Bull. Korean Math. Soc. 56 (2019), No. 1, pp. 13–22 https://doi.org/10.4134/BKMS.b170774 pISSN: 1015-8634 / eISSN: 2234-3016

SECOND COHOMOLOGY OF aff(1) ACTING ON n-ARY DIFFERENTIAL OPERATORS

Imed Basdouri, Ammar Derbali, and Soumaya Saidi

Abstract. We compute the second cohomology of the affine Lie algebra n aff(1) on the dimensional real space with coefficients in the space Dλ,µ of n-ary linear differential operators acting on weighted densities where λ = (λ1, . . . , λn). We explicitly give 2-cocycles spanning these cohomology.

1. Introduction In mathematical deformation theory one studies how an object in a certain category of spaces can be varied in dependence on the points of a parameter space. In other words, deformation theory thus deals with the structure of families of objects like varieties, singularities, vector bundles, coherent sheaves, algebras or differentiable maps. Deformation problems appear in various areas of mathematics, in particular in algebra, algebraic and analytic geometry, and mathematical physics. Cohomology is a useful tool in Poisson Geometry, plays an important role in Deformation and Quantization Theory, and attracts more and more interest among algebraists. In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges. Starting with a polynomial loop algebra over finite-dimensional simple Lie algebra and performing two extensions, a central extension and an extension by a derivation, one obtains a Lie algebra which is isomorphic with an untwisted affine Kac-Moody algebra. Using the centrally extended loop algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra ([2]).

Received September 6, 2017; Revised September 2, 2018; Accepted December 5, 2018. 2010 Mathematics Subject Classification. 17B56, 17B67, 53D55. Key words and phrases. cohomology, affine Lie algebras, n-ary linear differential operat- ors.

c 2019 Korean Mathematical Society 13 14 I. BASDOURI, A. DERBALI, AND S. SAIDI

Central extensions are needed in physics, because the symmetry group of a quantized system usually is a central extension of the classical symmetry group, and in the same way the corresponding symmetry Lie algebra of the quantum system is, in general, a central extension of the classical symmetry algebra (see [3]). In 1967, Victor Kac and Robert Moody independently generalized the notion of classical Lie algebras, resulting in a new theory of infinite-dimensional Lie algebras, now called Kac-Moody algebras (see [9, 12]). They generalize the finite-dimensional simple Lie algebras and can often concretely be constructed as extensions. Kac-Moody algebras have been conjectured to be a symmetry groups of a unified superstring theory (see [8]). The centrally extended Lie algebras play a dominant role in quantum field theory, particularly in conformal field theory, string theory and in M-theory (see [6]). Central extensions are the simplest extensions of Lie algebras. They appear both in geometry and in physics. Thus, they play an important role in symplectic geometry [10,13] and in various versions of quantization (see also [11]). A large portion towards the end is devoted to background material for appli- cations of Lie algebra extensions, both in mathematics and in physics, in areas where they are actually useful. A parenthetical link, (background material), is provided where it might be beneficial. Lie algebra extensions are most interesting and useful for infinite-dimensional Lie algebras. The theory of group extensions and their interpretation in terms of cohomology is well known, see, e.g., [7]. Let g be a Lie algebra and M a g-module. The second space H2(g,M) classify the nontrivial extensions of the Lie algebra g by the module M: 0 −→ M −→ · −→ g −→ 0, the Lie structure on g ⊕ M being given by

[(g1, α), (g2, β)]g⊕M = ([g1, g2], g1.β − g2.α + Ω(g1, g2)), where Ω is a 2-cocycle with values in M. In this paper we consider a natural class of “non-central” extensions of aff(1), namely extensions by the modules Dλ,µ of n-ary linear differential operators acting on weighted densities. The result is quite surprising: there exists a k Cn+k−1 extensions if and only if µ = λ1 + ··· + λn + k, where k ∈ N. We consider the one-parameter action of the Lie algebra of vector fields Vect(R) by the Lie derivative on the space C∞(R) of smooth functions on R defined by:

λ 0 0 (1.1) L d (f) = Xf + λfX , X dx ∞ d 0 where f ∈ C (R) and X dx ∈ Vect(R) and where the superscript stands for d ∞ dx . We denote by Fλ the Vect(R)-module structure on C (R) defined by the action (1.1). Geometrically, Fλ is the space weighted densities of weight λ on SECOND COHOMOLOGY OF aff(1) 15

R:  λ ∞ Fλ = fdx , f ∈ C (R) . Any differential operator A on R can be viewed as the linear mapping λ µ f(dx) 7→ (Af)(dx) from Fλ to Fµ (λ, µ in R). Thus the space of differ- 1 ential operators is a Vect(R)-module, denoted Dλ,µ := Homdiff (Fλ, Fµ). The Vect(R) action is: λ,µ µ λ (1.2) LX (A) = LX ◦ A − A ◦ LX . More generally we consider the structures Vect(R)-modules on the space n Dλ,µ of n-linear differential operators: A : Fλ1 ⊗ · · · ⊗ Fλn → Fµ. The Lie n algebra Vect(R) acting on the space Dλ,µ of n-ary Linear differential operators by:

(λ1,...,λn);µ µ (λ1,...,λn) (1.3) LX (A) = LX ◦ A − A ◦ LX ,

(λ1,...,λn) where LX is the Lie derivative defined by the Leibniz rule:

(λ1,...,λn) LX (Φ1 ⊗ Φ2 ⊗ · · · ⊗ Φn)(1.4)

λ1 λn = LX (Φ1) ⊗ Φ2 ⊗ · · · ⊗ Φn + ··· + Φ1 ⊗ · · · ⊗ Φn−1 ⊗ LX (Φn).

2. Definitions and notations In this section, we recall the main definitions and facts related to the geome- try of the space R. We also recall some fundamental concepts from cohomology theory (see, e.g., [5, 7]). 2.1. Cohomology theory Let us first recall some fundamental concepts from cohomology theory (see, e.g., [4,5,7,14]). Let g be a Lie algebra acting on a vector space V . The space of n-cochains of g with values in V is the g-module Cn(g,V ) := Hom(Λng; V ). n n+1 The coboundary operator δn : C (g,V ) −→ C (g,V ) is a g-map satisfying n δn ◦ δn−1 = 0. The kernel of δn, denoted Z (g,V ), is the space of n-cocycles, among them, the elements in the range of δn−1 are called n-coboundaries. We denote Bn(g,V ) the space of n-coboundaries. By definition, the nth cohomolgy space is the quotient space Hn(g,V ) = Zn(g,V )/Bn(g,V ).

We will only need the formula of δn (which will be simply denoted δ) in degrees 0, 1 and 2: for Ξ ∈ C0(g,V ) = V , δΞ(x) := x · Ξ, and for Λ ∈ C1(g,V ), (2.5) δ(Λ)(x, y) := g · Λ(y) − y · Λ(x) − Λ([x, y]) for any x, y ∈ g, for Ω ∈ C2(g,V ), δ(Ω)(x, y, z) := x · Ω(y, z) − y · Ω(x, z) + z · Ω(x, y)(2.6) 16 I. BASDOURI, A. DERBALI, AND S. SAIDI

− Ω([x, y], z) + Ω([x, z], y) − Ω([y, z], x), where x, y, z ∈ g.

2.2. Lie algebra aff(1)

The Lie algebra aff(1) is realized as a subalgebra of the Lie algebra Vect(R) (see [1]): d d aff(1) = Span(X = ,X = x ). 1 dx x dx The commutation relations are

[X1,Xx] = X1, [Xx,Xx] = 0, [X1,X1] = 0.

2.3. The space of tensor densities on R The Lie algebra, Vect(R), of vector fields on R naturally acts, by the Lie derivative, on the space  λ ∞ Fλ = fdx : f ∈ C (R) , λ of weighted densities of degree λ. The Lie derivative LX of the space Fλ along d the vector field X dx is defined by λ 0 (2.7) LX = X∂x + λX , ∞ 0 dX λ where X, f ∈ C (R) and X := dx . More precisely, for all fdx ∈ Fλ, we have λ λ 0 0 λ LX (fdx ) = (Xf + λfX )dx . In the paper, we restrict ourselves to the Lie algebra aff(1) which is isomorphic to the Lie subalgebra of Vect(R) spanned by

{X1,Xx}.

