Normal forms of vector fields on Poisson manifolds

We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.


Introduction
This paper is devoted to the study of normal formsà la Poincaré-Birkhoff for analytic or formal vector fields on Poisson manifolds. We will be interested in two kinds of vector fields, namely Hamiltonian vector fields, and "radial" vector fields, i.e. those vector fields X such that [X, Π] = L X Π = −Π, where Π denotes the Poisson structure, and the bracket is the Schouten bracket. Our motivation for studying radial vector fields comes from Jacobi structures [7], while of course the main motivation for studying Hamiltonian vector fields comes from Hamiltonian dynamics. We will assume that our vector field X vanishes at a point, X(0) = 0, and that the linear part of Π or of its transverse structure at 0 corresponds to a semisimple Lie algebra. In this case, it is well known [13,4] that Π admits a formal or analytic linearization in a neighborhood of 0. We are interested in a simultaneous linearization or normalization of Π and X.
In Section 2, we study the problem of simultaneous linearization of couples (Π, X) where Π is a Poisson structure and X is a vector field such that L X Π = −Π. Such couples are called homogeneous Poisson structures in the sense of Dazord, Lichnerowicz and Marle [7], and they are closely related to Jacobi manifolds. Such structures are closely related to Jacobi structures. More precisely, a 1-codimensional submanifold of a homogeneous Poisson manifold (M, Π, X) which is transverse to the vector field X has an induced Jacobi structure, and all Jacobi manifolds can be obtained in this way. On the other hand, a 1-codimensional submanifold of a Jacobi manifold (N, Λ, E) transverse to the structural vector field E has an induced homogeneous Poisson structure, and all homogeneous Poisson manifolds can be obtained in this way (see [7]). Our first result is the following (see Theorem 2.4): Theorem A. Let (Π, X) be a formal homogeneous Poisson structure on K n (where K is C or R) such that the linear part Π 1 of Π corresponds to a semisimple Lie algebra g. Suppose that its linear part (Π 1 , X (1) ) is semisimple nonresonant. Then there exists a formal diffeomorphism which sends (Π, X) to (Π 1 , X (1) ).
The semisimple nonresonant condition in the above theorem is a generic position on X (1) : the set of X (1) which does not satisfy this condition is of codimension 1, and moreover if X (1) − I is diagonalizable and small enough, where I = x i ∂ ∂xi denotes the standard radial (Euler) vector field, then the semisimple nonresonant is automatically satisfied.
For analytic linearization, due to possible presence of small divisors, we need a Diophantine-type condition. Here we choose to work with a modified Bruno's ω-condition [2,3] adapted to our case. See Definition 2.5 for the precise definition of our ω-condition. The set of (Π 1 , X (1) ) which satisfy this ω-condition is of full measure. We have (see Theorem 2.7): Theorem B. Let (Π, X) be an analytic homogeneous Poisson structure on K n (where K is C or R) such that the linear part Π 1 of Π corresponds to a semisimple Lie algebra. Suppose moreover that its linear part (Π 1 , X (1) ) is semisimple nonresonant and satisfies the ω-condition. Then there exists a local analytic diffeomorphism which sends (Π, X) to (Π 1 , X (1) ).
In Section 3, we study local normal forms of Hamiltonian systems on Poisson manifolds. According to Weinstein's splitting theorem [13], our local Poisson manifold ((K n , 0), Π), where K = R or C, is a direct product ((K 2l , 0), Π symp ) × ((K m , 0), Π trans ) of two Poisson manifolds, where the Poisson structure Π symp is nondegenerate (symplectic), and the Poisson structure Π trans (the transverse structure of Π at 0) vanishes at 0. If Π trans is trivial, i.e. the Poisson structure Π is regular near 0, then the problem local normal forms of Hamiltonian vector fields near 0 is reduced to the usual problem of normal forms Hamiltonian vector fields (with parameters) on a symplectic manifold. Here we are interested in the case when Π trans is not trivial. We will restrict our attention to the case when the linear part of Π trans corresponds to a semisimple Lie algebra g. According to linearization theorems of Weinstein [13] and Conn [4], we may identify ((K m , 0), Π trans ) with a neighborhood of 0 of the dual g * of g equipped with the associated linear (Lie-Poisson) structure. In other words, there is a local system of coordinates (x 1 , y 1 , . . . , x l , y l , z 1 , . . . , z m ) on K 2l+m such that Π symp = with c k ij being structural constants of g, and Such a coordinate system will be called a canonical coordinate system of Π near 0. Let H be a formal or analytic function on ((K n , 0), Π). We will assume that the Hamiltonian vector field X H of H vanishes at 0. Note that the differential of H does not necessarily vanish at 0 (for example, if n = 0 then we always have X H (0) = 0 for any H). We may assume that H(0) = 0.
We have the following generalization of Birkhoff normal form [1] (see Theorem 3.1): Theorem C. With the above notations and assumptions, there is a formal canonical coordinate system (x i ,ŷ i ,ẑ j ), in which H satisfies the following equation: where H ss is a (nonhomogeneous quadratic) function such that its Hamiltonian vector field X Hss is linear and is the semisimple part of the linear part X (1) H of X H (in this coordinate system). In particular, the semisimple part of the linear part of X H is a Hamiltonian vector field.
