� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 34, Num. 2 (2019), 269 – 283 doi:10.17398/2605-5686.34.2.269 Available online September 3, 2019 aff(1|1)-trivial deformations of aff(2|1)-modules of weighted densities on the superspace R1|2 Ismail Laraiedh Département de Mathématiques, Faculté des Sciences de Sfax, BP 802, 3038 Sfax, Tunisie Departement of Mathematics, College of Sciences and Humanities-Kowaiyia Shaqra University, Kingdom of Saudi Arabia Ismail.laraiedh@gmail.com , ismail.laraiedh@su.edu.sa Received March 15, 2019 Presented by Rosa Navarro Accepted June 14, 2019 Abstract: Over the (1|2)-dimensional real superspace, we study aff(1|1)-trivial deformations of the action of the affine Lie superalgebra aff(2|1) on the direct sum of the superspaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this action and we prove that any formal deformation is equivalent to its infinitisemal part. Key words: relative cohomology, trivial deformation, Lie superalgebra, symbol. AMS Subject Class. (2010): 17B56, 53D55, 58H15. 1. Introduction Let Vect(R) be the Lie algebra of polynomial vector fields on the real space. Consider the 1-parameter deformation of Vect(R)-action on C∞(R) Lλ X d dx (f) = Xf ′ + λX′f, where X, f ∈ C∞(R) and X′ := dX dx . Denote by Fλ the Vect(R)-module struc- ture on C∞(R) defined by Lλ for a fixed λ. Geometrically, Fλ = { fdxλ : f ∈ C∞(R)} is the space of polynomial weighted densities of weight λ ∈ R. The space Fλ coincides with the space of vector fields, functions and differential 1-forms for λ = −1 , 0 and 1, respectively. The superspace Dλ,µ := Homdiff (Fλ,Fµ) the linear differential operators with the natural Vect(R)-action denoted Lλ,µX (A) = L µ X ◦A−A◦L λ X. Each module Dλ,µ has a natural filtration by the order of differential operators; the graded module Sλ,µ := gr Dλ,µ is the space of symbols. The quotient-module Dkλ,µ / D k−1 λ,µ is isomorphic to the module of weighted densities Fµ−λ−k; the ISSN: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.2.269 mailto:Ismail.laraiedh@gmail.com mailto:ismail.laraiedh@su.edu.sa https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 270 i. laraiedh isomorphism is defined by the principal symbol map σpr defined by A = k∑ i=0 ai(x) ( ∂ ∂x )i 7−→ σpr(A) = ak(x)(dx)µ−λ−k, (see, e.g.,[15]). Therefore, as a Vect(R)-module, the space Sλ,µ depends on the difference β = µ−λ, so that Sλ,µ be written as Sβ, and Sβ = ∞⊕ k=0 Fβ−k as Vect(R)-modules. The space of symbols of order ≤ n is Snβ := n⊕ k=0 Fβ−k. The space Dλ,µ cannot be isomorphic as a Vect(R)-module to the space of symbols, but is a deformation of this space in the sense of Richardson and Neijenhuis [19]. Deformation theory plays a crucial role in all branches of physics. In physics the mathematical theory of deformations has been proved to be a powerful tool in modeling physical reality. The concepts symmetry and de- formations are considered to be two fundamental guiding principles for devel- oping the physical theory further. The notion of deformation was applied to Lie algebras by Nijenhuis and