ∈ C^{k−1}(L, F), n ∈ ℤ.

The subspace C^{∗}(L, F) ⊂ ∑_{k∈ℤ} Hom(C^{k}(L, F), F) is invariant with respect to the operators ∂^{1} and ∂^{2}:

∂^{1}(φ ⊗ (u_{1} ∧ ⋯ ∧ u_{m})) = φ ⊗ ∑_{i<j}(−1)^{i+j−1}[u_{i}, u_{j}] ∧

∧u_{1} ∧ ⋯ ∧ û_{i} ∧ … ∧ û_{j} ∧ … ∧ u_{m},

∂^{2}(φ ⊗ (u_{1} ∧ ⋯ ∧ u_{m})) = ∑_{i=1}^{m}(−1)^{i−1}(φ ∘ u_{i}) ⊗

⊗u_{1} ∧ ⋯ ∧ û_{i} ∧ … ∧ u_{m}.

The operator ∂^{2} − ∂^{1} will be denoted by d^{∗}.

1.7. Let us consider the exterior algebra of L : ∧(L) = ∑_{k=0}^{∞} ∧^{k}L which is a subalgebra of C^{∗}(L, F). The space ∧(L) is an invariant subspace with respect to the action of the operator ∂^{1}:

∂_{1}(u_{1} ∧ ⋯ ∧ u_{m}) = ∑_{i<j}(−1)^{i+j−1}[u_{i}, u_{j}] ∧ u_{1} ∧ ⋯ ∧ û_{i} ∧ … ∧ û_{j} ∧ … ∧ u_{m}.

1.8. Generally speaking, the operator ∂^{1} is not an antidifferentiation.

Definition. We define the map (Schoten bracket [2]) [ , ] : ∧(L) × ∧(L) ⟶ ∧(L) as follows: let [u, v] = ∂^{1}(u∧v)−(∂^{1}(u)∧v+(−1)^{m}u∧∂^{1}(v)) for u ∈ ∧^{m}L and v ∈ ∧(L).

1.9. The space ∧(L) is not an invariant subspace of C^{∗}(L, F) with respect to the action of the operator ∂^{2}:

∂^{2}(1 ⊗ (u_{1} ∧ ⋯ ∧ u_{m})) = ∑_{i=1}^{m}(−1)^{i−1}u_{i} ⊗ (u_{1} ∧ ⋯ ∧ û_{i} ∧ … ∧ u_{m}).

However it is easy to show that for each u ∈ ∧^{m}L and v ∈ ∧(L) we have

∂^{2}(u ∧ v) − (∂^{2}(u) · v + (−1)^{m}u · ∂^{2}(v)) = 0.

Therefore we can define the bracket as

[u, v] = (d^{∗}(u) · v + (−1)^{m}u · d^{∗}(v)) − d^{∗}(u · v).

1.10. It is easy to check that for each u ∈ ∧^{m}L, v ∈ ∧^{n}L, w ∈ ∧^{k}L, we have:

(a) [u, v] = (−1)^{mn}[v, u];

(b) [u, v ∧ w] = [u, v] ∧ w + (−1)^{mn+n}v ∧ [u, w];

(c) (−1)^{mk}[[u, v], w] + (−1)^{mn}[[v, w], u] + (−1)^{nk}[[w, u], v] = 0. Let L be an F -modulus and assume that for each (u, v, a, b) ∈ L × L ×

F × F we have:

(a) (au)b = a(ub);

(b) [u, av] = (ua)v + a[u, v]. For each k = 1, 2, …, ∞ let V^{k}(L, F) denote an exterior degree of L as

an F -modulus: for a ∈ F and {u_{1}, …, u_{k}} ⊂ L we have au_{1} ∧u_{2} ∧…∧u_{k} = u_{1} ∧ au_{2} ∧ u_{3} ∧ … ∧ u_{k}. Assume that V^{0}(L, F) = F and V^{k}(L, F) = {0} when k < 0.

The space V (L, F) = ∑_{k∈ℤ} V^{k}(L, F) is an aniticommutative graded algebra.

1.12. Let J : ∧(L) ⟶ V (L, F)/ be the natural homomorphism which is an epimorphism onto ∑_{k∈ℤ\{0}} V^{k}(L, F).

Proposition. If elements {u, u′, v, v′} ⊂ ∧(L) are such that J(u) = J(u′) and J(v) = J(v′), then J([u, v]) = J([u′, v′]).

It is easy to prove this using the formulas (b) (1.10) and (b) (1.11).

