# Normal forms of dispersive scalar Poisson brackets with two independent variables

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## Abstract

We classify the dispersive Poisson brackets with one dependent variable and two independent variables, with leading order of hydrodynamic type, up to Miura transformations. We show that, in contrast to the case of a single independent variable for which a well-known triviality result exists, the Miura equivalence classes are parametrised by an infinite number of constants, which we call numerical invariants of the brackets. We obtain explicit formulas for the first few numerical invariants.

## Keywords

Poisson brackets Poisson cohomology Hamiltonian operator Miura transformation## Mathematics Subject Classification

37K05## 1 Introduction

*u*, i.e. formal power series in the variables \(\partial _{x^1}^{k_1}\partial _{x^2}^{k_2} u\) with coefficients which are smooth functions of

*u*:

*u*and two independent variables \(x^1\), \(x^2\) of the form

*u*(

*x*). In the specific case considered here, where there is a single dependent variable and two independent variables, such conditions guarantee the existence of a change of coordinates in the dependent variable (a Miura transformation of the first kind), to a flat coordinate that we still denote with

*u*, in which the bracket assumes the form

As follows from the discussion so far, the classification of dispersive Poisson brackets of the form (1) (with non-degeneracy condition) under Miura transformations (3), diffeomorphisms of the dependent variable and linear changes of the independent variables reduces to the problem of finding the normal forms of the equivalence classes under Miura transformations of the second kind (3) of the Poisson brackets (1) with leading term (2).

We solve this problem in our main result:

## Theorem 1

## Remark 1

- i.
for any choice of constants \(c_k\) formula (4) defines a Poisson bracket which is a deformation of (2);

- ii.
two Poisson brackets of the form (4) are Miura equivalent if and only if they are defined by the same constants \(c_k\);

- iii.
and any Poisson bracket of the form (1) can be brought to the normal form (4) by a Miura transformation.

*numerical invariants*of the Poisson bracket.

## Example 1

*t*and

*y*coordinates play the role of times. However, it is possible to cast the equation in evolutionary form, with the introduction of the inverse derivative operator \(\partial _x^{-1}\). KP equation can be written as

The deformation theory of Hamiltonian—and, albeit not addressed in our paper, bi-Hamiltonian—structures plays an important role in the classification of integrable Hamiltonian PDEs [10, 14]. Most results in this field have been obtained for \((1+1)\)-dimensional systems, namely the ones that depend only on one space variable.

The main result in this line of research is the triviality theorem [9, 14, 16] of Poisson brackets of Dubrovin–Novikov type. Together with the classical results by Dubrovin and Novikov [12], this allows to conclude that the dispersive deformations of non-degenerate Dubrovin–Novikov brackets are classified by the signature of a pseudo-Riemannian metric. Similarly, deformations of bi-Hamiltonian pencils [1, 20] are parametrised by functions of one variable, the so-called central invariants [10, 11]; in a few special cases, the corresponding bi-Hamiltonian cohomology has been computed, in particular for scalar brackets [4, 5, 19], and in the semi-simple *n*-component case [3, 6] . The \((2+1)\)-dimensional case is much less studied: the classification of the structures of hydrodynamic type has been completed up to the four-component case [15], and the nontriviality of the Poisson cohomology in the two-component case has been established [7]. In our recent paper [2] we computed the Poisson cohomology for scalar—namely, one-component—brackets. Since such a cohomology is far from being trivial, the actual classification of the dispersive deformations of such brackets is a highly complicated task. We address and solve it in the present paper.

The outline of the paper is as follows: in Sect. 1 we quickly recall basic definitions and facts related with the theta formalism. In Sect. 2, we specialise some results from our previous work [2] to the \(D=2\) case to obtain an explicit description of the second Poisson cohomology. In Sect. 3, we prove our main result. The proof is split into three steps corresponding to the three parts in Remark 1. In Sect. 4.4, we prove some technical lemmas that are required in the proof of Proposition 2. Finally, in Sect. 4 we give an explicit expression of the first few numerical invariants of the Poisson bracket.

## 2 Theta formalism

We present here a short summary of the basic definitions of the theta formalism for local variational multivector fields, specialising the formulas to the scalar case with two independent variables, i.e. \(N=1\), \(D=2\). We refer the reader to [2] for the general *N*, *D* case.

