On quantization of quadratic Poisson structures

D. Manchon ^{1}, M. Masmoudi ^{2}, A. Roux ^{3}

Abstract : Any classical r-matrix on the Lie algebra of linear operators on a real vector space V gives rise to a quadratic Poisson structure on V which admits a deformation quantization stemming from the construction of V. Drinfel’d [Dr], [Gr]. We exhibit in this article an example of quadratic Poisson structure which does not arise this way.

I. Introduction

let V be a finite-dimensional real vector space. The linear action of the Lie group Gl(V) on V induces by differentiation a Lie algebra isomorphism from 𝔤 = 𝔤l(V) to the Lie algebra of linear vector fields on V. Given a basis (e_{1}, …, e_{n}) and then identifying 𝔤l(V) with the Lie algebra of real n × n matrices the isomorphism is given by :

J(E_{ij}) = x_{i}∂_{j},

where E_{ij} is the matrix with entries all vanishing except one equal to 1 on the i^{th} line and j^{th} column.

There is a unique way to extend the Lie bracket of 𝔤 to a graded Lie bracket, called the Schouten bracket on the shifted exterior algebra Λ(𝔤)[1] in a way compatible with the exterior product. The shift means that elements of Λ^{k}(𝔤) are of degree k − 1, and then the Schouten bracket maps Λ^{k}(𝔤) × Λ^{l}(𝔤) to Λ^{k+l−1}(𝔤). The exterior algebra Λ(𝔤) inherits then a structure of Gerstenhaber algebra (cf. for example [V], introduction).

The space T^{poly}(V) of polyvector fields on M is also endowed with a Gerstenhaber algebra structure, with Schouten bracket extending Lie bracket of vector fields [V]. The subalgebra (for exterior product) generated by linear vector fields is a Gerstenhaber subalgebra Λ̃(V) of T^{poly}(V). The isomorphism J extends to a surjective Gerstenhaber algebra morphism :

J^{•} : Λ^{•}(𝔤) ⟶ Λ̃^{•}(V).

Map J^{k} has nontrivial kernel for k ≥ 2 as long as V has dimension ≥ 2 : for example we have :

J^{2}(E_{ij} ∧ E_{kj}) = 0.

1 Institut Elie Cartan, CNRS, BP 239, 54506 Vandoeuvre CEDEX. manchon@iecn.u-nancy.fr 2 Université de Metz, Département de Mathématiques, île du Saulcy, 57045 Metz CEDEX 01. masmou-

di@poncelet.univ-metz.fr 3 Université de Metz, Département de Mathématiques, île du Saulcy, 57045 Metz CEDEX 01. roux@pon-

celet.univ-metz.fr

A classical r-matrix on 𝔤 is by definition an element r of 𝔤∧𝔤 such that [r, r] = 0. According to the discussion above the bivector field J^{2}(r) defines then a quadratic Poisson structure on V. A natural question arises then : can one recover this way any quadratic Poisson structure on V? It was claimed true in [BR] but Z.H. Liu and P. Xu discovered that the authors’ argument was not correct [LX]. They brought up a positive answer in the two-dimensional case ([LX] prop. 2.1) : namely the general quadratic Poisson structure (ax_{1}^{2} + 2bx_{1}x_{2} + cx_{2}^{2})∂_{1} ∧ ∂_{2} is equal to J^{2}(r) with :

r = ( \begin{matrix}b & −a \\ c & −b \end{matrix} ) ∧ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) .

We give here a negative answer to this question in general : after a somewhat lengthy but elementary computation we show in § III that bivector field (x_{1}^{2} + x_{2}x_{3})∂_{2} ∧ ∂_{3} on ℝ^{3} is a counterexample to this conjecture : it is outside the image of the set of r-matrices by J^{2}.