Richardson [19, 18]. This theory is developed by Ovsienko and by other authors [3, 9, 19]. We consider the superspace R1|2 endowed with its standard contact struc- ture defined by the 1-form α2, and K(2) of contact vector fields on R1|2. We introduce the K(2)-module F2λ of λ-densities on R 1|2 and the K(2)-module of linear differential operators, D2λ,µ := Homdiff (F 2 λ,F 2 µ), which are super analogs of the spaces Fλ and Dλ,µ, respectively. The Lie superalgebra aff(2|1), a su- per analog of aff(1), is a subalgebra of K(2). We classify the aff(1|1)-trivial deformations of the structure of the aff(2|1)-module S2µ−λ = ∞⊕ k=0 F2 µ−λ−k 2 , which is super analog of the space Sβ. We prove that any formal deformation is equivalent to its infinitesimal part and we give an example of deformation with one parameter. aff(1|1)-trivial deformations of aff(2|1)-modules 271 2. Definitions and notations We briefly give in this section the basics definitions of geometrical objects on R1|2 that will be needed for our purpose, for more details, see [7, 11, 6, 16, 15, 17]. 2.1. The Lie superalgebra of contact vector fields on R1|2. Let R1|2 be the superspace with coordinates (x, θ1, θ2), where θ1 and θ2 are odd indeterminates: θiθj = −θjθi. We consider the superspace C∞(R1|2) of polynomial functions. Any element of C∞(R1|2) has the form F = f0 + f1θ1 + f2θ2 + f12θ1θ2, where f0,f1,f2,f12 ∈ C∞(R). Even elements in C∞(R1|2) are the functions F(x,θ) = f0(x) + f12(x)θ1θ2, the functions F(x,θ) = f1(x)θ1 + f2(x)θ2 are odd elements. We denote by |F| the parity of a function F. Let Vect(R1|2) be the space of polynomial vector fields on R1|2: Vect(R1|2) = { F0∂x + F1∂1 + F2∂2 : Fi ∈ C∞(R1|2) } , where ∂i and ∂x stand for ∂ ∂θi and ∂ ∂x . The superbracket of two vector fields is bilinear and defined for two homogeneous vector fields by [X,Y ] = X ◦Y − (−1)|X||Y |Y ◦X. The supespace R1|2 is equipped with the contact structure given by the 1-form α2 = dx + θ1dθ1 + θ2dθ2. This contact structure is equivalently defined by the kernel of α2, spanned by the odd vector fields ηi = ∂i −θi∂x. We consider the superspace K(2) of contact vector fields on R1|2. That is, K(2) = { X ∈ Vect(R1|2) : ∃ F ∈ C∞(R1|2) such that LX(α2) = Fα2 } , where LX is the Lie derivative of a vector field, acting on the space of functions, forms, vector fields, . . . . Any contact vector field on R1|2 can be expressed as XF = F∂x − 1 2 (−1)|F| 2∑ i=1 ηi(F)ηi , where F ∈ C ∞(R1|2). 272 i. laraiedh Of course, K(2) is a subalgebra of Vect(R1|2), and K(2) acts on C∞(R1|2) through LXF (G) = FG ′ − 1 2 (−1)|F | 2∑ i=1 ηi(F) ·ηi(G), (2.1) where G ∈ C∞(R1|2). The contact bracket is defined by [XF , XG] = X{F,G}. The space C∞(R1|2) is thus equipped with a Lie superalgebra structure isomorphic to K(2). The explicit formula can be easily calculated: {F,G} = FG′ −F ′G− 1 2 (−1)|F| 2∑ i=1 ηi(F) ·ηi(G). (2.2) 2.2. The superalgebra aff(2|1). Recall that the Lie algebra aff(1) can be realized as a subalgebra of Vect(R): aff(1) = Span (X1, Xx) , and the affine Lie superalgebra aff(1|1) is realized as a subalgebra of