1.13. Definition. We define the Schoten bracket on V (L, F) as follows: for {x, y} ⊂ ∑_{k∈ℤ\{0}} V^{k}(L, F) the bracket [x, y] is defined as J([u, v]) where J(u) = x and J(v) = y. We extend the definition to the space V (L, F) using equalities (b) (1.10) and (b) (1.11), namely: if u ∈ V′(L, F) and a ∈ V^{0}(L, F) = F, then [u, a] = u(a); for u = u_{1} ∧ … ∧ u_{k} ∈ V^{k}(L, F) and a ∈ F we use formula (b) (1.10). Finally, we recall that elements au_{1} ∧ u_{2} ∧ … ∧ u_{k} form the basis of V (L, F).

1.14. In the special case where F = C^{∞}(M) is the algebra of smooth functions on a smooth manifold M, L = V′(M) is the Lie algebra of smooth vector fields on the manifold M and V^{k}(M) is the space of antisymmetric contravariant tensors of degree k (V^{k}(M) is locally isomorphic to ∧^{k}V′(M)). The bracket defined above coincides with the well-known Schoten bracket (see [2]).

In that case if u ∈ V^{m}(M), v ∈ V^{n}(M), and ω ∈ Hom(V^{m+n−1}(M), C^{∞}(M)) is a differential form, then the formula defining the bracket by means of d^{∗} (see 1.9) gives

ω([u, v]) = (−1)^{mn+n}(d(i_{v}ω))(u) + (−1)^{m}(d(i_{u}ω))(v) − (dω)(u ∧ v),

where d is the well-known exterior differentiation of differential form (see [3]).

The above formula can be used as yet another definition of the Schoten bracket.

2. Poisson Bracket and a Bivector Field

2.1. Thus we have:

M is a finite-dimensional smooth manifold;

V^{0}(M) = C^{∞}(M) is the algebra of real-valued smooth functions on M; V^{k}(M), k > 0, is the space of antisymmetric contravariant tensor fields

of degree k; V^{k}(M) = {0} when k < 0; V (M) = ∑_{k∈ℤ} V^{k}(M) is the exterior algebra of polyvector fields; A^{0}(M) = C^{∞}(M); A^{k}(M) = {0} when k < 0; A^{k}(M), k > 0, is the space of exterior differential forms of degree k.

At the same time it is clear that A^{k}(M) = Hom(V^{k}(M), C^{∞}(M)) and V^{k}(M) = Hom(A^{k}(M), C^{∞}(M)) for k ∈ ℤ (in the sense of homomorphisms of the C^{∞}(M)-moduli).

2.2. An element of the space V^{2}(M) will be called a bivector field on the manifold M.

Given any bivector field ξ, for f, g ∈ C^{∞}(M) the bracket {f, g} ∈ C^{∞}(M) is defined to be (df ∧ dg)(ξ).

It is easy to show that the bracket defined by ξ satisfies the following conditions:

(a) antisymmetricity: {f, g} = −{g, f};

(b) bilinearity: {f, c_{1}g_{1} +c_{2}g_{2}} = c_{1}{f, g_{1}}+c_{2}{f, g_{2}} for each c_{1}, c_{2} ∈ ℝ;

(c) Leibnitz’ rule: {f, g · h} = {f, g} · h + {f, h} · g;

(d) for f, g, h ∈ C^{∞}(M) we have

\{\{f, g\}, h\} + \{\{h, f\}, g\} + \{\{g, h\}, f\} = \frac{1}{2}(df ∧ dg ∧ dh)([ξ, ξ])

where [ , ] is the Schoten bracket (see 1.14).

2.3. Proposition. Let { , } be any bracket on C^{∞}(M), having properties (a), (b), (c) from 2.2. There is one and only one bivector field ξ on M, defining the bracket { , } as describe in 2.2.

The bracket { , } defines the structure of a Lie algebra on a subspace A ⊂ C^{∞}(M) when and only when for each f, g, h ∈ A we have (df ∧dg)(ξ) ∈ A and (df ∧ dg ∧ dh)([ξ, ξ]) = 0.

2.4. We can consider ξ as a homomorphism of exterior algebras: for f ∈ A^{0}(M), α, β ∈ A′(M) we have ξ̃(f) = f, β(ξ̃(α)) = (α ∧ β)(ξ).

As follows from 2.3, the bracket { , } defines in exact terms the structure of a Lie algebra on C^{∞}(M) when [ξ, ξ] = 0.

Proposition. If [ξ, ξ] = 0, then the map ξ̃∘ d : C^{∞}(M) ⟶ V′(M) is a homomorphism of Lie algebras; C^{∞}(M) is a central extension of I_{m}(ξ̃∘ d) and ℝ ⊂ Ker(ξ̃∘ d).

Proof. In that case the pair (C^{∞}(M), { , }) is called the Poisson structure on M and the map f ⟼ ξ̃(df) = {f, } is the so-called Hamiltonian map which is a homomorphism of Lie algebras (see [4]).