Note that the normalizing canonical coordinates given in the above theorem are only formal in general. The problem of existence of a local analytic normalization for a Hamiltonian vector field (even in the symplectic case) is much more delicate than for a general vector field, due to "auto-resonances" (e.g, if λ is an eigenvalue of a Hamiltonian vector field then −λ also is). However, there is one particular situation where one knows that a local analytic normalization always exists, namely when the Hamiltonian vector field is analytically integrable. See [16] for the case of integrable Hamiltonian vector fields on symplectic manifolds. Here we can generalize the main result of [16] to our situation (see Theorem 3.8): Theorem D. Assume that K = C, the Hamiltonian function H in Theorem C is locally analytic, and is analytically integrable in the generalized Liouville sense. Then the normalizing canonical coordinate system (x i ,ŷ i ,ẑ j ) can be chosen locally analytic.
We conjecture that the above theorem remains true in the real case (K = R). Recall (see, e.g., [15] and references therein) that a Hamiltonian vector field X H on a Poisson manifold (M, Π) of dimension n is called integrable in generalized Liouville sense if there are nonnegative integers p, q with p + q = n, p pairwise commuting Hamiltonian functions H 1 , . . . , H p ({H i , H j } = 0 ∀i, j) with H 1 = H and q first integrals F 1 , . . . , F p , such that X Hi (F j ) = 0 ∀ i, j, and dF 1 ∧. . .∧dF q = 0 and X H1 ∧ . . . ∧ X Hp = 0 almost everywhere. (The Liouville case corresponds to p = q = n/2 and F i = H i ). Analytic integrability means that all Hamiltonian functions and vector fields in question are analytic.

Homogeneous Poisson structures
Following [7], we will use the following terminology: a homogeneous Poisson structure on a manifold M is a couple (Π, X) where Π is a Poisson structure and X a vector field which satisfies the relation where the bracket is the Schouten bracket.
Remark 2.1. Poisson structures which satisfy the above condition are also called exact, in the sense that the Poisson tensor is a coboundary in the associated Lichnerowicz complex which defines Poisson cohomology. They have nothing to do with another kind of homogeneous spaces, namely those which admit a transitive group action.
An analog of Weinstein's splitting theorem for homogeneous Poisson structures is given in [7], and it reduces the study of normal forms of homogeneous Poisson structures to the case when both Π and X vanish at a point. So we will assume that (Π, X) is a homogeneous Poisson structure defined in a neighborhood of 0 in K n , where K = R or C, such that (2.2) Π(0) = 0 and X(0) = 0 .
We are interested in the linearization of these structures, i.e. simultaneous linearization of Π and X. Denote by Π 1 and X (1) the linear parts of Π and X respectively. Then the terms of degree 1 of Equation (2.1) imply that (Π 1 , X (1) ) is again a homogeneous Poisson structure.
In this paper, we will assume that the linear Poisson structure Π 1 corresponds to a semisimple Lie algebra, which we denote by g. Then, according to linearization results of Weinstein [13] (for the formal case) and Conn [4] (for the analytic case), the Poisson structure Π can be linearized. In other words, there is a local coordinate system (x 1 , . . . , x n ) on (K n , 0), in which where c k ij are structural constants of g. In order to linearize (Π, X) = (Π 1 , X), it remains to linearize X by local (formal or analytic) diffeomorphisms which preserve the linear Poisson structure Π 1 .

Formal linearization.
First consider the complex case (K = C). Let X be a formal vector field on C n such that (Π 1 , X) forms a homogeneous Poisson structure on C n . Denote by the Euler vector field written in coordinates (x 1 , . . . , x n ). Since this vector field satisfies the relation [I, Π 1 ] = −Π 1 , we can write X as where Y is a Poisson vector field with respect to Π 1 , i.e., [Y, Π 1 ] = 0. It is wellknown that, since the complex Lie algebra g is semisimple by assumptions, the first formal Poisson cohomology space of Π 1 is trivial (see, e.g., [5]), i.e. any formal Poisson vector field is Hamiltonian. In particular, we have for some formal function h. Writing the Taylor expansion h = h 1 + h 2 + h 3 + · · · where each h r is a polynomial of degree r, we have Denote by X (1) = I + X h1 the linear part of X. In order to linearize X (while preserving the linearity of Π = Π 1 ), we want to kill all the terms X hr with r ≥ 2, using a sequence of changes of coordinates defined by flows of Hamiltonian vector fields with respect to Π 1 . Working degree by degree, we want to find for each r a homogeneous polynomial g r of degree r such that (2.9) [X (1) , X gr ] = X hr .
Note that [X h1 , X gr ] = X {h1,gr} , and [I, X gr ] = (r − 1)X gr because X gr is homogeneous of degree r. Hence Relation (2.9) will be satisfied if g r satisfies the following relation: Remark that, h 1 can be viewed as an element of g, and h r , g r may be identified with elements of the symmetric power S r (g) of g. Under this identification, {h 1 , g r } is nothing but the result of the adjoint action of h 1 ∈ g on g r ∈ S r (g).