We recall in § II the construction by V.G. Drinfel’d of a translation-invariant deformation quantization on any Lie group G once given a classical r-matrix on the Lie algebra 𝔤 [Dr], [T]. The problem reduces to the case when r is non-degenerate, and the deformation quantization is then obtained by suitable restriction and transportation of Baker-Campbell-Hausdorff deformation quantization ([Ka]) of the dual 𝔤̃^{∗} of the central extension 𝔤̃ of 𝔤 defined by r. The construction works moreover for any Kontsevich-type star product [ABM] on 𝔤̃^{∗}. For 𝔤 = 𝔤l(V), such a star product on 𝔤̃^{∗} gives almost immediately through this construction a deformation quantization of quadratic Poisson structure J^{2}(r) on V.

Deformation quantization of some particular quadratic Poisson structures has been considered by several people namely Omori, Maeda and Yoshioka [OMY prop. 4.7]. Explicit computations for all quadratic Poisson structures in dimension 3 (then including our counterexample as well) have been performed by El Galiou and Tihami [ET], by a case-by-case method based on the classification of Dufour and Haraki [DH]. Let us recall that the existence of a deformation quantization for any Poisson manifold is a direct consequence of M.Kontsevich’s formality theorem. Acknowledgements : the authors thank Maxim Kontsevich and Daniel Sternheimer for useful comments and precisions.

II. Quantization of Poisson structures coming from r-matrices

Let 𝔤 be a Lie algebra, and let r ∈ 𝔤∧𝔤 a classical r-matrix. It defines an antisymmetric operator :

r̃: 𝔤^{∗} ⟶ 𝔤.

Classical Yang-Baxter equation [r, r] = 0 is equivalent to :

< ξ, [r̃(η), r̃(ζ)] > + < η, [r̃(ζ), r̃(ξ)] > + < ζ, [r̃(ξ), r̃(η)] >= 0  (∗)

for any ξ, η, ζ ∈ 𝔤^{∗}. The r-matrix r defines a left translation-invariant Poisson structure on any Lie group G with Lie algebra 𝔤.

II.1. A central extension

We can firstly suppose r nondegenerate, i.e. that r̃ is inversible with inverse ω̃, where ω belongs to 𝔤^{∗} ∧ 𝔤^{∗}. Classical Yang-Baxter equation is in this case equivalent to :

< ω̃X, [Y, Z] > + < ω̃Y, [Z, X] > + < ω̃Z, [X, Y] >= 0

for any X, Y, Z ∈ 𝔤, i.e is equivalent to the fact that ω is a 2-cocycle with values in the trivial representation. Let 𝔤̃ the central extension of 𝔤 by this cocycle, defined by 𝔤̃ = 𝔤⊕ℝ with bracket :

[X + α, Y + β] = [X, Y]+ < ω, X ∧ Y > .

The cocycle condition on ω is equivalent to de Jacobi identity for this bracket. Let X_{0} = (0, 1) ∈ 𝔤̃, and let H the hyperplane in 𝔤̃^{∗} defined by :

H = {ξ ∈ 𝔤̃^{∗}/ < ξ, X_{0} >= 1}.

It is the symplectic leaf through the point ξ_{0} defined by < ξ_{0}, 𝔤 >= 0 and < ξ_{0}, X_{0} >= 1. It is then the coadjoint orbit Ad^{∗} G̃.ξ_{0} for any Lie group G̃ with Lie algebra 𝔤̃.

II.2. Kontsevich star products (after [ABM])

The linear Poisson manifold 𝔤̃^{∗} admits a whole bunch of equivalent Ad^{∗} G̃-invariant deformation quantizations which can be built from the enveloping algebra U(𝔤̃), for example the Baker-Campbell-Hausdorff quantization or the Kontsevich quantization [K], [ABM], [Ka], [Di]. The baker-Campbell-Hausdorff quantization is given by the following integral formula [ABM] :

(u #_{BCH} v)(ξ) = ∫∫_{𝔤×𝔤} F^{−1}u(x)F^{−1}v(y)e^{−i<ξ, x ._{h̄} y>} dx dy,

where the inverse Fourier transform is given by :

F^{−1}u(x) = (2π)^{−n} ∫_{𝔤_{∗}} u(η)e^{i<x,η>}dη,

and x ._{h̄} y stands for the Baker-Campbell-Hausdorff expansion :

x + y + \frac{h̄}{2}[x, y] + \frac{h̄^{2}}{12} ([x, [x, y]] + [y, [y, x]]) + · · · .