2.5. Let ω be any differential 2-form on the manifold M, giving rise to the homomorphism of C^{∞}(M)-moduli: ω̃ : V′(M) ⟶ A′(M), ω̃(X) = ω(X, ), which is an isomorphism when ω is nondegenerate. In that case the induced map denoted similarly by ω̃ : V^{k}(M) ⟶ A^{k}(M), ω̃(u_{1} ∧…∧u_{k}) = ω̃(u_{1}) ∧ … ∧ ω̃(u_{k}), k = 1, …, ∞, is also an isomorphism. Let ξ_{ω} ∈ V^{2}(M) be ω^{−1}(ω).

More clearly, let ω = ∑_{i=1}^{n} a_{i} ∧ b_{i}, a_{i}, b_{i} ∈ A′(M), i = 1, …, n; the nondegeneracy of ω means that {a_{i}, b_{i} | i = 1, …, n} is a basis of A′(M) as a C^{∞}(M)-modulus. We introduce the following vector-fields on M: \frac{∂}{∂a_{i}}, \frac{∂}{∂b_{i}}, i = 1, …, n,

a_{k}(\frac{∂}{∂a_{i}} ) = b_{k}(\frac{∂}{∂b_{i}} ) = [ \begin{matrix}1, when k = i, \\ 0, when k ≠ i, \end{matrix} k = 1, …, n;

a_{p}(\frac{∂}{∂b_{q}} ) = b_{p}(\frac{∂}{∂a_{q}} ) = 0, p, q = 1, …, n.

With this notation and keeping in mind the definition of ω̃ we have ω̃(\frac{∂}{∂a_{i}} ) = b_{i}, ω̃(\frac{∂}{∂b_{i}} ) = −a_{i}, i = 1, …, n. Consequently, ξ_{ω} = ∑_{i=1}^{n} \frac{∂}{∂a_{i}} ∧

\frac{∂}{∂b_{i}}.

2.6. Theorem. ω̃([ξ_{ω}, ξ_{ω}]) = −2dω.

Proof. Using property (b) from 1.10 and the bilinearity of the Schoten bracket, we obtain

[ξ_{ω}, ξ_{ω}] = [ ∑_{i=1}^{n} \frac{∂}{∂a_{i}} ∧ \frac{∂}{∂b_{i}} , ∑_{i=1}^{n} \frac{∂}{∂a_{k}} ∧ \frac{∂}{∂b_{k}} ] =

= ∑_{i,k} [\frac{∂}{∂a_{i}} ∧ \frac{∂}{∂b_{i}} , \frac{∂}{∂a_{k}} ∧ \frac{∂}{∂b_{k}} ] = ∑_{i,k} ( − [\frac{∂}{∂a_{i}} , \frac{∂}{∂a_{k}} ] ∧

∧\frac{∂}{∂b_{i}} ∧ \frac{∂}{∂b_{k}} + [\frac{∂}{∂a_{i}} , \frac{∂}{∂b_{k}} ] ∧ \frac{∂}{∂b_{i}} ∧ \frac{∂}{∂a_{k}} +

+[\frac{∂}{∂b_{i}} , \frac{∂}{∂a_{k}} ] ∧ \frac{∂}{∂a_{i}} ∧ \frac{∂}{∂b_{k}} − [\frac{∂}{∂b_{i}} , \frac{∂}{∂b_{k}} ] ∧ \frac{∂}{∂a_{i}} ∧ \frac{∂}{∂a_{k}} .

By the definition of ω̃ (see 2.5) we have

ω̃([ξ_{ω}, ξ_{ω}]) = ∑_{i,m,k} (b_{m}([\frac{∂}{∂a_{i}} , \frac{∂}{∂a_{k}} ]) · a_{m} ∧ a_{i} ∧ a_{k} −

−a_{m}([\frac{∂}{∂a_{i}} , \frac{∂}{∂a_{k}} ]) · b_{m} ∧ a_{i} ∧ a_{k} + b_{m}([\frac{∂}{∂a_{i}} , \frac{∂}{∂b_{k}} ]) · a_{m} ∧ a_{i} ∧ b_{k} −

−a_{m}([\frac{∂}{∂a_{i}} , \frac{∂}{∂b_{k}} ]) · b_{m} ∧ a_{i} ∧ b_{k} + b_{m}([\frac{∂}{∂b_{i}} , \frac{∂}{∂a_{k}} ]) · a_{m} ∧ b_{i} ∧ a_{k} −

−a_{m}([\frac{∂}{∂b_{i}} , \frac{∂}{∂a_{k}} ]) · b_{m} ∧ b_{i} ∧ a_{k} + b_{m}([\frac{∂}{∂b_{i}} , \frac{∂}{∂b_{k}} ]) · a_{m} ∧ b_{i} ∧ b_{k} −

−a_{m}([\frac{∂}{∂b_{i}} , \frac{∂}{∂b_{k}} ]) · b_{m} ∧ b_{i} ∧ b_{k} ≡ Ω.