K(1): aff(1|1) = Span(X1, Xx, Xθ). The space aff(1|1)0 is isomorphic to aff(1), while (aff(1|1))1̄ = Span(Xθ). Similarly, the affine Lie superalgebra aff(2|1) can be realized as a subalge- bra of K(2): aff(2|1) = Span(X1, Xx, Xθ1, Xθ2, Xθ1θ2 ), where (aff(2|1))0̄ = Span(X1, Xx, Xθ1θ2 ), (aff(2|1))1̄ = Span(Xθ1, Xθ2 ). We easily see that aff(1|1) is subalgebra of the Lie superalgebra aff(2|1). aff(1|1)-trivial deformations of aff(2|1)-modules 273 2.3. The space of weighted densities on R1|2. We introduce a one- parameter family of modules over the Lie superalgebra K(2). As vector spaces all these modules are isomorphic to C∞(R1|2), but not as K(2)-modules. For every contact vector field XF , define a one-parameter family of first- order differential operators on C∞(R1|2): LλXF = XF + λF ′, λ ∈ R. (2.3) We easily check that [LλXF ,L λ XG ] = LλX{F,G}. (2.4) We thus obtain a one-parameter family of K(2)-modules on C∞(R1|2) that we denote F2λ, the space of all weighted densities on C ∞(R1|2) of weight λ with respect to α2: F2λ = { Fαλ2 : F ∈ C ∞(R1|2) } . (2.5) In particular, we have F0λ = Fλ. Obviously the adjoint K(2)-module is iso- morphic to the space of weighted densities on R1|2 of weight −1. 2.4. Differential operators on weighted densities. A differen- tial operator on R1|2 is an operator on C∞(R1|2) of the form: A = m∑ j=0 aj∂ j x + 2∑ i=1 ni∑ k=0 bk,i∂ k x∂i + n∑ `=0 c`∂ ` x∂1∂2, (2.6) where aj, bk,i, c` ∈ C∞(R1|2). Any differential operator defines a linear mapping Fαλ2 7→ (AF)α µ 2 from F 2 λ to F 2 µ for any λ, µ ∈ K; thus, the space of differential operators becomes a family of osp(2|2)-modules D2λ,µ for the natural action: XF ·A = L µ XF ◦A− (−1)|A||F|A◦LλXF . (2.7) Proposition 2.1. Every differential operator A ∈ D2λ,µ can be expressed in the form A(Fαλ2 ) = ∑ `,m a`,m(x,θ)η ` 1η m 2 (F)α µ 2, (2.8) where a`,m(x,θ) are arbitrary functions. Proof. Since −η2i = ∂x, and ∂i = ηi − θiη 2 i , every differential operator A given by (2.6) is a polynomial expression in η1 and η2. 274 i. laraiedh Proposition 2.2. As a aff(1|1)-module, we have D2λ,µ ' D 1 λ,µ ⊕D 1 λ+ 1 2 ,µ+ 1 2 ⊕ Π ( D1 λ,µ+ 1 2 ⊕D1 λ+ 1 2 ,µ ) . (2.9) Proof. Any element F ∈ C∞(R1|2) can be uniquely written as follows: F = F1 + F2θ2, where ∂2F1 = ∂2F2 = 0. Therefore, for any XH ∈ aff(1|1), we easily chek that LλXH (F) = L λ XH (F1) + L λ+ 1 2 XH (F2)θ2. Thus, the following map is an aff(1|1)-isomorphism: Φλ : F 2 λ −→ F 1 λ ⊕ Π(F 1 λ+ 1 2 ) Fαλ2 7−→ ( F1α λ 1 , Π(F2α λ+ 1 2 1 ) ) . (2.10) So, we deduce an aff(1|1)-isomorphism: Ψλ,µ : D 1 λ,µ ⊕D 1 λ+ 1 2 ,µ+ 1 2 ⊕ Π ( D1 λ,µ+ 1 2 ⊕D1 λ+ 1 2 ,µ ) −→ D2λ,µ (2.11) A 7−→ Φ−1µ ◦A◦ Φλ. We identify the aff(1|1)-modules the following isomorphisms: Π ( D1 λ,µ+ 1 2 ) −→ Homdiff ( F1λ, Π(F 1 µ+ 1 2 ) ) , Π(A) 7−→ Π ◦A, Π ( D1 λ+ 1 2 ,µ ) −→ Homdiff ( F1 λ+ 1 2 , Π(F1µ) ) , Π(A) 7−→ A◦ Π, D1 λ+ 1 2 ,µ+ 1 2 −→ Homdiff ( Π(F1 λ+ 1 2 ), Π(F1 µ+ 1 2 ) ) , Π(A) 7−→ Π ◦A◦ Π. 3. aff(1|1)-trivial deformation of aff(2|1)-modules Deformation theory of Lie algebra was first considered with one-parameter of deformation [13, 19, 12, 21, 4, 5]. Recently, deformations of Lie (su- per)algebras with multi-parameters were intensively studied (see, e.g., [1, 2, 3, 8, 20]). 