We will suppose that h 1 is a semisimple element of g, and denote by h a Cartan subalgebra of g which contains h 1 . According to the root decomposition of g with respect to h, we can choose a basis (x 1 , . . . , x n ) of g, and elements α 1 , . . . , α n of h * , such that (2.11) [y, Each α i is either 0 (in which case x i ∈ h) or a root of g (in which case x i belongs to the root subspace g αi of g).
We define for each r ≥ 2 the linear operator Each monomial i x λi i of degree |λ| = λ i = r is an eigenvector of this linear operator: Definition 2.2. With the above notations, we will say that (Π 1 , X (1) ) is semisimple nonresonant if h 1 is a semisimple element of g and the eigenvalues of Θ r don't vanish, i.e., for any r ≥ 2 and any (λ 1 , . . . , λ n ) ∈ Z n + such that λ i = r we have r − 1 + n i=1 λ i α i , h 1 = 0. Remark 2.3. It is easy to see that the above nonresonance condition is a generic position condition, and the subset of elements which do not satisfy this condition is of codimension 1. In fact, if the Cartan subalgebra h is fixed, then the set of elements h 1 ∈ h such that (Π 1 , I + X h1 ) is resonant is a countable union of affine hyperplanes in h which do not contain the origin, and there is a neighborhood of 0 in h such that if h 1 belongs to this neighborhood then (Π 1 , X (1) ) is automatically nonresonant.
The algorithm of formal linearization. We now show how to linearize (Π 1 , X), by killing the nonlinear terms of h step by step, provided that (Π 1 , X (1) ) is nonresonant. Actually, at each step, we will kill not just one term h d , but a whole block of 2 d consecutive terms. This "block killing" will be important in the next section when we want to show that, under some Diophantine-type condition, our formal linearization process actually yields a local analytic linearization.
For each q ≥ 0, denote byÔ q the space of formal power series on C n of order greater or equal to q, i.e. without terms of degree < q.
We begin with X = X (1) modÔ 2 , and will construct a sequence of formal vector fields (X d ) d and diffeomorphisms (ϕ d ) d , such that X 0 = X and, for all Assuming that we already have X d for some d ≥ 0, we will construct ϕ d (and H d is a sum of homogeneous polynomials of degrees between 2 d + 1 and 2 d+1 (we also could write d is homogeneous of degree u) and the same for H d , we have We then have where H d+1 is a polynomial of degree 2 d+2 inÔ 2 d+1 +1 .
Constructed in this way, it is clear that the successive compositions of the diffeomorphisms ϕ d converge in the formal category to a formal diffeomorphism Φ ∞ which satisfies Φ ∞ * X = X (1) and which preserves the linear Poisson structure Π 1 .
Consider now the real case (K = R, and g is a real semisimple Lie algebra). By complexication, we can view real objects as holomorphic objects with real coefficients, and then repeat the above algorithm. In particular, under the nonresonance condition, we will find homogeneous polynomials G is also real. This means that the coordinate transformations constructed above are real in the real case.
We have proved the following: Theorem 2.4. Let (Π, X) be a formal homogeneous Poisson structure on K n (where K is C or R) such that the linear part Π 1 of Π corresponds to a semisimple Lie algebra. Assume that its linear part (Π 1 , X (1) ) is semisimple nonresonant. Then there exists a formal diffeomorphism which sends (Π, X) to (Π 1 , X (1) ).

2.2.
Analytic linearization. Now we work in the local analytic context, i.e. the vector field X is supposed to be analytic on (K n , 0). In order to show that the algorithm given in the previous subsection leads to a local analytic linearization, in addition to the nonresonance condition we will need a Diophantine-type condition, similar to Bruno's ω-condition for the analytic linearization of vector fields [2,3].
Keeping the notations of the previous subsection, for each d ≥ 1, put (2.20) Definition 2.5. We will say that X (1) , or more precisely that a semisimple nonresonant linear homogeneous Poisson structure (Π 1 , X (1) ) satisfies the ω-condition if Remark that, similarly to other situations, the set of X (1) which satisfy the about ω-condition is of full measure. More precisely, we have: Proposition 2.6. The set of elements h of a given Cartan subalgebra h such that See the Appendix for a straightforward proof of the above proposition.
Using the same analytical tools as in the proof of Bruno's theorems about linearization of analytic vector fields [2,3], we will show the following theorem: Theorem 2.7. Let (Π, X) be an analytic homogeneous Poisson structure on (K n , 0) (where K is C or R) such that the linear part Π 1 of Π corresponds to a semisimple Lie algebra. Suppose that its linear part (Π 1 , X (1) ) is semisimple nonresonant and satisfies the ω-condition. Then there exists a local analytic diffeomorphism which sends (Π, X) to (Π 1 , X (1) ).