It is obvious that dω = ∑_{i=1}^{n}(da_{i} ∧ b_{i} − a_{i} ∧ db_{i}).

The monomials u′_{mik} = \frac{∂}{∂a_{m}} ∧ \frac{∂}{∂a_{i}} ∧ \frac{∂}{∂a_{k}}, u_{mik}^{2} = \frac{∂}{∂b_{m}} ∧ \frac{∂}{∂a_{i}} ∧ \frac{∂}{∂a_{k}}, u_{mik}^{3} = \frac{∂}{∂b_{m}} ∧ \frac{∂}{∂b_{i}} ∧ \frac{∂}{∂a_{k}}, u_{mik}^{4} = \frac{∂}{∂b_{m}} ∧ \frac{∂}{∂b_{i}} ∧ \frac{∂}{∂b_{k}}, \{m, i, k\} ⊂ \{1, …, n\} form the basis of V^{3}(M) as a C^{∞}(M)-modulus and it is easy to check that Ω(u_{mik}^{j}) = −2(dω)(u_{mik}^{j}) for each j ∈ {1, 2, 3, 4} and {m, i, k} ⊂ {1, …, n}.

We have therefore ascertained that Ω = −2dω.

2.7. Let (M, ω) be a symplectic manifold (see [3], [5]). For f ∈ C^{∞}(M) we define the vector field X_{f} by the formula df = ω( , X_{f}). It is a well-known fact (see [3], [5]) that ω defines a Poisson structure on M: for f, g ∈ C^{∞}(M) we have {f, g} = ω(X_{f}, X_{g}). It is easy to show that the corresponding bivector field is ξ_{ω}, i.e., (df ∧ dg)(ξ_{ω}) = ω(X_{f}, X_{g}).

As follows from 2.6, the equality dω = 0 is equivalent to [ξ_{ω}, ξ_{ω}] = 0.

2.8. Lemma. If ω ∈ A^{2}(M), α, β ∈ A′(M) and X, Y ∈ V^{2}(M), then we have (ω ∧ α ∧ β)(X ∧ Y) = ω(X) · (α ∧ β)(Y) + ω(Y) · (α ∧ β)(X) − ω(X̃(α), Ỹ(β)) + ω(X̃(β), Ỹ(α)).

Proof. It is sufficient to prove the lemma for the case ω = φ ∧ ψ where φ, ψ ∈ A′(M).

So, using the definition of the exterior product of differential forms (see [3]), we obtain

(φ ∧ ψ ∧ α ∧ β)(X ∧ Y) = (φ ∧ ψ)(X) · (α ∧ β)(Y) + +(φ ∧ α)(X) · (β ∧ ψ)(Y) + (φ ∧ β)(X) · (ψ ∧ α)(Y) + +(ψ ∧ α)(X) · (φ ∧ β)(Y) + (ψ ∧ β)(X) · (α ∧ φ)(Y) +

+(α ∧ β)(X) · (φ ∧ ψ)(Y) = ω(X) · (α ∧ β)(Y) +

+ω(Y) · (α ∧ β)(X) − ω(X̃(α), Ỹ(β)) + ω(X̃(β), Ỹ(α)).

2.9. A submodulus W ⊂ V′(M) is said to be an involutory differential system if for each pair X, Y ∈ W we have [X, Y] ∈ W (see [6]).

Theorem. If ξ̃: A′(M) ⟶ V′(M) is the homomorphism corresponding to the bivetor field ξ (see 2.4), then the differential system I_{m}ξ̃ is involutory in exact terms when [ξ, ξ] ∈ I_{m}ξ̃∧ I_{m}ξ̃∧ I_{m}ξ̃.

Proof. We can use any local coordinate system {x_{1}, …, x_{n}}. So, we want to show that for each pair {i, j} ⊂ {1, …, n} the vector field [ξ̃(dx_{i}), ξ̃(dx_{j})] is an element of I_{m}ξ̃ or, which is the same thing, that σ([ξ̃(dx_{i}), ξ̃(dx_{j})]) = 0 for each σ ∈ (I_{m}ξ̃)^{⊥} ⊂ A′(M).