3.1. Infinitesimal deformations and the first cohomology. Let ρ0 : g → End(V ) be an action of a Lie superalgebra g on a vector superspace V and let h be a subagebra of g (if h is omitted it assumed to be {0}). When aff(1|1)-trivial deformations of aff(2|1)-modules 275 studying h-trivial deformations of the g-action ρ0, one usually starts with infinitesimal deformations ρ = ρ0 + t Υ, (3.1) where Υ : g → End(V ) is a linear map vanishing on h and t is a formal parameter with p(t) = p(Υ). The homomorphism condition [ρ(x),ρ(y)] = ρ([x,y]), (3.2) where x, y ∈ g, is satisfied in order 1 in t if and only if Υ is a h-relative 1-cocycle. That is, the map Υ satisfies (−1)|x||Υ|[ρ0(x), Υ(y)] − (−1)|y|(|x|+|Υ|)[ρ0(y), Υ(x)] − Υ([x, y]) = 0. Moreover, two h-trivial infinitesimal deformations ρ = ρ0 + t Υ1, and ρ = ρ0 + t Υ2, are equivalents if and only if Υ1 − Υ2 is h-relative coboundary: (Υ1 − Υ2)(x) = (−1)|x||A|[ρ0(x),A] := δA(x), where A ∈ End(V )h and δ stands for differential of cochains on g with values in End(V ) (see, e.g., [14, 19]). So, the space H1(g,h; End(V )) determines and classifies infinitesimal deformations up to equivalence. If dim H1(g,h; End(V )) = m, then choose 1-cocycles Υ1, . . . , Υm representing a basis of H 1(g,h; End(V )) and consider the infinitesimal deformation ρ = ρ0 + m∑ i=1 ti Υi, (3.3) where t1, . . . , tm are independent parameters with |ti| = |Υi|. Since we are interested in the aff(1|1)-trivial deformations of the aff(2|1)- module structure on the space S 2,m β = 2m⊕ k=0 F2 β−k 2 , where m ∈ 1 2 N. (3.4) The first differential cohomology spaces H1diff (aff(2|1),aff(1|1),D 2 λ,µ) was computed in [10]. The result is as follows: dim(H1(aff(2|1),aff(1|1),D2λ,µ)) =   1 if µ = λ, 2 if µ−λ = k, k ∈{1, 2, . . .}, 0 otherwise. 276 i. laraiedh The following 1-cocycles span the corresponding cohomology spaces: Υλ,λ(XG) = η1η2(G), Γ1λ,λ+k(XG) = η1η2(G)∂ k x, Γ2λ,λ+k(XG) = η1η2(G)η1η2∂ k−1 x . We consider the space H1(aff(2|1), aff(1|1), End(S2,mβ )) spanned by the classes Υiλ,λ and Γ i λ,λ+k, i = 1, 2, where k ∈ {1, . . . , [m]}, [m] denoting the integer part of m, and 2(β − λ) ∈ {2k, . . . , 2m} for a generic β. Any infinitesimal aff(1|1)-trivial deformation of the aff(2|1)-module structure on S2,mβ is then of the form L̃XF = LXF + L (1) XF , (3.5) where LXF is the Lie derivate of S 2,m β along the vector XF defined by (2.3), and L (1) XF = ∑ λ tλ,λΥλ,λ(XF ) + ∑ λ [m]∑ k=1 2∑ i=1 tiλ,λ+kΓ i λ,λ+k(XF ), (3.6) where tλ,λ and t i λ,λ+k are independent parameters with |tλ,λ| = |Υλ,λ| and |tiλ,λ+k| = |Γ i λ,λ+k|. 