The Lebesgue measure dη on 𝔤^{∗} is normalized so that it is the dual mesure of Lebesgue measure dx on 𝔤. The quantizations we can consider here are the ones called “Kontsevich star products” in [ABM]. They are all equivalent to the BCH quantization. The equivalence is a formal series of differential operators with constant coefficients on 𝔤^{∗} precisely given by a formal series of G-invariant polynomials on 𝔤 of the following form :

F (x) = 1 + ∑_{k≥1} h̄^{2k} ∑_{c≥1} ∑_{(s_{1},...,s_{c})∈S_{2k}^{c}} a_{s_{1},...,s_{c}} Tr(ad x)^{s_{1}} ⋯ Tr(ad x)^{s_{c}},

where S_{2k}^{c} stands for those (s_{1}, …, s_{c}) in ℕ^{c} such that s_{1} +⋯+s_{c} = 2k, s_{1} ≤ s_{2} ≤ ⋯ ≤ s_{c} and s_{j} ≠ 1. The star product obtained this way admits the following integral form :

(u#v)(ξ) = ∫∫_{𝔤×𝔤} F^{−1}u(x)F^{−1}v(y)\frac{F (−ix)F (−iy)}{F (−i(x ._{h̄} y))}e^{−i<ξ, x ._{h̄} y>} dx dy.

II.3. Quantization of left-invariant Poisson structures

It is easy to derive from the fact that X_{0} is central that any of the deformation quantizations defined above does define by restriction a deformation quantization of H. Let G be the subgroup of G̃ with Lie algebra 𝔤. We clearly have :

Ad^{∗} G.ξ_{0} = Ad^{∗} G̃.ξ_{0} = H.

It is moreover easy to check that the stabilizer of ξ_{0} in G̃ is the one-dimensional subgroup with Lie algebra generated by X_{0}. It is a simple consequence of the nondegeneracy of the alternate bilinear form ω. The dimension of G is the equal to the dimension of H. The map :

\begin{matrix}φ : G′ & ⟶ H \\ g & ⟼ Ad^{∗} g.ξ_{0} \end{matrix}

is then a local G-equivariant diffeomorphism near the identity (with left translation on the left-hand side and coadjoint action on the right-hand side). We can then transport any deformation quantization of H and get a left translation-invariant deformation quantization of a neighbourhood of the identity in G. It extends by translation invariance to the whole group G, as well as to any Lie group G′ locally isomorphic to G.

The deformation quantization on G can be written :

u#v = ∑_{k≥0} h̄^{k}C_{k}(u, v),

where the C_{k}’s are left-invariant bidifferential operators on G. There exists then an element F = ∑ h̄^{k}F_{k} in (U(𝔤) ⊗ U(𝔤))[[h̄]] such that :

u#v(g) = F (u ⊗ v)(g, g).  (∗∗)

Let us now fix a basis x_{1}, …, x_{n} of 𝔤, and consider elements of U(𝔤) as polynomials F (x) = F (x_{1}, …, x_{n}) of the n noncommuting variables x_{1}, …, x_{n}, which satisfy the relations :

x_{i}x_{j} − x_{j}x_{i} = [x_{i}, x_{j}] = ∑_{k} c_{ij}^{k}x_{k}.

Introducing a second identical set of noncommuting variables y = (y_{1}, …, y_{n}) commuting with th x′_{j}s we can write any element A ∈ U(𝔤) ⊗ U(𝔤) as A(x, y). The element F defined above can then be written F (x, y) as a formal series with coefficients F_{k}(x, y).

Proposition II.1. The formal series F = F (x, y) ∈ (U(𝔤) ⊗ U(𝔤))[[h̄]] above verifies :

1) F_{0}(x, y) = 0.

