Symmetry, Integrability and Geometry: Methods and Applications sl(2)-Trivial Deformations of VectPol(R)-Modules of Symbols ⋆

We consider the action of Vect_{Pol}(R) by Lie derivative on the spaces of symbols of differential operators. We study the deformations of this action that become trivial once restricted to sl(2). Necessary and sufficient conditions for integrability of infinitesimal deformations are given.


Introduction
Let Vect Pol (R) be the Lie algebra of polynomial vector fields on R. Consider the 1-parameter action of Vect Pol (R) on the space R[x] of polynomial functions on R defined by where X, f ∈ R[x] and X ′ := dX dx . Denote by F λ the Vect Pol (R)-module structure on R[x] defined by this action for a fixed λ. Geometrically, F λ is the space of polynomial weighted densities of weight λ on R The space F λ coincides with the space of vector fields, functions and differential 1-forms for λ = −1, 0 and 1, respectively.
Denote by D ν,µ := Hom diff (F ν , F µ ) the Vect Pol (R)-module of linear differential operators with the Vect Pol (R)-action given by the formula Each module D ν,µ has a natural filtration by the order of differential operators; the graded module S ν,µ := grD ν,µ is called the space of symbols. The quotient-module D k ν,µ /D k−1 ν,µ is isomorphic to the module of tensor densities F µ−ν−k , the isomorphism is provided by the principal symbol σ defined by x → σ(A) = a k (x)(dx) µ−ν−k ⋆ This paper is a contribution to the Special Issue on Deformation Quantization. The full collection is available at http://www.emis.de/journals/SIGMA/Deformation Quantization.html (see, e.g., [10]). As a Vect Pol (R)-module, the space S ν,µ depends only on the difference δ = µ−ν, so that S ν,µ can be written as S δ , and we have as Vect Pol (R)-modules. The space of symbols of order ≤ n is The space D ν,µ cannot be isomorphic as a Vect Pol (R)-module to the corresponding space of symbols, but is a deformation of this space in the sense of Richardson-Neijenhuis [12]; however, they are isomorphic as sl(2)-modules (see [9]). In the last two decades, deformations of various types of structures have assumed an ever increasing role in mathematics and physics. For each such deformation problem a goal is to determine if all related deformation obstructions vanish and many beautiful techniques had been developed to determine when this is so. Deformations of Lie algebras with base and versal deformations were already considered by Fialowski [5]. It was further developed, with introduction of a complete local algebra base (local means a commutative algebra which has a unique maximal ideal) by Fialowski [6]. Also, in [6], the notion of miniversal (or formal versal) deformation was introduced in general, and it was proved that under some cohomology restrictions, a versal deformation exists. Later Fialowski and Fuchs, using this framework, gave a construction for the versal deformation [7].
We use the framework of Fialowski [6] (see also [1] and [2]) and consider (multi-parameter) deformations over complete local algebras. We construct the miniversal deformation of this action and define the complete local algebra related to this deformation.
According to Nijenhuis-Richardson [12], deformation theory of modules is closely related to the computation of cohomology. More precisely, given a Lie algebra g and a g-module V , the infinitesimal deformations of the g-module structure on V , i.e., deformations that are linear in the parameter of deformation, are related to H 1 (g; End(V )). The obstructions to extension of any infinitesimal deformation to a formal one are related to H 2 (g; End(V )). More generally, if h is a subalgebra of g, then the h-relative cohomology space H 1 (g, h; End(V )) measures the infinitesimal deformations that become trivial once the action is restricted to h (h-trivial deformations), while the obstructions to extension of any h-trivial infinitesimal deformation to a formal one are related to H 2 (g, h; End(V )) (see, e.g., [3]).
Denote D := D(n, δ) the Vect Pol (R)-module of differential operators on S n δ . The infinitesimal deformations of the Vect Pol (R)-module S n δ are classified by the first differential cohomology space, was calculated by Bouarroudj and Ovsienko [4]; and the second space was calculated by Bouarroudj [3]. We give explicit expressions of some 2-cocycles that span the cohomology group H 2 (Vect P (R), sl(2); D λ,λ+k ). This paper is organized as follows. In Section 2 we study some properties of the sl(2)-invariant differential operators. These properties are related to the sl(2)-relative cohomology. In Section 3 we study the first and the second sl(2)-relative cohomology spaces which are closely related to the deformation theory. Especially we explain some sl(2)-relative 2-cocycles which naturally appear as obstructions to integrate any sl(2)-trivial infinitesimal deformation to a formal one. In Section 4 we give an outline of the general deformation theory: definitions, equivalence, integrability conditions and miniversal deformations. In Section 5 we give the first main result of this paper: Theorem 2. That is, we explain all second-order integrability conditions for any infinitesimal sl(2)-trivial deformation of the Vect Pol (R)-module S n δ . In Section 6 we complete the list of integrability conditions by computing those of third and fourth-order. We prove that these conditions are necessary and sufficient to integrate any infinitesimal sl(2)-trivial deformation to a formal one. Moreover, we prove that any sl(2)-trivial deformation is, in fact, equivalent to a polynomial one of degree ≤ 2: Theorem 3. Finally, in Section 7, we complete our study by giving a few examples of deformations.