*u*. The standard gradation \(\deg \) on \(\mathcal {A}\) is given by \(\deg u^{(s,t)} = s+t\). We denote \(\mathcal {A}_d\) the homogeneous component of degree

*d*.

*p*-vector

*P*is a linear

*p*-alternating map from \(\mathcal {F}\) to itself of the form

*p*-vectors by \(\Lambda ^p \subset \mathrm {Alt}^p(\mathcal {F}, \mathcal {F})\).

*u*, i.e.

*d*, resp. super-degree

*p*, while \(\hat{\mathcal {A}}_d^p := \hat{\mathcal {A}}_d \cap \hat{\mathcal {A}}^p\). Clearly \(\hat{\mathcal {A}}^0 =\mathcal {A}\). The derivations \(\partial _x\) and \(\partial _y\) are extended to \(\hat{\mathcal {A}}\) in the obvious way.

We denote by \(\hat{\mathcal {F}}\) the quotient of \(\hat{\mathcal {A}}\) by the subspace \(\partial _{x} \hat{\mathcal {A}}+ \partial _{y} \hat{\mathcal {A}}\), and by a double integral \(\int \mathrm{d}x \ \mathrm{d}y\) the projection map from \(\hat{\mathcal {A}}\) to \(\hat{\mathcal {F}}\). Since the derivations \(\partial _x\), \(\partial _y\) are homogeneous, \(\hat{\mathcal {F}}\) inherits both gradations of \(\hat{\mathcal {A}}\).

It turns out, see Proposition 2 in [2], that the space of local multivectors \(\Lambda ^p\) is isomorphic to \(\hat{\mathcal {F}}^p\) for \(p\not =1\), while \(\Lambda ^1\) is isomorphic to the quotient of \(\hat{\mathcal {F}}^1\) by the subspace of elements of the form \( \int (k_1 u^{(1,0)} + k_2 u^{(0,1)} ) \theta \) for two constants \(k_1\), \(k_2\). Moreover \(\hat{\mathcal {F}}^1\) is isomorphic to the space \({{\mathrm{Der}}}'(\mathcal {A})\) of derivations of \(\mathcal {A}\) that commute with \(\partial _x\) and \(\partial _y\).

A bivector \(P\in \hat{\mathcal {F}}^2\) is a Poisson structure when \([P,P]=0\). In such case \(d_P := \mathrm{ad}_P = [P, \cdot ]\) squares to zero, as a consequence of the graded Jacobi identity, and the cohomology of the complex \((\hat{\mathcal {F}}, d_P)\) is called Poisson cohomology of *P*.

*u*. The action of a general Miura transformation of the second kind on a local multivector

*Q*in \(\hat{\mathcal {F}}\) is given by the exponential of the adjoint action with respect to the Schouten–Nijenhuis bracket

## 3 Poisson cohomology

In our previous paper [2], we gave a description of the Poisson cohomology of a scalar multidimensional Poisson bracket in terms of the cohomology of an auxiliary complex with constant coefficients. Our aim here is to give an explicit description of a set of generators of the Poisson cohomology in the \(D=2\) case, which will be used in the proof of the main theorem in the next Section.

Let us begin by recalling without proof a few results from our paper [2], specialising them to the case \(D=2\).

This sequence allows us to write the Poisson cohomology \(H^p(\hat{\mathcal {F}})\) as a sum of two homogeneous subspaces of \(\varTheta \slash \partial _x \varTheta \) in super-degree *p* and \(p+1\), respectively, where the first one is simply injected, while the second one has to be reconstructed via the inverse to the Bockstein homomorphism.

We have therefore shown that

## Lemma 1

\({H}^{p}_{d}(\hat{\mathcal {F}}) = \left( \frac{\varTheta }{\partial _x \varTheta }\right) ^{p}_d \oplus \mathcal {B}\left( \frac{\varTheta }{\partial _x \varTheta }\right) ^{p+1}_d .\)

We remark that this lemma gives an explicit description of representatives of the cohomology classes in \(H^p_d(\hat{\mathcal {F}})\). In particular, the only non-trivial classes in \(\varTheta \slash \partial _x \varTheta \) in super-degree \(p=2\) are given by \(\theta \theta ^{(2k+1,0)}\) for \(k\geqslant 1\) and correspond to the deformations of the Poisson brackets in Theorem 1. The following reformulation of this observation will be useful in the proof of Proposition 2:

## Corollary 1

Moreover, we can define an explicit basis of \(\left( \frac{\varTheta }{\partial _x \varTheta }\right) ^{3}_{d}\) and \(\mathcal {B}\left( \frac{\varTheta }{\partial _x \varTheta }\right) ^{3}_{d}\):

## Lemma 2

## Proof

*lexicographic order*, that is, we say that \(\theta ^{i_1} \cdots \theta ^{i_p} > \theta ^{j_1} \cdots \theta ^{j_p}\) if \(i_1 > j_1\), or if \(i_1 = j_1\) and \(i_2 > j_2\), and so on.