2) F_{1}(x, y) = \frac{1}{2} ∑_{i,j} r_{ij}x_{i}y_{j}.

3) F_{k}(x, 0) = F_{k}(0, y) = 0 for k ≥ 1.

4) F (x + y, z)F (x, y) = F (x, y + z)F (y, z).

Conversely any F (x, y) endowed with those 4 properties defines by formula (∗∗) a left translation deformation quantization of G.

Proof. It is well-known : see for example [Dr], [T] : first condition comes from the fact that C_{0}(u, v) is the ordinary product uv. Second property comes from the expression of left-invariant Poisson bracket on G defined from the r-matrix, third property expresses the fact that 1#u = u#1 = u, and last property is an expression of the associativity of star product #. Let us elaborate a bit on that last point : any element X_{j} of the basis corresponds to polynomial expression G(x) = x_{j}. The Leibniz rule :

X_{j}.(φψ) = (X_{j}.φ)ψ + φ.(X_{j}.ψ)

can be written as :

G(x) ∘ m = m ∘ G(x + y),

where m : C^{∞}(G × G) → C^{∞}(G) stands for multiplication, here the restriction to the diagonal. The formula above extends to any polynomial expression G representing any element of the enveloping algebra. We have then :

(u#v)#w = (m ∘ F (u ⊗ v))#w
= m ∘ F ((m ∘ F (u ⊗ v)) ⊗ w)
= m ∘ F ∘ (m ⊗ I) ∘ (F ⊗ I)(u ⊗ v ⊗ w)
= m ∘ F (x, z) ∘ (m ⊗ I) ∘ F (x, y)(u ⊗ v ⊗ w)
= m ∘ (m ⊗ I) ∘ F (x + y, z)F (x, y)(u ⊗ v ⊗ w).

Similarly we have :

u#(v#w) = m ∘ (I ⊗ m) ∘ F (x, y + z)F (y, z)(u ⊗ v ⊗ w).

The associativity condition for product # is then equivalent to property 4) of the proposition.

•

let us now look at the case when r is degenerate. Then the image 𝔤_{0} of r̃ is a subspace strictly contained in 𝔤. By skew-symmetry 𝔤_{0} is also the orthogonal of the kernel of r̃, and classical Yang-Baxter equation [r, r] = 0 ensures thanks to (∗) that 𝔤_{0} is a Lie subalgebra of 𝔤. We get this way a nondegenerate r_{0} ∈ 𝔤_{0} ∧ 𝔤_{0} such that [r_{0}, r_{0}] = 0. Applying the procedure above we get an F = ∑ h̄^{k}F_{k} in (U(𝔤_{0}) ⊗ U(𝔤_{0}))[[h̄]] which can be seen as an element of (U(𝔤) ⊗ U(𝔤))[[h̄]].

II.4. A class of easily quantizable Poisson structures

Let G be a Lie group with Lie algebra 𝔤. Let F (x, y) a formal series in U(𝔤)⊗U(𝔤)[[h̄]] satisfying properties 1-4 of proposition II.1 (for example that one constructed from an rmatrix along the lines above). Let M be any differentiable manifold endowed with an action of G. The differentiation of this action induces a Lie algebra morphism from 𝔤 to the vector fields on M, which extends to an algebra morphism from U(𝔤) to the algebra of differential operators on M. Similarly it induces an algebra morphism from U(𝔤) ⊗ U(𝔤) to the algebra of differential operators on M ×M. The formal series of bidifferential operators defined by the formula :

∗ = m ∘ F (x, y)

(where m : C^{∞}(M × M) → C^{∞}(M) stands for ordinary multiplication of functions on M) defines then a star product on M, the associated Poisson bivector being defined by F_{1}(x, y) − F_{1}(y, x). The proof of this fact goes the same way as that of proposition II.1. It is easily seen that if F (x, y) comes from a classical r-matrix r ∈ 𝔤 ∧ 𝔤 then the Poisson structure on M is J^{2}(r) where J^{•} is the Gerstenhaber algebra morphism from Λ(𝔤) to multivector fields on M extending the action of 𝔤.