2.4. The space of n-ary linear differential operators as a aff(1)-mod- ule

The space of n-ary linear differential operators is a Vect(R)-module, denoted n Dλ,µ := Homdiff (Fλ1 ⊗ · · · ⊗ Fλn , Fµ). The Vect(R) action is: λ,µ µ λ (2.8) LX (A) = LX ◦ A − A ◦ LX ,

(λ1,...,λn) where λ = (λ1, . . . , λn) and LX is the Lie derivative on Fλ1 ⊗ · · · ⊗ Fλn defined by the Leibnitz rule:

λ λ1 λn LX (f1dx ⊗ · · · ⊗ fndx )

λ1 λn λ1 λn λn = LX (f1) ⊗ · · · ⊗ fndx + ··· + f1dx ⊗ · · · ⊗ LX (fndx ). SECOND COHOMOLOGY OF aff(1) 17

2 3 3. The space Hdiff (aff(1), Dλ,µ) The first main result of paper is the following.

2 3 Theorem 3.1. The space Hdiff (aff(1), Dλ,µ) has the following structure:

( Ck k+2 if µ = λ + λ + λ + k, (3.9) H2 (aff(1), D3 ) ' R 1 2 3 diff λ,µ 0 otherwise,

p q! where Cq = (q−p)!p! . The following 2-cocycles span the corresponding cohomology spaces: d d C(X ,Y )(f dxλ1 , f dxλ2 , f dxλ3 ) dx dx 1 2 3 k−1 X 0 0 (i) (j) (i+j−k) λ1+λ2+λ3+k = ci,j(XY − X Y )f1 f2 f3 dx , i+j=0

λ1 λ2 λ3 d where ci,j are constants, f1dx ∈ Fλ1 , f2dx ∈ Fλ2 , f3dx ∈ Fλ3 and X dx , d Y dx ∈ aff(1). We need the following lemma. Lemma 3.2. Let b ∈ C1(aff(1), D3 ) be defined as follows: for λ,λ1+λ2+λ3+k d λ1 λ2 λ3 X dx ∈ aff(1), f1dx ∈ Fλ1 , f2dx ∈ Fλ2 and f3dx ∈ Fλ3 X (i) (j) (n) µ (3.10) b(X)(f1, f2, f3) = ci,j,nXf1 f2 f3 dx i+j+n=k X 0 (i) (j) (n) µ + αi,j,nX f1 f2 f3 dx , i+j+n=k−1 where the coefficients ci,j,n and αi,j,n are constants. Then the map δb : aff(1) × aff(1) → D3 is given by λ,λ1+λ2+λ3+k

δb(X,Y )(f1, f2, f3)(3.11)

λ,λ1+λ2+λ3+k λ1 λ2 λ3 = LX b(Y )(f1dx , f2dx , f3dx )

λ,λ1+λ2+λ3+k λ1 λ2 λ3 − LY b(X)(f1dx , f2dx , f3dx )

λ1 λ2 λ3 − b([X,Y ])(f1dx , f2dx , f3dx ) X 0 0 (i) (j) (n) = ci,j,n(µ − λ1 − λ2 − λ3 − k)(X Y − XY )f1 f2 f3 . i+j+n=k 3.1. Proof of theorem 1 3 Any 2-cocycle C ∈ Zdiff (aff(1), Dλ,µ) should be retain the following general form: X 0 0 (i) (j) (n) µ (3.12) C(X,Y )(f1, f2, f3) = ci,j,n(XY − X Y )f1 f2 f3 dx , i+j+n=k−1 18 I. BASDOURI, A. DERBALI, AND S. SAIDI where ci,j,n are constants. λ1 λ2 The 2-cocycle condition reads as follows: for all f1dx ∈ Fλ1 , f2dx ∈ Fλ2 , λ3 f3dx ∈ Fλ3 and for all X,Y ∈ aff(1), we have