By the definition of the Schoten bracket (see 1.14) we obtain (dσ ∧ dx_{i} ∧ dx_{j})(ξ ∧ ξ) = 2(dσ)(ξ) · (dx_{i} ∧ dx_{j})(ξ) − (σ ∧ dx_{i} ∧ dx_{j})([ξ, ξ]). Using Lemma 2.8, we have (dσ ∧ dx_{i} ∧ dx_{j})(ξ ∧ ξ) = 2(dσ)(ξ) · (dx_{i} ∧ dx_{j})(ξ) − 2(dσ)(ξ̃(dx_{i}), ξ̃(dx_{j})). Thus (σ ∧ dx_{i} ∧ dx_{j})([ξ, ξ]) = 2(dσ)(ξ̃(dx_{i}), ξ̃(dx_{j})). Clearly, (dσ)(ξ̃(dx_{i}), ξ̃(dx_{j})) = −σ([ξ̃(dx_{i}), ξ̃(dx_{j})]). Keeping in mind these identities, we obtain (σ ∧ dx_{i} ∧ dx_{j})([ξ, ξ]) = −2σ([ξ̃(dx_{i}), ξ̃(dx_{j})]).

2.10. Definition. An integer 2k ≥ 0 is said to be a rank of the bivector field ξ at a point a ∈ M if (∧^{k}ξ_{a}) ≠ 0 and ∧^{k+1}ξ_{a} = 0.

Let e = {e_{1}, …, e_{n}} be a basis of T_{a}(M) and e′ = {e^{1}, …, e^{n}} be the corresponding dual basis of T_{a}^{∗}(M). As is known (see [3]), a basis e can be chosen so that ξ_{a} = e_{1} ∧ e_{2} + ⋯ + e_{2k−1} ∧ e_{2k}. From the definition of ξ̃ (see 2.4) it follows that {e^{1}, …, e^{2k}} is a basis of I_{m}ξ̃_{a}. Also, it is clear that ∧^{k}ξ_{a} = e_{1} ∧ … ∧ e_{2k} and ∧^{k+1}ξ_{a} = 0. We have therefore ascertained that dim(I_{m}ξ̃_{a}) = rank ξ_{a}.

2.11. If the rank ξ = const and [ξ, ξ] ∈ ∧^{3}I_{m}ξ̃, then Theorem 2.9 and Frobenius’ theorem imply that the differential system I_{m}ξ̃ is integrable (see [3]), i.e., for each point a ∈ M there is a submanifold N ⊂ M such that a ∈ N and for each X ∈ N we have I_{m}ξ̃_{x} = T_{x}(N). It is clear that dim N = rank ξ.

2.12. Proposition. If [ξ, ξ] = 0, then the differential system I_{m}ξ̃ is integrable.

The proof follows from Hermann’s generalization of Frobenius’ theorem (see [7]) and the fact that for each function f ∈ C^{∞}(M) the one-parameter group corresponding to ξ̃(df) preserves ξ. Consequently, the rank ξ̃ is invariant under the action of this group.

2.13. Definition. The bivector field ξ is said to be nondegenerate at a point a ∈ M if the rank ξ_{a} = dim M. It is said to be nondegenerate on the manifold M if it is nondegenerate at each point of M.

2.14. If ξ is nondegenerate on M, then ξ̃ is an isomorphism defining the differential 2-form ω = ξ̃^{−1}(ξ), which is a symplectic form exactly when [ξ, ξ] = 0.

The Poisson bracket defined by ξ coincides with that defined by ω.

As mentioned in 2.12, if [ξ, ξ] = 0, then ξ defines the foliation on M perhaps with fibers of different dimensions. Let N be any fiber from this foliation and ξ_{N} be the restriction of ξ on the manifold N. It is easy to check that

(a) ξ_{N} ∈ V^{2}(N);

(b) ξ_{N} is nondegenerate on N.

Consequently,

(c) N is a symplectic manifold with the differential 2-form ω_{N} = ξ̃_{N}^{−1}(ξ_{N}).

3. Some Cohomology Properties of Bivector Fields

3.1. Let ξ be a bivector field on the manifold M. Setting u = ξ in equality (b) of 1.10, we obtain

[ξ, v ∧ w] = [ξ, v] ∧ w + (−1)^{n}v ∧ [ξ, w]

which implies that the endomorphism

[ξ, ] : V (M) ⟶ V (M)

is an antidifferentiation of degree 1:

[ξ, V^{m}(M)] ⊂ V^{m+1}(M), m ∈ ℤ.

Let [ξ, ξ] = 0. Then by (c) from 1.10 we obtain [ξ, [ξ, X]] = 0 for each X ∈ V (M). So the endomorphism [ξ, ] can be regarded as a coboundary operator defining some cohomology algebra H_{ξ}(M).

To investigate bivector fields from this standpoint we have to prove some propositions.