We will be interested in the sequel by the following particular situation : the manifold M is a vector space V, the action of G is linear, and there is a classical r-matrix r on 𝔤. We can as in the introduction view J as a Lie algebra morphism from 𝔤 to the space of linear vector fields on V, and extend J to a morphism J^{•} of Gerstenhaber algebras from Λ(𝔤) to Λ̃(V). In particular J^{2}(r) defines a quadratic Poisson structure on V, and formula just above gives a quantization of this particular quadratic Poisson structure.

III. Quadratic Poisson structures and r-matrices

III.1. Some definitions

We keep the notations of the introduction. The Gerstenhaber algebra Λ̃(V) can be written as :

Λ̃(V) = ⨁_{n≥0}(S^{n}(V) ⊗ Λ^{n}(V))[1] = ⨁_{n≥0} Λ̃^{n}(V)[1].

A quadratic Poisson structure on V can be defined as a bivector field Λ in Λ̃^{2}(V) such that :

[Λ, Λ] = 0.

Let Λ be an element of Λ̃^{2}(V), and let r an element of Λ^{2}(𝔤) such that J^{2}(r) = Λ. It is then obvious that [Λ, Λ] = 0 if and only if J^{3}([r, r]) = 0. If n ≥ 2 then J^{2} and J^{3} have nontrivial kernels : Precisely we have :

dim ker J^{2} = \frac{n^{2}(n^{2} − 1)}{4} and dim ker J^{3} = \frac{n^{2}(n^{2} − 1)(5n^{2} − 8)}{36}.

III.2. A counterexample in dimension 3

With the notations of § I, an element of 𝔤 ∧ 𝔤 can be written as :

r = ∑_{i,j,k,l=1}^{n} r_{ik}^{jl}E_{ij} ∧ E_{kl}.

Whe shall need for further calculations the following result :

Proposition III.1.

Let r = ∑_{i,j,k,l=1}^{n}r_{ik}^{jl}E_{ij} ∧ E_{kl} be an element of 𝔤 ∧ 𝔤 then [r, r] = 0 if and only if for any

i, j, k, l, m, p ∈ {1, …, n} such that (i, j) < (k, l) < (m, p) according to lexicographical order we have :

∑_{d=1}^{n} r_{ik}^{dl}r_{dm}^{jp} − r_{mk}^{dl}r_{di}^{pj} + r_{km}^{dp}r_{di}^{lj} − r_{im}^{dp}r_{dk}^{jl} + r_{mi}^{dj}r_{dk}^{pl} − r_{ki}^{dj}r_{dm}^{lp} = 0.

Proof. This proposition is a direct consequence of formula :

[E_{ij}, E_{kl}] = δ_{jk}E_{il} − δ_{li}E_{kj}

and the following lemma :

Lemma III.1. Let 𝔥 be a finite-dimensional Lie algebra and let X_{1}, …, X_{N} be a basis of 𝔥.

If r = ∑_{I,J=1}^{N}r^{IJ}X_{I} ∧ X_{J} is an element of 𝔥 ∧ 𝔥 ( r^{IJ} = −r^{JI}) then

[r, r] = 4 ∑_{I,J,K,L=1}^{N} r^{IJ}r^{KL}[X_{I}, X_{K}] ∧ X_{J} ∧ X_{L}.

•

Remark : We can directly show proposition III.1 using relation (∗) of beginning of § II applied to elements of the dual basis of X_{1}, …, X_{n}.

Proposition III.2. The Poisson structure Λ on ℝ^{3} given by

Λ = (x_{1}^{2} + αx_{2}x_{3})∂_{2} ∧ ∂_{3}

with α ≠ 0 is not the image of a classical r-matrix by J^{2}.

Proof. An element r = ∑_{i,j,k,l=1}^{n}r_{ik}^{jl}E_{ij} ∧ E_{kl} of 𝔤 ∧ 𝔤 is parametrized by 36 coefficients r_{ik}^{jl}.