Proof. Due to Conn's theorem [4], we can assume that Π = Π 1 is already linear. The process to linearize the vector field X is the same as in the formal case, noting that if we start with an analytic vector field, the diffeomorphisms ϕ d that we constructed will be analytic too (as is the vector fields X d ). We just have to check the convergence of the sequence Φ d = ϕ d • . . . • ϕ 1 in the analytic setup.
We will assume that K = C (the real case can be reduced to the complex case by the same argument as given in the previous subsection). Denote by O q the vector space of local analytic functions of (K n , 0) of order greater or equal to q (i.e. without terms of degree < q).
For each positive real number ρ > 0, denote by D ρ the ball {x = (x 1 , . . . , x n ) ∈ C n ; |x i | < ρ} and if f = λ∈N n a λ x λ is an analytic function on D ρ we define the following norms: In the same way, if F = (F 1 , . . . , F n ) is a vector-valued local map then we put |F | ρ := max{|F 1 | ρ , . . . , |F n | ρ } and similarly for ||F || ρ . These norms satisfy the following properties.
Lemma 2.8. Let ρ and ρ ′ be two real numbers such that 0 < ρ ′ < ρ. If f ∈ O q is an analytic function on D ρ , then a) c) Let R > 0 be a positive constant. Then there is a natural number N such that for The proof of the above lemma is elementary (see the Appendix).
It is important to remark that, with the same notations as in the formal case, for ρ > 0, we have, by (2.12): Put ρ 0 = 1, and define the following two decreasing sequences of radii (r d ) d and (ρ d ) d by and it is clear, by the ω-condition (2.21), that the sequences (r d ) d and (ρ d ) d converge to a strictly positive limit R > 0. Moreover, they satisfy the following properties: The proof of Lemma 2.9 is elementary (see the Appendix).
and moreover, we have Proof . • We first prove the second inclusion : We have where H d is a polynomial formed by homogenous terms of degree between 2 d + 1 and 2 d+1 . By (2.27), we write Then, by (2.25), we get And, using the assumption This map is continuous and is the identity on the boundary of D r d thus, by As a conclusion, if y is in D r d+1 then, by the surjectivity ofφ d , y =φ d (z) with, a priori, z in D r d . We saw above that in fact z cannot be in D r d /D ρ d . Therefore, We write the obvious inequality By (2.25), we have To do that, we use the inequalities of Lemma 2.8. The drawback of these inequalities is that they sometimes induce a change of radius. Therefore, we define the following intermediar radii (between ρ d and r d ) : Let us explain a little bit the definitions of these radii : d is defined in order to use inequality (2.26). -Finally, if d is sufficiently large, the differences ρ d are strictly smaller than r d 5d 2 and then, We can write ϕ t d = Id + ξ t d where the n components of ξ t d are functions in O 2 d +1 . We have the estimates We then deduce by (2.43) that Finally, we just have to estimate [X G d , X d ]) r d . We first have by (2.24), which gives, by (2.26), recalling that ω d ≤ 1 2d , Using (2.27) and (2.25), we get and then In the same way, one can prove that In addition, by (2.25), we get We deduce finally that This gives the following estimate , and the conclusion follows.
End of the proof of Theorem 2.7. Let d 0 be a positive integer such that Lemmas 2.9 and 2.10 are satisfied for d ≥ d 0 . By the homothety trick (dilate a given coordinate system by appropriate linear transformations), we can assume that |X d0 − X (1) | ρ d 0 −1 < 1.
By recurrence, for all d ≥ d 0 , we have We consider the sequence (Ψ d ) d given by Let x be an element of D R ; recall that R > 0 is the limit of the decreasing sequences (r d ) d and (ρ d ) d . Then x belongs to the ball D r d+1 for any d and if d > d 0 , we get by (2.58), and iterating this process, we obtain we then obtain, for all x in D R and all d > d 0 , The theorem follows.

Hamiltonian vector fields on Poisson manifolds
In this section, we study normal forms of formal or analytic Hamiltonian vector fields in the neighborhood of the origin on the Poisson manifold (K 2l+m , Π), where Here Π symp = l i=1 ∂ ∂xi ∧ ∂ ∂yi is the standard symplectic Poisson structure on K 2n , and Π trans = Π g = 1 2 i,j,k c k ij z k ∂ ∂zi ∧ ∂ ∂zj is the associated linear Poisson structure on the dual of a given semisimple Lie algebra g of dimension m over K.
Let H : (K 2l+m , 0) → (K, 0) be a formal or local analytic function with H(0) = 0, and consider the Hamiltonian vector field X H of H with respect to the above Poisson structure Π = Π symp + Π g . If X H (0) = 0, then it is well-known that it can be rectified, i.e. there is a local canonical coordinate system (x 1 , y 1 , . . . , x l , y l , z 1 , . . . , z m ) in which H = x 1 and X H = ∂ ∂y1 . Here we will assume that X H (0) = 0 3.1. Formal Poincaré-Birkhoff normalization. In this subsection, we will show that the vector field X H can be put formally into Poincaré-Birkhoff normal form.