By specialising to the case \(p=3\), and spelling out the allowed sets of indexes, we obtain the statement of the lemma. \(\square \)

*a*,

*b*,

*c*chosen as in the basis above.

## 4 Proof of the main theorem

Let us first reformulate our main statement in the \(\theta \)-formalism.

The proof of Theorem 1 reduces to prove the three statements listed in Remark 1.

### 4.1 Proof of the statement in Remark 1.i

Our first observation is:

## Lemma 3

## Proof

The Poisson bivectors \(\mathfrak {p}_k\) do not depend on *u* and its derivatives; therefore, the variational derivatives w.r.t. *u* appearing in the definition of Schouten–Nijenhuis bracket are vanishing. \(\square \)

It clearly follows that \(\mathfrak {p}(c)\) is a Poisson bivector for any choice of the constants \(c=(c_1, c_2, \ldots )\).

### 4.2 Proof of the statement in Remark 1.ii

Next we show that for any distinct choice of the constants \(c=(c_1, c_2, \ldots )\) the corresponding bivector *P* belongs to a different equivalence class under Miura transformations.

## Proposition 1

Let \(\mathfrak {p}(c)\), resp. \(\mathfrak {p}(\tilde{c})\), be the Poisson bivector of the normal form (13) corresponding to a choice \(c=(c_1, c_2, \ldots )\), resp. \(\tilde{c}=(\tilde{c}_1, \tilde{c}_2, \ldots )\), of constants. If the two sequences *c* and \(\tilde{c}\) are not identically equal, then there is no Miura transformation of the second kind which maps \(\mathfrak {p}(c)\) to \(\mathfrak {p}(\tilde{c})\).

## Proof

*c*and \(\tilde{c}\) are not identically equal, and hence there exists a smallest index

*k*for which \(c_k \not = \tilde{c}_k\). It follows that

*k*, i.e.

*X*. Let us consider in particular the constraints with odd degree

The Lemma is proved. \(\square \)

### 4.3 Proof of the statement in Remark 1.iii

Finally, we prove that any Poisson bivector with leading order \(\mathfrak {p}_1\) given by (12) can always be brought to the form (13) by a Miura transformation of the second kind.

## Proposition 2

Let \(P \in \hat{\mathcal {F}}^2_{\geqslant 1}\) be a Poisson bivector with degree one term equal to \(\mathfrak {p}_1\). Then there is a Miura transformation that maps *P* to a \(\mathfrak {p}(c)\) for a choice of constants \(c=(c_1, c_2, \ldots )\).

## Proof

The Poisson bivector \(P\in \hat{\mathcal {F}}^2_{\geqslant 1}\) has to satisfy \([P,P]=0\). We want to show by induction that, taking into account this equation, it is possible, by repeated application of Miura transformations, to put all terms in normal form and to remove all terms that come from the Bockstein homomorphism.

*s*, respectively, even or odd, where \(Q_l \in \mathcal {B}\left( \frac{\varTheta }{\partial _x \varTheta } \right) ^3_l\) , \(P_l \in \hat{\mathcal {F}}^2_l\), the dots denote higher-order terms, and

Let us now show that by a Miura transformation a Poisson bivector of the form \(\mathfrak {p}_{(s)}\) can be made of the form \(\mathfrak {p}_{(s+1)}\).

*y*, while the second term has degree zero.

By Corollary 1, the cohomology \(H^2_{2k}(\hat{\mathcal {F}})\) is given only by elements coming from the Bockstein homomorphism, and therefore exists \(Q_{2k} \in \mathcal {B}\left( \frac{\varTheta }{\partial _x \varTheta } \right) ^3_{2k}\) such that \(P_{2k} +\mathrm {ad}_{\mathfrak {p}_1} X_{2k-1} = Q_{2k}\) for some \(X_{2k-1} \in \hat{\mathcal {F}}^1_{2k-1}\).