3.2. Lemma. If ξ is a bivector field with [ξ, ξ] = 0, then for each closed 1-form α we have [ξ, ξ̃(α)] = 0.

Proof. Using the local coordinate system x_{1}, …, x_{m}, the formula from 1.14, and the definition of ξ̃ (see 2.4), we find that for each i, j = 1, …, n we have (dx_{i} ∧ dx_{j})([ξ, ξ̃(α)]) = −(d((α ∧ dx_{i})(ξ)((·dx_{j} − (α ∧ dx_{j})(ξ) · dx_{i}))(ξ) + α ∧ d((dx_{i} ∧ dx_{j})(ξ)) = −(d((dx_{i} ∧ dx_{j})(ξ) · α − (dx_{i} ∧ α)(ξ) · dx_{j} + (dx_{j} ∧ α)(ξ) · dx_{i}))(ξ) = −\frac{1}{2}(dx_{i} ∧ dx_{j} ∧ α)([ξ, ξ]) = 0. Consequently, [ξ, ξ̃(α)] = 0.

3.3. Theorem. If ξ is a bivector field with [ξ, ξ] = 0, then the diagram

A(M) ⟶^{d} A(M)
_{ξ}↓  ↓_{ξ}
V (M) ⟶^{[ξ, ]} V (M)

is commutative.

Proof. So, the aim is to show that for each form ω we have ξ̃(dω) = [ξ, ξ̃(ω)]. It is suffiecient to show this for ω = f · dx_{1} ∧ … ∧ dx_{m}, where f, x_{1}, …, x_{m} are smooth functions on M:

ξ̃(ω) = f · ξ̃(dx_{1}) ∧ … ∧ ξ̃(dx_{m});
[ξ, ξ̃(ω)] = [ξ, f · ξ̃(dx_{1}) ∧ … ∧ ξ̃(dx_{m})] =
= [ξ, f · ξ̃(dx_{1})] ∧ ξ̃(dx_{2}) ∧ … ∧ ξ̃(dx_{m}) ±
±f · ξ̃(dx_{1}) ∧ [ξ, ξ̃(dx_{2}) ∧ … ∧ ξ̃(dx_{m})] =

= f · [ξ, ξ̃(dx_{1})] ∧ ξ̃(dx_{2}) ∧ … ∧ ξ̃(dx_{m}) +
+ξ̃(df) ∧ ξ̃(dx_{1}) ∧ … ∧ ξ̃(dx_{m}) ±
±f · ξ̃(dx_{1})] ∧ [ξ, ξ̃(dx_{2}) ∧ … ∧ ξ̃(dx_{m})].

The preceding lemma and formula (b) from 1.10 give

[ξ, ξ̃(dx_{i})] = [ξ, ξ̃(dx_{1}) ∧ … ∧ ξ̃(dx_{m})] = 0.

Eventually, [ξ, ξ̃(ω)] = ξ̃(df) ∧ ξ̃(dx_{1}) ∧ … ∧ ξ̃(dx_{m}) = ˜(ξ(dω).

3.4. To say otherwise, we have the following homomorphism of cochain complexes:

ℝ ⟶ A^{0}(M) = C^{∞}(M) ⟶^{d} A′(M) ⟶^{d} ⋯
Id↓  _{_{ξ}=Id}↓  _{ξ}↓
ℝ ⟶ V^{0}(M) = C^{∞}(M) ⟶^{[ξ, ]} V′(M) ⟶^{[ξ, ]} ⋯

where the top complex is that of De-Rham.

The above homomorphism defines the homomorphism between the De-Rham cohomology algebra H(M, ℝ) and the cohomology algebra H_{ξ}(M), which will also be denoted by ξ̃.

3.5. Example. Let M = T^{∗}(X) where X is any smooth manifold. As known, there is a canonical symplectic form ω on M (see [3], [4], [5]), defining the Poisson structure on C^{∞}(M). Consider the corresponding bivector field ξ_{ω} = ω^{−1}(ω) (see 2.5, 2.7). It is clear that ξ̃_{ω}(ω) = ξ_{ω}. Since ω = dλ, where λ is the Liouville form (see [3]), by the theorem from 3.3 we obtain ξ_{ω} = ξ̃(dλ) = [ξ_{ω}, ξ̃_{ω}(λ)].

It is easy to show that the vector field ξ̃_{ω}(λ) is the vector field corresponding to the one-parameter group φ_{t}(u) = e^{−t} · u, t ∈ ℝ, u ∈ T^{∗}(X). Otherwise, ξ̃_{ω}(λ)|_{u} = −u.