λ1,λ2,λ3,µ λ1 λ2 λ3 LX C(Y,Z)(f1dx , f2dx , f3dx )(3.13) − C([X,Y ],Z)+ (X,Y,Z) = 0, where (X,Y,Z) denotes summation over the cyclic permutation on X, Y , Z. A direct computation, proves that we have

λ1,λ2,λ3,µ Φ1 = LX C(Y,Z)(f1, f2, f3)

µ λ1 = LX C(Y,Z)(f1, f2, f3) − C(Y,Z)(LX (f1), f2, f3)

λ2 λ3 − C(Y,Z)(f1,LX (f2), f3) − C(Y,Z)(f1, f2,LX (f3)) X 0 0 0 (i) (j) (n) = ci,j,nX(YZ − Y Z)f1 f2 f3 i+j+n=k−1 X 0 0 0 (i) (j) (n) + ci,j,n(µ − λ1 − λ2 − λ3 − k + 1)X (YZ − Y Z)f1 f2 f3 , i+j+n=k−1

λ1,λ2,λ3,µ Φ2 = LY C(X,Z)(f1, f2, f3) X 0 0 0 (i) (j) (n) = ci,j,nY (XZ − X Z)f1 f2 f3 i+j+n=k−1 X 0 0 0 (i) (j) (n) + ci,j,n(µ − λ1 − λ2 − λ3 − k + 1)Y (XZ − X Z)f1 f2 f3 , i+j+n=k−1

λ1,λ2,λ3,µ Φ3 = LZ C(X,Y )(f1, f2, f3) X 0 0 0 (i) (j) (n) = ci,j,nZ(XY − X Y )f1 f2 f3 i+j+n=k−1 X 0 0 0 (i) (j) (n) + ci,j,n(µ − λ1 − λ2 − λ3 − k + 1)Z (XY − X Y )f1 f2 f3 , i+j+n=k−1

X 0 0 0 (i) (j) (n) Ψ1 = C([X,Y ],Z) = ci,j,nZ (XY − X Y )f1 f2 f3 , i+j+n=k−1 X 0 0 0 (i) (j) (n) Ψ2 = C([X,Z],Y ) = ci,j,nY (XZ − X Z)f1 f2 f3 , i+j+n=k−1 X 0 0 0 (i) (j) (n) Ψ3 = C([Y,Z],X) = ci,j,nX (YZ − Y Z)f1 f2 f3 . i+j+n=k−1 Then, by considering the equation (3.13), we can write

λ1,λ2,λ3,µ λ1 λ2 λ3 LX C(Y,Z)(f1dx , f2dx , f3dx )(3.14) SECOND COHOMOLOGY OF aff(1) 19

3 X − C([X,Y ],Z)+ (X,Y,Z) = (Φt − Ψt) = 0. t=1 0 We show that the equation (3.14) is satisfied only if ci,j,n = 0 and ci,j,n(µ − λ1 − λ2 − λ3 − k) = 0. Any trivial 2-cocycle should be of the form

λ,µ λ1 λ2 λ3 λ,µ λ1 λ2 λ3 LX b(Y )(f1dx , f2dx , f3dx ) − LY b(X)(f1dx , f2dx , f3dx )

λ1 λ2 λ3 − b([X,Y ])(f1dx , f2dx , f3dx ). By using Lemma 3.2, we have

δb(X,Y )(f1, f2, f3)(3.15)

λ,λ1+λ2+λ3+k λ1 λ2 λ3 = LX b(Y )(f1dx , f2dx , f3dx )

λ,λ1+λ2+λ3+k λ1 λ2 λ3 − LY b(X)(f1dx , f2dx , f3dx )

λ1 λ2 λ3 − b([X,Y ])(f1dx , f2dx , f3dx ) X 0 0 (i) (j) (n) = ci,j,n(µ − λ1 − λ2 − λ3 − k)(X Y − XY )f1 f2 f3 . i+j+n=k

2 n 4. The space Hdiff (aff(1), Dλ,µ) The second main result of paper is the following.