It has image Λ if and only if the following 18 equations are satisfied :

r_{11}^{12} = r_{11}^{13} = r_{22}^{12} = r_{22}^{13} = r_{22}^{23} = r_{33}^{12} = r_{33}^{13} = r_{33}^{23} = 0, r_{11}^{23} = 1
r_{12}^{12} = r_{12}^{21},  r_{12}^{13} = r_{12}^{31},  r_{13}^{12} = r_{13}^{21}
r_{13}^{13} = r_{13}^{31},  r_{12}^{23} = r_{12}^{32},  r_{13}^{23} = r_{13}^{32}
r_{23}^{12} = r_{23}^{21},  r_{23}^{13} = r_{23}^{31},  r_{23}^{32} = r_{23}^{23} − α

To lighten writing we rename the 18 remaining unknowns as follows :

r_{12}^{11} = a, r_{12}^{12} = b, r_{12}^{13} = c, r_{13}^{11} = d, r_{13}^{12} = e, r_{13}^{13} = f,
r_{12}^{22} = g, r_{12}^{23} = h, r_{13}^{22} = i, r_{13}^{23} = j, r_{12}^{33} = k, r_{13}^{33} = l,
r_{23}^{11} = m, r_{23}^{12} = n, r_{23}^{13} = p, r_{23}^{22} = q, r_{23}^{23} = r, r_{23}^{33} = s.

The unknown r must not be confused with the classical r-matrix on the whole. The context will not lead to any confusion. We have then :

r_{11}^{23} = 1,
r_{12}^{21} = b, r_{13}^{21} = e, r_{12}^{31} = c,
r_{12}^{32} = h, r_{13}^{31} = f, r_{13}^{32} = j,
r_{23}^{21} = n, r_{23}^{31} = p, r_{23}^{32} = r − α,

and other r_{ik}^{jl} are equal to 0.

If r is such an element the equation [r, r] = 0 develops according to Proposition III.2. into a system of 84 equations involving our 18 unknowns a, b, …, s, given by the vanishing of the 84 coefficients of elements of the basis E_{ij} ∧ E_{kl} ∧ E_{mn} of 𝔤 ∧ 𝔤 ∧ 𝔤. The 84 equations reduce to 66 thanks to the fact that we already have [Λ, Λ] = 0. But we shall only consider 20 of them, which will be sufficient for exhibiting the counterexample :

Let us order the E_{ij}’s lexicographically from first to 9th, rename them accordingly (A_{1} = E_{11}, A_{2} = E_{12}, …, A_{9} = E_{33}), and labelize by (x, y, z) the equation obtained by the vanishing of the coefficient of A_{x} ∧ A_{y} ∧ A_{z}. We shall consider precisely the following equations :

(1, 2, 5) ci − eh + n = 0 (1, 2, 9) eh − ci + d = 0
(1, 3, 5) cj − ek − a = 0 (1, 3, 7) ce + f^{2} − ja − ld + m = 0
(1, 3, 9) ek − cj + p = 0 (1, 4, 5) mh − nc = 0
(1, 4, 6) mk − pc = 0 (1, 5, 6) αc + nk − ph = 0
(1, 7, 8) en − im = 0 (1, 7, 9) ep − jm = 0
(1, 8, 9) αe + ip − jn = 0 (2, 3, 9) − 3f + r + ik − jh = 0
(2, 4, 5) ag − b^{2} + nh − qc = 0 (2, 5, 6) 2αh + bh − rh + qk − cg = 0
(2, 8, 9) ej + ir + αi − jq − fi = 0 (3, 8, 9) is + 2αj + el − fj − jr = 0

(4, 5, 7) pn − rm − bm + na = 0 (5, 6, 9) − nk − rs + hp + s(r − α) = 0

(5, 8, 9) nj − ip + αq = 0 (6, 8, 9) ln − pj + qs − (r − α)^{2} = 0.