Acting with the Miura transformation \(e^{\mathrm {ad}_{X_{2k-1}}}\) on \(\mathfrak {p}_{(2k)}\), we get a new Poisson bivector, where the terms of degree less or equal to \(2k-1\) are unchanged, the term \(P_{2k}\) has been replaced with the term \(Q_{2k}\), and the terms of higher order are in general different. We have therefore that \(\mathfrak {p}_{(2k+1)} = e^{\mathrm {ad}_{X_{2k-1}}} \mathfrak {p}_{(2k)}\) is of the form above, as required.

The second and third terms in (22) have also to be both zero. This follows from the fact that they have different degree in the number of \(u^{(s,t)}\). As we have seen in Sect. 3, the elements \(Q_k\) are linear in the variables \(u^{(s,t)}\), while the elements \(\mathfrak {p}_{k}\) do not contain them.

From the vanishing of the last term, \([Q_{k+1}, Q_{k+1}]= 0\), we finally derive that \(Q_{k+1}\) is zero. This is guaranteed by Lemma 4. The proof of this Lemma, being quite technical, is given in Sect. 4.4.

Taking into account this vanishing, the action of the Miura transformation \(e^{\mathrm {ad}_{X_{2k}}}\) on \(\mathfrak {p}_{(2k+1)}\) gives exactly the term \(\mathfrak {p}_{(2k+2)}\).

By induction, we see that we can continue this procedure indefinitely; therefore, we conclude that we cannot have any non-trivial deformation coming from \(\left( \frac{\varTheta }{\partial _x \varTheta } \right) ^3\) via the Bockstein homomorphism, and that the Miura transformation \(\cdots e^{\mathrm {ad}_{X_2}} e^{\mathrm {ad}_{X_1}}\) given by the composition of the Mira transformations defined above, sends the original Poisson bivector \(P = \mathfrak {p}_1 + \cdots \) to a Poisson bivector of the form \(\mathfrak {p}(c)\) for a choice of constants \(c_1, c_2, \ldots \).

The Proposition is proved. \(\square \)

### 4.4 Some technical lemmas

In this section, we prove the following statement, which is essential in the proof of Proposition 2:

## Lemma 4

Let \(\chi \in \left( \frac{\varTheta }{\partial _x \varTheta } \right) ^3_d\) and \(\mathcal {B}(\chi )\) its image through the map (9) in \(\hat{\mathcal {F}}^2_d\). If \([\mathcal {B}(\chi ),\mathcal {B}(\chi ) ]=0\), then \(\chi =0\).

## Proof

^{1}

Let \(\mathrm {sq}:\varTheta ^2_k\rightarrow \varTheta ^4_{2k}\) be the map that sends an element \(\alpha \in \varTheta ^2_k\) to \(\alpha ^2\in \varTheta ^4_{2k}\). In the rest of this section, we will use the notation \(\theta ^d = \theta ^{(d,0)}\).

## Lemma 5

The intersection of \(\mathrm {sq}(\varTheta ^2_k)\) and \(\partial _x\varTheta ^4_{2k-1}\) is equal to zero. In other words, if \(\alpha \in \varTheta ^2_k\) and \(\alpha ^2\) is \(\partial _x\)-exact, then \(\alpha ^2=0\) and, therefore, \(\alpha \) is proportional to a monomial \(\theta ^i\theta ^{k-i}\) for some \(i=1,\ldots ,\lfloor \frac{k-1}{2} \rfloor \).

## Proof

A basis in \(\varTheta _{2k-1}^4\) is given by standard monomials \(\theta ^{i_1} \theta ^{i_2} \theta ^{i_3} \theta ^{i_4}\) with total degree \(i_1 + i_2+ i_3+i_4 = 2k-1\). By *standard* monomial, we indicate a monomial where the indices are ordered as \(i_1> i_2> i_3 > i_4 \geqslant 0\) to avoid duplicates.

We can write \(\varTheta _{2k-1}^4 = \mathcal {V}_1 \oplus \mathcal {V}_2\), where a basis for \(\mathcal {V}_1\) is given by standard monomials with the restriction \(i_1+i_4 \leqslant k-1\), and a basis for \(\mathcal {V}_2\) is given by standard monomials with \(i_1 +i_4 \geqslant k\).