More precisely, we have: Theorem 3.1. With the above notations, for any formal or local analytic function H : (K 2l+m , 0) → (K, 0), there is a formal canonical coordinate system (x i ,ŷ i ,ẑ j ), in which the Poisson structure Π has the form and in which we have where H ss is a function such that X Hss is the semisimple part of the linear part of X H , and Proof. For any function f on K 2l+m , we can write the Hamiltonian vector field of f with respect to Π symp (resp. Π g ). We can write H = p,q H p,q where H p,q is a polynomial of degree p in x, y and of degree q in z.
A difficulty of our situation comes from the fact that Π is not homogeneous. If p > 0 then X H p,q is not a homogeneous vector field but the sum of a homogeneous vector field of degree p + q (given by X g H p,q ) and a homogeneous vector field of degree p + q − 1 (given by X symp H p,q ). Note that X g H 0,q is homogeneous of degree q and of course X symp H 0,q = 0. Denoting by X (1) the linear part of X H , we have This linear vector field X (1) is not a Hamiltonian vector field in general, but we will show that its semisimple part is Hamiltonian. By complexifying the system if necessary, we will assume that K = C. By a linear canonical change of coordinates, we can suppose that the semisimple part of X H 2,0 is X h2 where h 2 (x, y) = l j=1 γ i x j y j (γ j ∈ C) and that the semisimple part of X H 0,1 is X h1 where h 1 belongs to a Cartan subalgebra h of g. We write : Remark that we can assume that α s+1 = . . . = α m = 0 where m−s is the dimension of the Cartan subalgebra h. Denote α = (α 1 , . . . , α m ) and γ = (γ 1 , . . . , γ l ). If λ, µ ∈ Z l + and ν ∈ Z m + then where, for example, α, ν = α j ν j denotes the standard scalar product of α and ν. In particular, {h 1 + h 2 , .} acts in a "diagonal" way on monomials.
We can arrange so that, written as a matrix, the terms coming from X H 0,1 −h1 , X H 2,0 −h2 and X symp H 1,1 in the expression of X (1) are off-diagonal upper-triangular (and the terms coming from X h1+h2 are on the diagonal).
If {h 1 + h 2 , H 1,1 } = 0 then we can apply some canonical changes of coordinates to make (the new) H 1,1 commute with h 1 + h 2 as follows. According to (3.7), there exist two polynomials G 1,1 (1) and G 1,1 (1) of degree 1 in x, y and 1 in z such that Remark that, for any homogeneous polynomials K 0,1 , K 2,0 , K 1,1 of corresponding degrees in (x, y) and z, we have Change the coordinate system by the push-forward of the time-1 flow ϕ (1) = exp X G 1,1 (1) of the Hamiltonian vector field X G 1,1 (1) , i.e., x new i = x i • ϕ (1) and so on. The new coordinate system is still a canonical coordinate system, because ϕ (1) preserves the Poisson structure Π. By this canonical change of coordinates, we can replace H by H • ϕ (1) , and X H by (3.11) X new , , , X H ]] + . . .

It follows from (3.11) and (3.10) that the linear part of
(1) ,F1} . In particular, by the above canonical change of coordinates, we have replaced (1) , F 1 } is homogeneous of degree 1 in (x, y) and degree 1 in z).
Since F 1 is "nilpotent", by iterating the above process a finite number of times, By recurrence, assume that, for some r ≥ 2, we have {H ss , H k } = 0 for all k ≤ r − 1. We will change H r by a canonical coordinate transformation to get the same equality for k = r.
In order to put H r in normal form, we use the same method that we used to normalize H 1,1 . Similarly to (3.8), we can write where K r andK r are of the same type as H r (i.e., they are sums of monomials of bidegrees (0, r) and (p, r + 1 − p) with p > 0), {H ss ,K r } = 0. Note K r can be written as K r = {H ss , K (2)r } for some K (2)r .
The canonical coordinate transformation given by the time-1 flow exp X Kr of X Kr leaves H 1 , . . . , H r−1 intact, and changes H r =K r + {H ss , K r } to the sum of K r with the terms of appropriate bidegrees in {K r , F 1 + H 1,1 }. We will write it as (3.20)K r + {K r , F 1 + H 1,1 } mod (terms of higher bidegrees).
It can also be written as (3.21)K r + {H ss , {K (2)r , F 1 + H 1,1 }} mod (terms of higher bidegrees). Now apply the canonical coordinate transformation given by exp X {K (2)r ,F1+H 1,1 } , and so on. Since F 1 + H 1,1 is "nilpotent", after a finite number of coordinate transformations like that, we can change H r toK r , which commutes with H ss . Denote the composition of these coordinate changes (for a given r) as φ r . Note that φ r is of the type Thus, the sequence of local or formal Poisson-structure-preserving diffeomorphisms (Φ r ) r≥2 , where Φ r = φ r • . . . • φ 2 , converges formally and gives a formal normalization of H.
Finally, notice that, in the real case (K = R), by an argument similar to the one given in the previous section, all canonical coordinate transformations constructed above can be chosen real.
Theorem 3.1 is proved.