Invariant dif ferential operators
In this paper we study the sl(2)-trivial deformations of the space of symbols of differential operators which is a Vect Pol (R)-module, so we begin by studying some properties of sl(2)invariant bilinear differential operators.
Let us consider the space of bilinear differential operators c : F λ × F µ → F τ . The Lie algebra, Vect Pol (R), acts on this space by the Lie derivative: A bilinear differential operator c : That is, the set of such sl(2)-invariant bilinear differential operators is the subspace on which the subalgebra sl(2) acts trivially. Now, let us consider a linear map c : Vect Pol (R) → D λ,µ , then we can see c as a bilinear differential operator c : . So, the sl(2)-invariance property (2) of c reads, for all X ∈ sl(2) and Y ∈ Vect Pol (R), or equivalently The sl(2)-invariant bilinear differential operators were calculated by Gordan. We recall here the results and we need to add some precision concerning the space of the sl(2)-invariant differential operators from Vect Pol (R) to D λ,µ vanishing on sl(2). Proposition 1 ( [11]). There exist sl(2)-invariant bilinear differential operators, called transvectants, given by where k ∈ N and the coefficients c i,j are characterized as follows: Moreover, the space of solutions of the system (3) is two-dimensional if 2λ = −s and 2τ = −t with t > k − s − 2, and one-dimensional otherwise.
The coboundary operator ∂ n : C n (g, h; V ) → C n+1 (g, h; V ) is a g-map satisfying ∂ n • ∂ n−1 = 0. The kernel of ∂ n , denoted Z n (g, h; V ), is the space of h-relative n-cocycles, among them, the elements in the range of ∂ n−1 are called h-relative n-coboundaries. We denote B n (g, h; V ) the space of n-coboundaries.
By definition, the n th h-relative cohomolgy space is the quotient space We will only need the formula of ∂ n (which will be simply denoted ∂) in degrees 0, 1 and 2: for and for Ω ∈ C 2 (g; h, V ), where (X, Y, Z) denotes the summands obtained from the two written ones by the cyclic permutation of the symbols X, Y , Z.
If Ω is a sl(2)-relative 2-coboundary then (up to a scalar factor) we have Ω = ∂J −1,λ k+1 . Proof . i) The 1-cocycle relation reads: (2). Since c(X) = 0, one easily sees that The equation (6) expresses the sl(2)-invariance property of the bilinear map c. Thus, according to Proposition 1, the map c coincides with the transvectant This last relation is nothing but the sl(2)-invariance property of the bilinear map Ω. iii Since ∂b(X, Y ) = b(X) = 0 for all X ∈ sl(2) we deduce that b is sl(2)-invariant: According to Proposition 1, the space of sl(2)-invariant linear differential operator from Vect Pol (R) to D λ,λ+k vanishing on sl (2) is one dimensional and it is spanned by J −1,λ k+1 . Thus, up to a scalar factor, b = J −1,λ k+1 . Proposition 2 is proved.

The first cohomology space
Note that, by Proposition 2, we can describe the space H 1 diff (Vect Pol (R), sl(2); D λ,λ+k ). This space is, in fact, one-dimensional if and only if the corresponding transvectant J −1,λ k+1 is a nontrivial sl(2)-relative 1-cocycle, otherwise it is trivial. However, this space was computed by Bouarroudj and Ovsienko, the result is as follows: (2); D λ,λ+k ) are generated by the cohomology classes of the sl(2)-relative 1-cocycles, C λ,λ+k : Vect Pol (R) → D λ,λ+k that are collected in the following table. Table 1.
The maps C λ,λ+j (X) are naturally extended to S n δ = n j=0 F δ−j .