Consider the following two sums :

\begin{matrix}(1, 2, 5) + (1, 2, 9) : & n + d = 0 \\ (1, 3, 5) + (1, 3, 9) : & p − a = 0. \end{matrix}

Hence n = −d and p = a. We will discuss the four cases a = d = 0, a = 0 and d ≠ 0, a ≠ 0 and d = 0, a ≠ 0 and d ≠ 0. First case : a = d = 0. Then looking successively at the following equations we get :

(5, 8, 9) ⟹ q = 0 (6, 8, 9) ⟹ r = α (1, 5, 6) ⟹ c = 0
(1, 8, 9) ⟹ e = 0 (2, 4, 5) ⟹ b = 0 (2, 5, 6) ⟹ h = 0
(4, 5, 7) ⟹ m = 0 (5, 6, 9) ⟹ s = 0 (1, 3, 7) ⟹ f = 0
(3, 8, 9) ⟹ j = 0 (2, 8, 9) ⟹ i = 0 (2, 3, 9) ⟹ α = 0,

hence a contradiction to the hypothesis α ≠ 0. Second case : a = 0 and d ≠ 0 (hence n ≠ 0).

(1, 4, 6) ⟹ mk = 0.

First subcase : m = 0. Then :

(1, 4, 5) ⟹ c = 0 (1, 7, 8) ⟹ e = 0 (1, 2, 5) ⟹ n = 0,

hence a contradiction. Second subcase : m ≠ 0, hence k = 0.

(1, 5, 6) ⟹ c = 0 (1, 4, 5) ⟹ h = 0 (1, 2, 5) ⟹ n = 0,

hence a contradiction again. Third case : a ≠ 0 and d = 0 (hence p ≠ 0).

(1, 4, 5) ⟹ mh = 0.

First subcase : m = 0. Then :

(1, 4, 6) ⟹ c = 0 (1, 7, 9) ⟹ e = 0 (1, 3, 5) ⟹ a = 0,

hence a contradiction. Second subcase : m ≠ 0, hence h = 0.

(1, 5, 6) ⟹ c = 0 (1, 4, 6) ⟹ k = 0 (1, 3, 5) ⟹ a = 0,

hence a contradiction again. Fourth case : a ≠ 0 and d ≠ 0. First subcase : m = 0.

(1, 4, 5) ⟹ c = 0 (1, 7, 8) ⟹ e = 0 (1, 3, 5) ⟹ a = 0,

contradiction. Second subcase : m ≠ 0.

(1, 4, 6) ⟹ k = \frac{a}{m}c (1, 7, 9) ⟹ j = \frac{a}{m}e (1, 3, 5) ⟹ a = 0,

contradiction. This proves proposition III.2.

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III.3. Cartan-type quadratic Poisson structures

Recall from [DH] that the curl of a Poisson stucture Λ = ∑_{i,j} Λ^{ij}∂_{i} ∧ ∂_{j} is defined by :

rot Λ = ∑_{i,j} ∂_{j}Λ^{ij}.∂_{i}.

It is a linear vector field (and hence can be viewed as an n×n matrix) when Λ is quadratic. A quadratic Poisson structure is of Cartan type if it can be written for some choice of coordinates as :

Λ = ∑_{i,j=1}^{n} c_{ij}x_{i}x_{j}∂_{i} ∧ ∂_{j}

with c_{ji} = −c_{ij}. J.P. Dufour and A. Haraki proved the following result :

Theorem III.3 (Dufour - Haraki). Any quadratic Poisson structure the curl of which has eigenvalues λ_{i} such that λ_{i} + λ_{j} ≠ λ_{r} + λ_{s} for any (i, j, r, s) with r ≠ s and {i, j} ≠ {r, s} is of Cartan type.

Such a Cartan-type Poisson structure is image by J^{(2)} of a classical r-matrix, namely :

r = ∑_{i,j} c_{ij}E_{ii} ∧ E_{jj}.

References

[ABM] Arnal, D., Ben Amar, N., Masmoudi, M.: Cohomology of good graphs and Kontsevich

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