We denote by \(\varTheta _k^2 \cdot \varTheta _k^2\) the subspace of \(\varTheta _{2k}^4\) spanned by standard monomials \(\theta ^{i_1} \theta ^{i_2} \theta ^{i_3} \theta ^{i_4}\) with \(i_1> i_2> i_3 > i_4 \geqslant 0\) and \(i_1 + i_2+ i_3+i_4 = 2k\) with \(i_1+i_4 = k\) and \(i_2 + i_3 = k\). It is indeed the subspace given by the product of two arbitrary elements of \(\varTheta _k^2\).

Clearly, both \(\partial _x \mathcal {V}_1\) and \(\varTheta _k^2 \cdot \varTheta _k^2\) are subspaces of \(\mathcal {W}\).

Let us now prove that \(\partial _x \mathcal {V}_2\) has zero intersection with \(\mathcal {W}\). Let \(v = \sum _\gamma v_\gamma \, \gamma \) be an element in \(\mathcal {V}_2\), where \(\gamma \) is in the standard basis of \(\mathcal {V}_2\) described above. Let \(\partial _x v = \sum _\gamma v_\gamma \, \partial _x \gamma \in \mathcal {W}\). We have already seen that the elements \(\partial _x \gamma \) are linearly independent. If \(\gamma = \theta ^{i_1} \theta ^{i_2} \theta ^{i_3} \theta ^{i_4}\) then \(\partial _x \gamma \) is equal to \(\theta ^{i_1+1} \theta ^{i_2} \theta ^{i_3} \theta ^{i_4}\) plus lexicographically lower terms. The lexicographically leading order term is therefore of a standard monomial \(\theta ^{j_1} \theta ^{j_2} \theta ^{j_3} \theta ^{j_4}\) with \(j_1+j_4 \geqslant k+1\). But all basis elements in \(\mathcal {W}\) are standard monomials with \(j_1+j_4 \leqslant k\). It follows that, if \(\gamma \) is the lexicographically highest term in *v*, we must have \(v_\gamma =0\). By induction *v* vanishes.

Let \(\alpha = \sum _{i = \lceil \frac{k+1}{2} \rceil }^{k} \alpha _i\, \theta ^i \theta ^{k-i}\) be an element of \(\varTheta _k^2\) whose square is in \(\partial _x \mathcal {V}_1\). We want to show that at most one of the coefficients \(\alpha _i\) is nonzero. We therefore assume that at least two such coefficients are nonzero and show that it leads to a contradiction. Let *s* be the higher index for which \(\alpha _s \not = 0\) and \(t <s\) the second higher index for which \(\alpha _t \not =0\).

The lexicographically higher term in \(\beta \), i.e. for \(i=k-1\), is sent by \(\partial _x\) to a term proportional to \(\theta ^{k} \theta ^{t} \theta ^{k-t} \theta ^{0}\), which does not appear in \((\alpha ^2)_t\), therefore \(\beta _{k-1}=0\). Proceeding like this, we set to zero all the constants \(\beta _{k-1}, \ldots , \beta _{s}\). Similarly, we can proceed from the lower part of the chain and set to zero all the remaining constants \(\beta _{t+1}, \ldots , \beta _{s-1}\). But then \(\beta =0\), therefore \(\alpha _s \alpha _t =0\) and we are led to a contradiction.

We have proved that at most one of the constants \(\alpha _i\) can be nonzero. In such case, \(\alpha ^2 =0\). The Lemma is proved. \(\square \)

## Lemma 6

Consider an arbitrary element \(\chi \in \varTheta ^3_d\). If \(\frac{\delta \chi }{\delta \theta } = c\cdot \theta ^i\theta ^{d-i}\) for some \(i=0,1,\ldots ,\lfloor \frac{d-1}{2}\rfloor \), then \(c=0\).

## Proof

*d*separately.

^{2}that the variational derivative \(\frac{\delta }{\delta \theta }\) of a basis element \( \theta ^{k-l+1} \theta ^{k-l} \theta ^{2l} \), with \(3l < k\), is equal to

Observe that \(\frac{\delta }{\delta \theta } \theta ^{k+1}\theta ^k\theta ^0\) contains the monomials \(\theta ^d\theta ^0\) and \(\theta ^{d-1}\theta ^1\), while the variational derivatives of all other basis elements with \(l\geqslant 1\) cannot contain \(\theta ^d\theta ^0\) and \(\theta ^{d-1}\theta ^1\). Thus, if \(\frac{\delta \chi }{\delta \theta } = c\cdot \theta ^i\theta ^{d-i}\) for some *i*, then the coefficient of \(\theta ^{k+1}\theta ^k\theta ^0\) in \(\chi \) has to be equal to zero.