3.2. Integrability conditions and deformations over supercom- mutative algebras. Consider the superalgebra with unity C[[t1, . . . , tm]] and consider the problem of integrability of infinitesimal deformations. Start- ing with the infinitesimal deformation (3.3), we look for a formal series ρ = ρ0 + m∑ i=1 ti Υi + ∑ i,j titj ρ (2) ij + · · · , (3.7) where the higher order terms ρ (2) ij , ρ (3) ijk, . . . are linear maps from g to End(V ) with |ρ(2)ij | = |titj|, |ρ (3) ijk| = |titjtk|, . . . such that the map ρ : g → C[[t1, . . . , tm]] ⊗ End(V ) (3.8) satisfies the homomorphism condition (3.2). Quite often the above problem has no solution. Following [13] and [2], we will impose extra algebraic relations on the parameters t1, . . . , tm. Let R be an ideal in C[[t1, . . . , tm]] generated by some set of relations, and we can aff(1|1)-trivial deformations of aff(2|1)-modules 277 speak about deformations with base A = C[[t1, . . . , tm]]/R, (for details, see [13]). The map (3.8) sends g to A⊗ End(V ). Setting ϕt = ρ−ρ0, ρ(1) = ∑ ti Υi, ρ (2) = ∑ titj ρ (2) ij , . . . , we can rewrite the relation (3.2) in the following way: [ϕt(x),ρ0(y)] + [ρ0(x),ϕt(y)] −ϕt([x,y]) + ∑ i,j>0 [ρ(i)(x),ρ(j)(y)] = 0. (3.9) The first three terms are (δϕt)(x,y). For arbitrary linear maps γ1, γ2 : g → End(V ), consider the standard cup-product : [[γ1,γ2]] : g⊗g → End(V ) defined by: [[γ1,γ2]](x,y) = (−1)|γ2|(|γ1|+|x|)[γ1(x),γ2(y)] + (−1)|γ1||x|[γ2(x),γ1(y)]. (3.10) The relation (3.9) becomes now equivalent to δϕt + 1 2 [[ϕt,ϕt]] = 0. (3.11) Expanding (3.11) in power series in t1, . . . , tm, we obtain the following equa- tion for ρ(k): δρ(k) + 1 2 ∑ i+j=k [[ρ(i),ρ(j)]] = 0. (3.12) The first non-trivial relation δρ(2) + 1 2 [[ρ(1),ρ(1)]] = 0 gives the first ob- struction to integration of an infinitesimal deformation. Thus, considering the coefficient of ti tj, we get δρ (2) ij + 1 2 [[Υi, Υj]] = 0. (3.13) It is easy to check that for any two 1-cocycles γ1 and γ2 ∈ Z1(g,h; End(V )), the bilinear map [[γ1,γ2]] is a h-relative 2-cocycle. The relation (3.13) is precisely the condition for this cocycle to be a coboundary. Moreover, if one of the cocycles γ1 or γ2 is a h-relative coboundary, then [[γ1,γ2]] is a h-relative 2- coboundary. Therefore, we naturally deduce that the operation (3.10) defines a bilinear map: H1(g,h; End(V )) ⊗ H1(g,h; End(V )) −→ H2(g,h; End(V )). (3.14) All the obstructions lie in H2(g,h; End(V )) and they are in the image of H1(g,h; End(V )) under the cup-product. 278 i. laraiedh 3.3. Equivalence. Two deformations ρ and ρ′ of a g-module V over A are said to be equivalent (see [13]) if there exists an inner automorphism Ψ of the associative superalgebra A⊗ End(V ) such that Ψ ◦ρ = ρ′ and Ψ(I) = I, where I is the unity of the superalgebra A⊗ End(V ). The following notion of miniversal deformation is fundamental. It as- signs to a g-module V a canonical commutative associative algebra A and a canonical deformation over A. A deformation (3.7) over A is said to be miniversal if (i) for any other deformation ρ′ with base (local) A′, there exists a homo- morphism ψ : A′ →A satisfying ψ(1) = 1, such that ρ = (ψ ⊗ Id) ◦ρ′; (ii) under notation of (i), if ρ is infinitesimal, then ψ is unique. If ρ satisfies only the condition (i), then it is called versal. This definition does not depend on the choice 1-cocycles Υ1, . . . , Υm representing a basis of H1(g,h; End(V )). The miniversal deformation corresponds to the smallest ideal R. We