Remark 3.2. In Theorem 3.1, if we forget the Lie algebra g and just keep the symplectic structure, then we recover the classical Birkhoff normalization for Hamiltonian vector fields on symplectic manifolds (see, e.g., [1,3,11,16]). On the other hand, if we forget the symplectic part and just deal with g * then we get the following result as a particular case: Corollary 3.3. Let h be a local analytic or formal function , with h(0) = 0 and dh(0) = 0, on the dual g * of a semisimple Lie algebra with the associated Lie-Poisson structure. Then the Hamiltonian vector field X h admits a formal Poincaré-Birkhoff normalization, i.e., there exists a formal coordinate system in which the Poisson structure is linear and in which we have where h ss is the semisimple part of dh(0) in g.
Example 3.4. The monomials x λ y µ z ν such that γ, µ − λ + α, ν = 0 in (3.7) may be called resonant terms. In the two following examples we give the set of all resonant terms in the case of a trivial symplectic part. a) g = sl (2). In this case, a Cartan subalgebra h of g is of dimension 1 and there are only two roots {α, −α}. Denote by z 1 , z 2 , z 3 a basis of g (or a coordinate system on g * ) such that z 1 (resp. z 2 ) spans the root space associated to α (resp. −α) and z 3 spans the Cartan subalgebra. We suppose that in the decomposition (3.16) we have h 1 = z 3 . Then the resonant terms are formal power expansion in the variables ω = z 1 z 2 and z 3 . b) g = sl(3). Here a Cartan subalgebra h is of dimension 2 (see for instance [8]). There are 6 roots {α 1 , α 2 , α 3 , −α 1 , −α 2 , −α 3 } and the relations between these roots are of type If {ξ 1 , ξ 2 , ξ 3 , ζ 1 , ζ 2 , ζ 3 , z 1 , z 2 } is a basis of g such that ξ j (resp. ζ j ) spans the root space associated to α j (resp. −α j ) and {z 1 , z 2 } spans h, then supposing that in the decomposition (3.16) h 1 is a linear combination of z 1 and z 2 we may write the resonant terms as formal power expansion formed by monomials of type

Analytic normalization for integrable Hamiltonian systems.
Here, we assume that we work in the complex analytic setup.
Of course, the elements (λ, µ, ν) of R correspond to the resonant monomials i.e. terms of type x λ y µ z ν such that {H ss , x λ y µ z ν } = 0. The dimension of R may be called the degree of resonance of H. Now, we consider the sublattice Q of Z 2l+m formed by vectors a ∈ Z 2l+m such that a | u = 0 for all u in R. Let {ρ (1) , . . . , ρ (r) } be a basis of Q. The dimension r of Q is called the toric degree of X H at 0. We then put for all k = 1, . . . , r The vector fields iZ 1 , . . . , iZ r (i = √ −1) are periodic with a real period in the sense that the real part of these vector fields is a periodic real vector field in C 2l+m = R 2(2l+m) ; they commute pairwise and are linearly independent almost everywhere. Moreover, the vector field X Hss is a linear combination (with coefficients in C a priori) of the iZ k . We also have the trivial following property Proof : We just give here the idea of the proof of this lemma supposing that Λ is a 2-vector for instance ; but it works exactly in the same way for other multivectors. If Y is a vector field of type l j=1 a j x j ∂ ∂xj + l j=1 a l+j y j ∂ ∂yj + m j=1 a 2l+j z j ∂ ∂zj and Λ of type Λ = x λ y µ z ν ∂ ∂xu ∧ ∂ ∂xv , then = (0, . . . , 1, . . . , 0) is the vector of Z l whose unique nonzero component is the u-component. Of course we get the same type of relation with 2-vectors in ∂ ∂x ∧ ∂ ∂z , ∂ ∂x ∧ ∂ ∂x , etc.... Using this remark and the definition of the vectors ρ (1) , . . . , ρ (r) , the equivalence of the lemma is direct.
According to this Lemma, since X Hss preserves the Poisson structure, the vector fields Z 1 , . . . , Z r will be Poisson vector fields for (C 2l × C m , { , } symp + { , } g * ). But according to Proposition 4.1 (see the Appendix), the Poisson cohomology space is trivial therefore, these vector fields are actually Hamiltonian : Finally, we have r periodic Hamiltonian linear vector fields iZ k which commute pairwise, are linearly independent almost everywhere. The real parts of these vector fields generate a Hamiltonian action of the real torus T r on (C 2n × C m , { , } symp + { , } g ). With all these notations, we can state the following proposition : Proposition 3.6. With the above notation, the following conditions are equivalent : a) There exists a holomorphic Poincaré-Birkhoff normalization of X H in a neighborhood of 0 in C 2l+m . b) There exists an analytic Hamiltonian action of the real torus T r in a neighborhood of 0 in C 2l+m , which preserves X H and whose linear part is generated by the (Hamiltonian) vector fields iZ k , k = 1, . . . , r.
Proof : Suppose that H is in holomorphic Poincaré-Birkhoff normal form. By Lemma 3.5, since {H, H ss } = 0, the vector fields iZ k preserve X H .