We can continue this process by induction. Assume that we have already proved that the first *l* elements of the basis cannot appear in \(\chi \). Then the variational derivative of the basis element \( \theta ^{k-l+1} \theta ^{k-l} \theta ^{2l} \) is the only one that contains \(\theta ^{d-2l} \theta ^{2l} \) and \( \theta ^{d-2l-1} \theta ^{2l+1} \). It follows from the same reason as above, that such basis element cannot appear in \(\chi \).

*d*, mutatis mutandis. \(\square \)

Now let us consider an arbitrary element \(\chi \in \varTheta ^3_d\), such that \((\frac{\delta \chi }{\delta \theta } )^2\) belongs to the image of \(\partial _x\). From Lemma 5, it follows that \(\frac{\delta \chi }{\delta \theta } =c\cdot \theta ^i\theta ^{d-i}\) for some \(i=0,1,\ldots ,\lfloor d/2\rfloor \). Then Lemma 6 implies that \(\frac{\delta \chi }{\delta \theta } =0\); hence, \(\chi \) belongs to the image of \(\partial _x\).

We have proved that \(\chi =0\) as element of \(\left( \frac{\varTheta }{\partial _x \varTheta } \right) ^3_d\). Lemma 4 is proved. \(\square \)

## 5 The numerical invariants of the Poisson bracket

In principle all the numerical invariants of a Poisson bracket of the form (1), namely the sequence \((c_1,c_2,\ldots )\), can be extracted iteratively solving order by order for the Miura transformation which eliminates the coboundary terms. Providing a general formula for the invariants of a Poisson bivector is hard, since the elimination of each coboundary term affects in principle all the higher-order ones and it is necessary to give an explicit form for the Miura transformation. However, the lowest invariants can be computed as follows.

## Proposition 3

Notice that \(A_{2;3,0}\) is implied to be a constant.

## Proof

*P*of form (1), it can be expanded according to its differential order. For notational compactness, we will denote

*x*,

*y*) as we did in the previous sections; moreover, with a slight abuse of notation we identify the Dirac’s delta derivatives with the corresponding elements of \(\hat{\mathcal {F}}\) previously used

*f*(

*u*). Since \(H^2_2(\hat{\mathcal {F}})=0\), we have \(P_2=[X_1,\mathfrak {p}_1]\) and the Miura transformation that eliminates \(P_2\) from

*P*is \(e^{-\mathrm {ad}_{\epsilon X_1}}\). The evolutionary vector field \(X_1\) has characteristic

*P*and get

*x*and \(x^2\) corresponds to

*y*, in the notation of Sects. 3 and 4. Hence, we can write

*x*derivatives is \(1/2\,A_{2;2,1}(u)\partial _{x}^2u+\tilde{A}(u)\left( \partial _{x}u\right) ^2\). Here we are interested only in first summand because it is the one that gives the highest number of

*x*-derivatives in \([X_2,\mathfrak {p}_r]\), for any

*r*.

*y*-degree bigger or equal to 1. Thus we focus on the summands

## Example 2

*P*gives

## Example 3

*stream function*\(\psi (x,y)\) such that \(u=\psi _y\) and \(v=-\psi _x\), for which \(\varDelta \psi =-\omega \) and \(\delta H/\delta \omega =-\psi \).

## Footnotes

- 1.
Notice that this fact, in the case of standard differential polynomials in commuting variables, follows from a simple observation: the derivative in

*x*of a differential polynomial cannot be a square, since it has to be linear in the highest derivative. In the case of anticommuting variables, however, a quite involved proof is necessary. - 2.
Note that the computation is slightly different in the case \(3l=k-1\).

## Notes

### Acknowledgements

We would like to thank Jenya Ferapontov for several useful observations and Dario Merzi for suggesting a clever identity in Example 2. M. C. was supported by the INdAM-COFUND-2012 Marie Curie fellowship “MPoisCoho—Poisson cohomology of multidimensional Hamiltonian operators”.

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