refer to [13] for a construction of miniversal deformations of Lie algebras and to [2] for miniversal deformations of g-modules. Superization of these results is immediate by the Sign Rule. 3.4. Integrability conditions. In this subsection, we obtain the in- tegrability conditions for the infinitesimal deformation (3.5). Theorem 3.1. (i) The following conditions are necessary and sufficient for integrability of the infinitesimal deformation (3.5): t1λ,λ+ktλ,λ − tλ+k,λ+k t 1 λ,λ+k = 0 t2λ,λ+ktλ,λ − tλ+k,λ+k t 2 λ,λ+k = 0 } for 2(β −λ) ∈{2k,. . . , 2m} . (3.15) (ii) Any formal aff(1|1)-trivial deformation of the aff(2|1)-module S 2,m β is equivalent to a deformation of order 1, that is, to a deformation given by (3.5). aff(1|1)-trivial deformations of aff(2|1)-modules 279 The super-commutative algebra defined by relations (3.15) corresponds to the miniversal deformation of the Lie derivative LX. Note that the super- commutative algebra defined in Theorem 3.1 is infinite-dimensional. The proof of Theorem 3.1 consists in two steps. First, we compute ex- plicitly the obstructions for existence of the second-order term, this will prove that relations (3.15) are necessary. Second we show that under relations (3.15) the highest-order terms of the deformation can be chosen identically zero, so that relations (3.15) are indeed sufficient. Proof. Assume that the infinitesimal deformation (3.5) can be integrated to a formal deformation L̃X = LX + L (1) X + L (2) X + · · · , where L (1) X is given by (3.6) and L (2) X is a quadratic polynomial in t with coefficients in S 2,m β . Considering the homomorphism condition, we compute the second order term L(2) which is a solution of the Maurer-Cartan equation: ∂(L(2)) = − 1 2 [[L(1), L(1)]]. (3.16) For arbitrary λ, the right hand side of (3.16) yields the following aff(1|1)- relative 2-cocycles: Bλ,λ = [[Υλ,λ, Υλ,λ]] : aff(2|1) ⊗aff(2|1) −→ D2λ,λ, B̃iλ,λ+k = [[Γ i λ,λ+k, Υλ,λ]] : aff(2|1) ⊗aff(2|1) −→ D 2 λ,λ+k, i ∈{1, 2}, B i λ,λ+k = [[Υλ+k,λ+k, Γ i λ,λ+k]] : aff(2|1) ⊗aff(2|1) −→ D 2 λ,λ+k, i ∈{1, 2}. By a straightforward computation, we check that Bλ,λ = 0, B̃ 1 λ,λ+k = −B 1 λ,λ+k and B̃ 2 λ,λ+k = −B 2 λ,λ+k, with B̃1λ,λ+k(XF ,XG) = ( η1η2(F)G ′ −F ′η1η2(G) ) ∂kx, B̃2λ,λ+k(XF ,XG) = ( η1η2(F)G ′ −F ′η1η2(G) ) η1η2∂ k−1 x . We will need the following: 280 i. laraiedh Proposition 3.2. Each of the bilinear map B̃1λ,λ+k, B̃ 2 λ,λ+k define generically nontrivial cohomology class. Moreover, these cohomology classes are linearly independent. Proof. Each map B̃iλ,λ+k, i = 1, 2, is a aff(1|1)-relative 2-cocycle on aff(2|1) since it is the Kolmogorov-Alexander product of two aff(1|1)-relative 1-cocycles. Assume that, for some differential 1-cochain bλ,λ+k on aff(2|1) with coefficients in D2λ,λ+k, we have B̃ i λ,λ+k = ∂(bλ,λ+k). The general form of such a cochain is bλ,λ+k(XF ) = ∑ a`1`2m1m2 (x,θ1,θ2)η `1 1 η `2 2 (F)η m1 1 η m2 2 , where the coefficients a`1`2m1m2 (x,θ1,θ2) are arbitrary functions. To complete the proof of the proposition we will need the following: Lemma 3.3. The condition B̃iλ,λ+k = ∂(bλ,λ+k) implies that the coeffi- cients a`1`2m1m2 (x,θ1,θ2) are functions of θi, not depending on x. Proof. The condition B̃iλ,λ+k = ∂(bλ,λ+k) reads B̃iλ,λ+k(XF ,XG) = XF · bλ,λ+k(XG) − (−1) |G||F |XG · bλ,λ+k(XF ) − bλ,λ+k([XF , XG]). (3.17) We choose a constant function F = 1, and prove that X1 ·bλ,λ+k = 0. Indeed, we have (X1 · bλ,λ+k)(XG) = X1 · bλ,λ+k(XG) − bλ,λ+k([X1, XG]). Since `1 + `2 ≥ 1 in the expression of bλ,λ+k, it follows that bλ,λ+k(X1) = 0, and thus the last equality gives (X1 · bλ,λ+k)(XG) = X1 · bλ,λ+k(XG) −XG · bλ,λ+k(X1) − bλ,λ+k([X1, XG]) = ∂(bλ,λ+k)(X1,XG). By assumption, B̃iλ,λ+k = ∂(bλ,λ+k), from the explicit formula for B̃ i λ,λ+k, we obtain B̃iλ,λ+k(X1,XG) = 0 for all XG ∈ aff(2|1). Therefore, X1 · bλ,λ+k = 0. Lemma 3.3 is proved. aff(1|1)-trivial deformations of aff(2|1)-modules 281 Now, by direct computation, we check that equation (3.17) has no solution. This is contradiction with the assumption, B̃iλ,λ+k = ∂(bλ,λ+k). Thus, the maps B̃iλ,λ+k are nontrivial aff(1|1)-relative 2-cocycles. Moreover, by a direct computation, we can check that, for some differential 1-cochain bλ,λ+k on aff(2|1) with coefficients in D2λ,λ+k, the system( B̃1λ,λ+k, B̃ 2 λ,λ+k, ∂(bλ,λ+k) ) is linearly independent. Thus, the cohomology classes of B̃iλ,λ+k are linearly independent. This completes the proof of Proposition 3.2. Proposition 3.2 implies that equation (3.16) has solutions if and only if the quadratic polynomials given by (3.15) vanish simultaneously. We thus proved that conditions (3.15) are, indeed, necessary. To prove that the conditions (3.15) are sufficient, we will find explicitly a deformation of LXF , whenever the conditions (3.15) are satisfied. The solution L(2) of (3.16) can be chosen identically zero. Choosing the hightest-order terms L(m) with m ≥ 3, also identically zero, one obviously obtains a deformation which is of order 1 in t. Theorem 3.1, part (i) is proved. The solution L(2) of (3.16) is defined up to a aff(1|1)-relative 1-cocycle and it has been shown in [13, 2] that 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 hightest-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 part (ii). Example 3.1. For m ∈ 1 2 N and for arbitrary generic λ ∈ R, the following example is a 1-parameter aff(1|1)-trivial deformation of the aff(2|1)-module S 2,m λ+m L̃XF = LXF + t 2m∑ `=0 [ 2m−` 2 ]∑ k=1 2∑ i=1 Γi λ+ ` 2 ,λ+ ` 2 +k , that is , we put ti λ+ ` 2 ,λ+ ` 2 +k = t and tλ,λ = 0. Of course it is easy to give many other examples of true deformations with one parameter or with several parameters. 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Geom. 3 (1969), 289 – 308. arXiv:hep-th/9702120 Introduction Definitions and notations The Lie superalgebra of contact vector fields on R1|2. The superalgebra aff(2|1). The space of weighted densities on R1|2. Differential operators on weighted densities. aff(1|1)-trivial deformation of aff(2|1)-modules Infinitesimal deformations and the first cohomology. Integrability conditions and deformations over supercommutative algebras. Equivalence. Integrability conditions.