3.6. Example. Let L be a finite-dimensional real vector space and s : L ∧ L ⟶ L be any linear map. We have the bivector field ξ on the manifold M = L^{∗} defined by means of s. Clearly, T^{∗}(M) = L^{∗} × L and for each point a ∈ L^{∗} we have ∧^{2}T_{a}^{∗}(M) = L ∧ L. Now we define ξ as follows: let α(ξ_{a}) = a(s(α)) for a ∈ L^{∗} and α ∈ ∧^{2}T_{α}^{∗}(M).

3.7. Theorem. The equality [ξ, ξ] = 0 for the above-defined bivector field holds if and only if the linear map s defines the structure of a Lie algebra on L, i.e., we have

s(s(u ∧ v) ∧ w) + s(s(w ∧ u) ∧ v) + s(s(v ∧ w) ∧ u) = 0

for each u, v, w ∈ L.

Proof. Let {u, v, w} ⊂ L and ω = u ∧ v ∧ w be an element of V^{3}(L^{∗}). Clearly, dω = 0 and for p ∈ L^{∗} we have (i_{ξ}ω)|_{p} = ((u ∧ v)(ξ) · w + (w ∧ u)(ξ) · v + (v ∧ w)(ξ) · u)|_{p} = p(s(u ∧ v)) · w + p(s(w ∧ u)) · v + p(s(v ∧ w)) · u. As one can see, the form i_{ξ}ω depends linearly on p and therefore d(i_{ξ}ω) = s(u ∧ v) ∧ w + s(w ∧ u) ∧ v + s(v ∧ w) ∧ u.

Using the formula from 1.14, we obtain ω([ξ, ξ])|_{p} = 2d(i_{ξ}ω)(ξ)|_{p} = p(s(s(u ∧ v) ∧ w + s(s(w ∧ u)) ∧ v + s(s(v ∧ w)) ∧ u)), p ∈ L^{∗}.

Thus [ξ, ξ] = 0 exactly when ω([ξ, ξ])|_{p} = 0 for each ω = u ∧ v ∧ w and p ∈ L^{∗}; otherwise, p(s(s(u ∧ v) ∧ w) + s(s(w ∧ u) ∧ v) + s(s(v ∧ w) ∧ u)) = 0 for each p ∈ L^{∗}, which is the same as s(s(u ∧ v) ∧ w + s(s(w ∧ u)) ∧ v + s(s(v ∧ w)) ∧ u) = 0.

3.8. We have ascertained that ξ defines the Poisson structure on C^{∞}(L^{∗}) if and only if the bracket [u, v] = s(u ∧ v) defines the structure of a Lie algebra on L.

Clearly, L is a subspace of C^{∞}(L^{∗}). Moreover, L is a Lie subalgebra of the Poisson algebra C^{∞}(M) and the bracket [ , ] coincides with the Poisson bracket { , } on L: for u, v ∈ L we have {u, v}(p) = (u, v)(ξ)|_{p} = p([u, v]), p ∈ L^{∗}. Finally, we find that the element [u, v] as a linear function on L^{∗} coincides with {u, v}.

3.9. Let us consider the exterior algebra ∧(L^{∗}) = ∑_{k∈ℤ} ∧^{k}L^{∗}. Clearly, ∧(L^{∗}) is a subalgebra of the exterior algebra V (L^{∗}).

Theorem. The subalgebra ∧(L^{∗}) in V (L^{∗}) is an invariant subspace of the operator [ξ, ], and [ξ, ] : ∧(L^{∗}) ⟶ ∧(L^{∗}) is the Chevalley–Eilenberg operator (see the operator ∂_{1} in 1.3) defining the cohomology of the Lie algebra L with coefficients in ℝ.

Proof. Let α ∈ ∧^{k}L^{∗}. Then [ξ, α] ∈ ∧^{k+1}L^{∗}. We must prove that for u_{1} ∧ … ∧ u_{k+1} ∈ ∧^{k+1}L ⊂ A^{k+1}(L^{∗}) we have (u_{1} ∧ … ∧ u_{k_{1}})([ξ, α]) = ∑_{i<j}(−1)^{i+j−1}α([u_{i}, u_{j}], u_{1}, …, u_{k+1}) (recall that for X ∈ ∧^{m}L^{∗} ⊂ V^{m}(L^{∗}) and λ ∈ ∧^{m}L ⊂ A^{m}(L^{∗}) we have λ(X) = X(λ)) : (u_{1} ∧ … ∧ u_{k+1})([ξ, α]) = (−1)^{k}(d(i_{α}(u_{1} ∧ … ∧ u_{k+1})))(ξ) + (d(i_{ξ}(u_{1} ∧ … ∧ u_{k+1})))(α) − (d(u_{1} ∧ … ∧ u_{k+1}))(ξ ∧ α). Clearly,

d(i_{α}(u_{1} ∧ … ∧ u_{k+1})) = d(u_{1} ∧ … ∧ u_{k+1}) = 0;

i_{ξ}(u_{1} ∧ … ∧ u_{k+1})|_{p} = ∑_{i<j}(−1)^{i+j−1}p([u_{i}, u_{j}]) · u_{1} ∧ … ∧ u_{k+1}

p ∈ L^{∗} and d(i_{ξ}(u_{1} ∧ … ∧ u_{k+1})) = ∑_{i<j}(−1)^{i+j−1}[u_{i}, u_{j}]) ∧ u_{1} ∧ … ∧ û_{i} ∧ … ∧ û_{j} ∧ … ∧ u_{k+1}.