The second cohomology space
Let g be a Lie algebra, h a subalgebra of g and V a g-module, the cup-product is defined, for arbitrary linear maps c 1 , c 2 : g → End(V ), by It is easy to check that for any two h-relative 1-cocycles c 1 and c 2 ∈ Z 1 (g, h; End(V )), the bilinear map [[c 1 , c 2 ]] is a h-relative 2-cocycle. Moreover, if one of the cocycles c 1 or c 2 is a h-relative 1-coboundary, then [[c 1 , c 2 ]] is a h-relative 2-coboundary. Therefore, we naturally deduce that the operation (7) defines a bilinear map Thus, by computing the cup-products of the 1-cocycles C λ,λ+k generating the spaces we can exhibit explicit expressions of some sl(2)-relative 2-cocycles Proposition 3.

The general framework
In this section we define deformations of Lie algebra homomorphisms and introduce the notion of miniversal deformations over complete local algebras. Deformation theory of Lie algebra homomorphisms was first considered with only one-parameter deformation [7,12,15]. Recently, deformations of Lie (super)algebras with multi-parameters were intensively studied (see, e.g., [1,2,13,14]). Here we give an outline of this theory.

Infinitesimal deformations
Let ρ 0 : g → End(V ) be an action of a Lie algebra g on a vector space V and let h be a subagebra of g. When studying h-trivial deformations of the g-action ρ 0 , one usually starts with infinitesimal deformations Moreover, two h-trivial infinitesimal deformations ρ = ρ 0 +t C 1 , and ρ = ρ 0 +t C 2 , are equivalents if and only if C 1 − C 2 is a h-relative coboundary: where A ∈ End(V ) h and ∂ stands for differential of cochains on g with values in End(V ). So, the space H 1 (g, h; End(V )) determines and classifies the h-trivial infinitesimal deformations up to equivalence. (see, e.g., [8,12]). If H 1 (g, h; End(V )) is multi-dimensional, it is natural to consider multi-parameter h-trivial deformations. More precisely, if dimH 1 (g, h; End(V )) = m, then choose h-relative 1-cocycles C 1 , . . . , C m representing a basis of H 1 (g, h; End(V )) and consider the htrivial infinitesimal deformation with independent parameters t 1 , . . . , t m .
In our study, we are interested in the infinitesimal sl(2)-trivial deformation of the Vect Pol (R)action on S n δ = n j=0 F δ−j , the space of symbols of differential operators, where n ∈ N and δ ∈ R. Thus, we consider the sl(2)-relative cohomology space H 1 diff (Vect Pol (R), sl(2); D). Any infinitesimal sl(2)-trivial deformation is then of the form where L X is the Lie derivative of S n δ along the vector field X d dx defined by (1), and and where t λ,λ+j are independent parameters, δ−λ ∈ N, δ−n ≤ λ, λ+j ≤ δ and the sl(2)-relative 1-cocycles C λ,λ+j are defined in Table 1.

Integrability conditions
Consider the problem of integrability of infinitesimal deformations. Starting with the infinitesimal deformation (9), we look for a formal series where L (k) X is an homogenous polynomial of degree k in the parameters (t λ,λ+j ) and with coefficients in D such that L (k) (2). This formal series (11) must satisfy the homomorphism condition in any order in the parameters (t λ,λ+j ) The homomorphism condition (12) gives the following (Maurer-Cartan) equations However, quite often the above problem has no solution. Note here that the right side of (13) must be a coboundary of a 1-cochain vanishing on sl(2), so, the obstructions for integrability of infinitesimal deformations belong to the second sl(2)-relative cohomology space (2); D). Following [7] and [2], we will impose extra algebraic relations on the parameters (t λ,λ+j ). Let R be an ideal in C[[t λ,λ+j ]] generated by some set of relations, the quotient is a complete local algebra with unity, and one can speak about deformations with base A, see [7] for details. Given an infinitesimal deformation (9), one can always consider it as a deformation with base (14), where R is the ideal generated by all the quadratic monomials. Our aim is to find A which is big as possible, or, equivalently, we look for relations on the parameters (t λ,λ+j ) which are necessary and sufficient for integrability (cf. [1,2]).