2 n Theorem 4.1. The space Hdiff (aff(1), Dλ,µ) has the following structure:

( Ck n+k−1 if µ = λ + ··· + λ + k, (4.16) H2 (aff(1), Dn ) ' R 1 n diff λ,µ 0 otherwise. The following 2-cocycles span the corresponding cohomology spaces: d d C(X ,Y )(f dxλ1 , . . . , f dxλn ) dx dx 1 n k−1 X 0 0 (i1) (in) λ1+···+λn+k = ci1,...,in (XY − X Y )f1 ··· fn dx , i1+···+in=0

λi d d where ci1,...,in are constants, fidx ∈ Fλi and X dx ,Y dx ∈ aff(1). We need the following lemma. Lemma 4.2. Let b ∈ C1(aff(1), Dn ) be defined as follows: for λ,λ1+···+λn+k d λi X dx ∈ aff(1), fidx ∈ Fλi .

X (i1) (in) µ (4.17) b(X)(f1, f2, . . . , fn) = ci1,...,in Xf1 ··· fn dx i1+···+in=k

X 0 (i1) (in) µ + αi1,...,in X f1 ··· fn dx , i1+···+in=k−1 20 I. BASDOURI, A. DERBALI, AND S. SAIDI

where the coefficients ci1,...,in and αi1,...,in are constants. Then the map δb : aff(1)⊗2 → Dn is given by λ,λ1+···+λn+k

δb(X,Y )(f1, . . . , fn)(4.18)

λ1,...,λn,λ1+···+λn+k λ1 λn = LX b(Y )(f1dx , . . . , fndx )

λ1,...,λn,λ1+···+λn+k λ1 λn − LY b(X)(f1dx , . . . , fndx )

λ1 λn − b([X,Y ])(f1dx , . . . , fndx )

X 0 0 (i1) (in) = ci1,...,in (µ − λ1 − · · · − λn − k)(X Y − XY )f1 ··· fn . i1+···+in=k 4.1. Proof of theorem Any 2-cocycle C ∈ Z1 (aff(1), Dn ) should be retain the following diff λ,λ1+···+λn+k general form:

C(X,Y )(f1, . . . , fn)(4.19)

X 0 0 (i1) (in) µ = ci1,...,in (XY − X Y )f1 ··· fn dx , i1+···+in=k−1 where ci1,...,in are constants. λi The 2-cocycle condition reads as follows: for all fidx ∈ Fλi , and for all X,Y ∈ aff(1), we have

λ1,...,λn;µ λ1 λn LX C(Y,Z)(f1dx , . . . , fndx ) − C([X,Y ],Z)+ (X,Y,Z) = 0.

(i1) (in) A direct computation, proves that the coefficient of the component f1 ··· fn in the 2-cocycle condition above is equal to

(4.20) µ − λ1 − · · · − λn − k = 0. Any trivial 2-cocycle should be of the form

λ1,...,λn,µ λ1 λn λ1,...,λn,µ λ1 λn LX b(Y )(f1dx , . . . , fndx ) − LY b(X)(f1dx , . . . , fndx )

λ1 λn − b([X,Y ])(f1dx , . . . , fndx ). By using Lemma 4.2, we have

δb(X,Y )(f1, . . . , fn)(4.21)

λ1,...,λn;λ1+···+λn+k λ1 λn = LX b(Y )(f1dx , . . . , fndx )

λ1,...λn;λ1+···+λn+k λ1 λn − LY b(X)(f1dx , . . . , fndx )

λ1 λn − b([X,Y ])(f1dx , . . . , fndx )

X 0 0 (i1) (in) = ci1,...,in (µ − λ1 − · · · − λn − k)(X Y − XY )f1 ··· fn . i1+···+in=k SECOND COHOMOLOGY OF aff(1) 21

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Imed Basdouri Departement´ de Mathematiques´ Faculte´ des Sciences de Gafsa Tunisie Email address: [email protected]

Ammar Derbali Universite´ de Gafsa Faculte´ des Sciences Departement´ de Mathematiques´ Tunisie Email address: [email protected] 22 I. BASDOURI, A. DERBALI, AND S. SAIDI

Soumaya Saidi Universite´ de Gafsa Faculte´ des Sciences Departement´ de Mathematiques´ Tunisie Email address: [email protected]