Conversely, if the point b) is satisfied, then according to the holomorphic version of the Splitting Theorem (see [10]) we can consider that the action of the torus is "diagonal", i.e. the product of an action on (C 2l , { , } symp ) by an action on (C m , { , } g ) and moreover that the action on the symplectic part is linear. According to Proposition 4.2 (in Appendix), we can linearize the second part of the action by a Poisson diffeomorphism. We then can consider that the action of T r is generated by the vector fields iZ k , k = 1, . . . , r. This action preserves X H then we have [iZ k , X H ] = 0 for all k. To conclude, just recall that X Hss is a linear combination of the Z k . Now, we are going to use Proposition 3.6 to clarify a link between the integrability of a Hamiltonian vector field X H on an analytic Poisson manifold (K n , { , }) and the existence of a convergent Poincaré-Birkhoff normalization. Recall first the definition (see for instance [15]) of the word integrability used here : (3.32) b) The functions are common first integrals for X 1 , . . . , X p : and they are functionally independent almost everywhere : Of course this definition has a sense in the smooth category as well as in the analytic category. We can speak about smooth or analytic integrability.
where { , } symp is a symplectic Poisson structure and g is a semisimple Lie algebra and { , } g the standard Lie-Poisson structure on g * . If X H is integrable then, forgetting one moment the Hamiltonian feature, Theorem 1.1 and Proposition 2.1 in [14] give the existence of an action of a real torus T r on (K 2l+m , 0) generated by vector fields Y 1 , . . . , Y r (r is the toric degree of X H ) where the linear parts of these vector fields are the iZ k (see 3.28), and which preserves X H . Moreover, the semisimple part X ss H of X H is a linear combination of the Y j : X ss H = j β j Y j without any resonance relation between the β j . Now, let us recall that we work in a Poisson manifold with a Hamiltonian vector field. Since the vector field X H preserves the Poisson structure, its semisimple part also does and then we will have [Y j , Π] = 0 for all j = 1, . . . , r. Therefore, the action of the torus also preserves the Poisson structure. Proposition 3.6 allows to conclude. Remark 3.9. If we suppose that H and the Poisson structure are real then it is natural to ask if all that we made is still valid. Note that in this case, we can consider H (and the Poisson structure) as complex analytic, with real coefficients.
Actually, in the same way as in [14,16], we conjecture that we have the equivalence: A real analytic Hamiltonian vector field X H with respect to a real analytic Poisson structure admits a local real analytic Poincaré-Birkhoff normalization iff it admits a local holomorphic Poincaré-Birkhoff normalization.

Appendix
In this appendix, we give a proof of auxiliary results used in the previous sections. We first compute the first Poisson cohomology space of the Poisson manifold we consider in Section 3. Suppose that Π S is a symplectic (i.e. nondegenerate) Poisson structure on K 2l (K is R or C). If (x 1 , . . . , x l , y 1 , . . . , y l ) are coordinates on K 2l , we can write Let g be a m-dimensional (real or complex) semisimple Lie algebra and consider Π g the corresponding linear Poisson structure on K m . Suppose that (z 1 , . . . , z m ) are coordinates on K m . We then show the following : Under the hypotheses above, if H 1 (K 2l × K m , Π S + Π g ) denotes the first (formal or analytic) Poisson cohomology space of the product of (K 2l , Π S ) by (K m , Π g ) then Proof : If X is a (formal or analytic) vector field on K 2l × K m we write X = X S + X g where X S is a vector field which only has components in the ∂ ∂xi and ∂ ∂yi and, in the same way, X g only has components in the ∂ ∂zi . Before computing the Poisson cohomology space, let us make the following two remarks : If [X S , Π S ] = 0 then X S = [f, Π S ] where f is a (formal or analytic) function on K 2l+m . Indeed, recalling that (because Π S is symplectic) the Poisson cohomology of (K 2l , Π S ) is isomorphic to the de Rham cohomology of K 2l (see for instance [12]), the relation [X S , Π S ] = 0 may be translated as dα = 0 where α is a 1-form on K 2l depending (formally or analytically) on parameters z 1 , . . . , z m . Then we can write α = df where f is a function on K 2l depending (formally or analytically) on parameters z 1 , . . . , z m .
In the same way, if [X g , Π S ] = 0 then, writing X g = i X g i (x, y, z) ∂ ∂zi , we get [X g i , Π S ] = 0 for all i. Thus, each X g i depends only on z. Indeed, here X g i may be seen as a function on K 2l depending (formally or analytically) on parameters z 1 , . . . , z m such that dX g i = 0.
Now if X = X S + X g is a vector field on K 2l × K m , it is easy to see that [X, Π S + Π g ] = 0 is equivalent to the three equations According to the first remark we made above, equation (4.1) gives X S = [f, Π S ] where f is a (formal or analytic) function on K 2l+m . Now, replacing X S by [f, Π S ] in (4.2) and using the graded Jacobi identity of the Schouten bracket, we get (4.4) X g − [f, Π g ], Π S = 0 .