Equivalence and the miniversal deformation
The notion of equivalence of deformations over complete local algebras has been considered in [6]. The following notion of miniversal deformation is fundamental. It assigns to a g-module V a canonical commutative associative algebra A and a canonical deformation with base A.
(ii) in the notations of (i), if A is infinitesimal then ψ is unique.
If ρ satisfies only the condition (i), then it is called versal.
We refer to [7] for a construction of miniversal deformations of Lie algebras and to [2] for miniversal deformations of g-modules.
2) For k = 9, the integrability conditions follow from the fact that any corresponding cupproduct of 1-cocycle is nontrivial. Moreover, we have only singular cases and we also show that 3) For k = 10 and λ = a i , a i − 4 we have B λ,λ+10 = 0. For λ = a i , a i − 4 we have Like in the previous case we prove that the 2-cocycles Ω a i ,a i +10 and Ω a i −4,a i +6 are nontrivial and then we deduce the corresponding integrability conditions.
Our main result in this section is the following .
Of course, any t λ,λ+k appears in the expressions of L (1) or L (2) if and only if δ − λ and k are integers satisfying δ − n ≤ λ, λ + k ≤ δ. Theorem 2 is proved.
6 Third and fourth-order integrability conditions

Computing the third-order Maurer-Cartan equation
Now we reconsider the formal deformation (11) which is a formal power series in the parameters t λ,λ+j with coefficients in D. We suppose that the second-order integrability conditions are satisfied. So, the third-order terms of (11) are solutions of the (Maurer-Cartan) equation As in the previous section we can write where E λ,λ+k are maps from Vect Pol (R)×Vect Pol (R) to D λ,λ+k . The third-order term L (3) of the sl(2)-trivial formal deformation (11) is a solution of (20). So, the 2-cochains E λ,k must satisfy E λ,k = ∂J −1,λ k+1 and then the third-order integrability conditions are deduced from this fact. It is easy to see that E λ,λ+k = 0 for k ≤ 6 or k ≥ 13, so we compute successively the E λ,λ+k for k = 7, . . . , 12 and we resolve E λ,λ+k = ∂J −1,λ k+1 to get the corresponding third-order integrability conditions.
Here, we mention that the maps E λ,λ+k are 2-cochains, but they are not necessarily 2cocycles because they are not cup-products of 1-cocycles like the maps B λ,λ+k . Indeed, L (2) is not necessarily a 1-cocycle.

Third-order integrability conditions
Proposition 9. For k = 7, 8, we have the following third-order integrability conditions of the infinitesimal deformation (9), for all λ 6.3 Fourth-order integrability conditions Proposition 14. For generic λ, the fourth-order integrability conditions of the infinitesimal deformation (9) are the following: Proof . These conditions come from the fact that the fourth term L (4) must satisfy: Indeed, we can always reduce L (3) to zero by equivalence.
The following theorem is our main result.
Theorem 3. The second-order integrability conditions (17) and (18) together with the third and the fourth-order conditions (21)-(26) are necessary and sufficient for the complete integrability of the infinitesimal deformation (9). Moreover, any formal sl(2)-trivial deformation of the Lie derivative L X on the space of symbols S n δ is equivalent to a polynomial one of degree equal or less than 2.
Proof . Clearly, all these conditions are necessary. So, let us prove that they are also sufficient. As in the proof of Theorem 2, the solution L (3) of the Maurer-Cartan equation (19) is defined up to a 1-coboundary, thus, we can always reduce L (3) to zero by equivalence. Moreover, by recurrence, the highest-order terms L (m) satisfy the equation ∂L (m) = 0 and can also be reduced to the identically zero map. This completes the proof of Theorem 3. Remark 1. The majority of integrability conditions concern some parameters t λ,λ+k with singular values of λ. All these singular values of λ are negatives. So, let us consider the space S n δ with generic δ, for example, δ − n > 0. In this case, the second-order integrability conditions are reduced to the following equations: bt λ,λ+3 t λ+3,λ+7 − at λ,λ+4 t λ+4,λ+7 = t λ,λ+4 t λ+4,λ+8 = 0.

Examples
Example 1. Let us consider the space of symbols S 4 λ+4 . Proposition 15. Any formal sl(2)-trivial deformation of the Vect Pol (R)-action on the space S 4 λ+4 is equivalent to his infinitesimal part, without any conditions on the parameters (independent parameters). That is, the miniversal deformation here has base C[[t]] where t designates the family of all parameters.
There are no conditions to integrate this infinitesimal deformation to a formal one. The solution L (2) of (13) is defined up to a 1-coboundary and different choices of solutions of the Maurer-Cartan equation correspond to equivalent deformations. Thus, we can always reduce L (2) to zero by equivalence. Then, by recurrence, the highest-order terms L (m) satisfy the equation ∂L (m) = 0 and L (m) can also be reduced to the identically zero map.
The formal deformation (27) is defined without any condition on the parameters (independent parameters). That is, the miniversal deformation here has base C[[t]] where t designates the family of all parameters.