Therefore we obtain

(u_{1} ∧ … ∧ u_{k+1})([ξ, α]) = [ξ, α](u_{1}, …, u_{k+1}) =

= α( ∑_{i<j}(−1)^{i+j−1}[u_{i}, u_{j}] ∧ u_{1} ∧ … ∧ û_{i} ∧ … ∧ û_{j} ∧ … ∧ u_{k+1}).

3.10. Let Ā(M) be a sheaf of local differential forms on M and V̄ (M) be a sheaf of local polyvector fields on M. Since the diagram in 3.3 is commutative, the diagram of morphisms of sheaves

Ā(M) ⟶^{d} Ā(M)
_{ξ}↓  ↓_{ξ}
V̄ (M) ⟶^{[ξ, ]} V̄ (M)

will also be commutative. Therefore we can talk about the sheave Īmξ̃, with the coboundary operator [ξ, ] : Ī_{m}ξ̃ ⟶ Ī_{m}ξ̃. On the global sections of Ī_{m}ξ̃ the operator [ξ, ] defines some cohomology algebra which will be denoted by h_{ξ}(M). The homomorphism ξ̃ induces a homomorphism from H(M, ℝ) into h_{ξ}(M). The element ξ ∈ V^{2}(M) defines some cohomology class [ξ] ∈ h_{ξ}(M).

3.11. Let N be any integral manifold of the differential system Imξ̃. Then the restriction map I_{m}ξ̃ ∋ X ⟶ X_{N} ∈ V (N) induces a homomorphism from h_{ξ}(M) into H_{ξ_{N}} (N). Since the bivector field ξ_{N} is nondegenerate, there is an isomorphism ξ̃_{N} : H(N, ℝ) ⟶ H_{ξ_{N}} (N) and therefore we have a homomorphism from h_{ξ}(M) into H(N, ℝ). Finally, we find that for each N which is an integral manifold of Imξ̃ there is a homomorphism

r_{N} : h_{ξ}(M) ⟶ H(N, ℝ).

3.12. Let us return to 3.6, 3.7, 3.8. As was proved, the canonical bivector field ξ on L^{∗}, where L is a Lie algebra, is such that [ξ, ξ] = 0. Therefore ξ defines the foliation in L^{∗}. One can show that if L is a Lie algebra of the connected Lie group G, then the orbits of the Ad^{∗}G-representation (see [1], [4]) are just the fibers of the foliation defined by ξ, while for each fiber N the symplectic form ξ_{N}^{−1}(ξ_{N}) is just the Souriau–Kostant form on the orbits of the coadjoint representation.

If the cohomology class [ξ] ∈ h_{ξ}(M) is zero, then, as follows from 3.11, each orbit satisfies the Souriau–Kostant prequantization condition (see [8]).

References

1. D. B. Fuks, Cohomology of infinite-dimensional Lie algebras. (Russian) Nauka, Moscow, 1984.

2. A. Lichnerowicz, Les variéteés de Poisson et leurs algébres de Lie associées. J. Diff. Geom. 12(1977), 253–300.

3. C. Godbillon, Géometrie différentielle et mécanique analytique. Collection Methods. Hermann, Paris, 1966.

4. P. J. Olver, Applications of Lie groups to differential equations. Springer-Verlag, New York, 1986.

5. R. Abraham and J. E. Marsden, Foundations of mechanics, 2nd ed. Benjamin–Cummings, Reading, Mass., 1978.

6. P. A. Griffiths, Exterior differential systems and the calculus of variations. Birkhäuser, Boston–Basel–Stuttgart, 1983.

7. R. Hermann, The differential geometry of foliations. J. Math. Mech. 11(1962), 303–315.

8. V. Guillemin and S. Sternberg, Geometric asymptotics. Mathematical Surveys, No. 14. Amer. Math. Soc., Providence, R.I. 1977.

(Received 21.10.1993)

Author’s address: Department of Applied Mathematics Georgian Technical University 77, M. Kostava St., Tbilisi 380075 Republic of Georgia