Since X g − [f, Π g ] is a vector field which only has component in ∂ ∂z , the second remark we made above gives where Y is a vector field on K m (i.e. only has components in ∂ ∂z and whose coefficients are functions of z). Finally, (4.3) gives [Y, Π g ] = 0 i.e. Y is a 1-cocycle for the Poisson cohomology of (K m , Π g ). Since the Lie algebra g is semisimple, the Poisson cohomology space H 1 (K m , Π g ) is trivial (see for instance [4]). We then obtain Y = [h, Π g ] where h is a function on K m .
To resume, we get which means that X is a 1-cobord for the Poisson cohomology of (K 2l × K m , Π S + Π g ).
The second result is an analytic version of a smooth linearization theorem due to V. Ginzburg. In the Appendix of [6], he states that the G-action of a compact Lie group on a Poisson manifold (P, Π) (everything is smooth here), fixing a point x of P and such that the Poisson structure is linearizeable at x, can be linearized by a diffeomorphism which preserves the Poisson structure. Here, we state the following : Consider an analytic action of a compact (analytic) Lie group on (K n , Π) (K is R or C), where Π is an analytic Poisson structure on K n . Suppose that the action fix the origin 0 and that the Poisson structure is linearizable at 0. Then, the action can be linearized by a Poisson diffeomorphim.
Proof : The proof is the same as in the smooth case : we use Moser's path method. If g is an element of G, we put ϕ g the corresponding diffeomorphism of K n and ϕ g lin its linear part at 0. We construct a path of analytic actions of G on (K n , Π) given by the following diffeomorphisms : for any g in G and x in K n . These actions preserve Π and fix 0. We want now to show that there exists a path of diffeomorphisms ψ t , with ψ 0 = Id, preserving the Poisson structure Π and such that (4.7) ψ t • ϕ g t • ψ −1 t = ϕ g 0 = ϕ g lin , for all t ∈ [0, 1] and all g in G.
Let C t (g) be the time-depending vector field associated to ϕ g t : (4.8) C t (g)(ϕ g t (x)) = ∂ϕ g t ∂t (x) .
Derivating the condition (4.7), we are led to look for a time-depending vector field X t (corresponding to ψ t ) verifying (4.9) C t (g) = ϕ g t * X t − X t , for all t ∈ [0, 1] and all g in G.
We put (4.10) dh is a bi-invariant Haar measure on G such that the volume of G is 1. This vector field is analytic and depends smoothly on t. Moreover, since each C t (h) preserves the Poisson structure Π, so does X t . Finally, one can check that X t satisfies the condition (4.9).
Proof of Proposition 2.6. We denote by α the linear application from the Cartan subalgebra h to K n defined by α(h) = (α 1 (h), . . . , α n (h)) for any h in h and by W its image. We show that the subset of W formed by the elements γ such that the ω d (γ) (defined as in (2.20) replacing α i , h 1 by γ i ) do not satisfy the ω-condition is of measure 0 (in W ). Since α is a linear surjection from h to W , it will show Proposition 2.6.
For any positive integer k and any positive real number c, if denotes the norm associated to , , we put Actually, we show here that W 1 ∩ V is of measure 0 but the same technic works to prove that W k ∩ V is also of measure 0 for each k. Therefore ∪ k (W k ∩ V ) is of measure 0 too, which proves the proposition. Now, for any λ in Z n + we consider the affine subspace V λ of K n formed by the vectors γ such that γ , λ = 1 − |λ| and we put for c > 0, (4.12) V λ,c = γ ∈ K n ; |λ| − 1 + γ , λ ≤ c |λ| s . This last set is like a tubular neighborhood of V λ of thickness 2c |λ| s . We look now at K λ,c = V λ,c ∩W 1 . If it is not empty, it is a kind of "band" in W 1 of thickness smaller than S 2c |λ| s where S is a positive constant which only depends on the dimension of W (and on the metric). Therefore, we get This latest sum converges (because s > n) and we then get V ol(W 1 ∩ V ) = V ol(∩ c>0 W 1 ∩ V c ) = 0.
The point b) is obvious.
It is easy to see that these numbers are smaller than 1, provided that d is large enough.
Proof of Lemma 2.9. a) Since the sequence (r d ) d decreases and converges to a positive real number R > 0, we have r d > R for all d. We write r d −ρ d = r d 1 d 2 > R d 2 , thus for d sufficiently large, we get r d − ρ d > 1 2 d . b) We have ρ d − r d+1 = ρ d 1 − ω d+1 2 d+1 1 2 d+1 +1 . Since the sequence (ρ d ) d decreases and converges to R > 0, we have ρ d > R > 0 for all d. We then show that if d is sufficiently large, then We have ). By the ω-condition, the sequence (γ d ) d converges to 0 and is negative for all d sufficiently large. Then, if ε is a small positive real number (for instance ε = 1/2), we have for all d sufficiently large, 1 − e γ d > −(1 − ε)γ d . We deduce that (4.14) R Therefore, for d sufficiently large, R(1 − e γ d ) > 1 2 d .