Lie conformal algebra cohomology and the variational complex

Lie conformal algebra cohomology and the v ariational complex Alb erto De Sole 1 and Victor G. Kac 2 1 Dip artimento di Matematic a, Universit´ a di R oma “L a Sapienza” Citt´ a U niversitaria, 00185 Roma, Italy E-mail: desole @mat.u niroma1.it 2 Department of Mathematics, MIT 77 Mas sach usetts Aven ue, Ca m br idge, MA 021 39, USA E-mail: kac@ma th.mit .edu De dic ate d to Corr ado De Concini on his 60-th birthday. Abstract W e find an in terpretation of the complex o f v ariatio na l calculus in terms of the Lie confor mal algebra cohomology theory . This leads to a b etter understanding of b oth theories. In par- ticular, w e giv e an explicit construction of the Lie conformal alg e bra cohomo logy complex, and endow it with a structure of a g -c o mplex. On the other ha nd, we give an explicit con- struction of the complex of v a riational calculus in terms of skew-symmetric p oly-differe ntial op erators. In tr o duction. Lie conformal algebras enco de the prop erties of operator pro duct expansions in conformal field theory , and, at the same time, of local P oisson brac kets in the th eory of in tegrable ev olution equations. Recall [K] that a Lie c onformal algebr a o v er a field F is an F [ ∂ ]-mo dule A , endow ed with a λ - br acket , that is an F -linear map A ⊗ A → F [ λ ] ⊗ A denoted b y a ⊗ b 7→ [ a λ b ], satisfying the t wo sesquiline arity pr op erties (0.1) [ ∂ a λ b ] = − λ [ a λ b ] , [ a λ ∂ b ] = ( ∂ + λ )[ a λ b ] , suc h that the skew-symmetry (0.2) [ a λ b ] = − [ b − ∂ − λ a ] and the Jac obi identity (0.3) [ a λ [ b µ c ]] − [ b µ [ a λ c ]] = [[ a λ b ] λ + µ c ] hold for an y a, b, c ∈ A . It is assum ed in (0.2) that ∂ is mo ve d to the left. A mo dule o ve r a Lie conformal algebra A is an F [ ∂ ]-mo du le M , endo w ed w ith a λ - action , that is an F -linear map A ⊗ M → F [ λ ] ⊗ M , denoted b y a ⊗ b → a λ b , such that sesqu ilinearit y (0.1) holds for a ∈ A , b ∈ M and Jacobi identi t y (0.3) h olds for a, b ∈ A , c ∈ M . A cohomology theory f or Lie conformal algebras was dev elop ed in [BKV]. Giv en a Lie conformal algebra A and an A -mo du le M , one first defines the b asic c ohomolo gy c omplex 1 e Γ • ( A, M ) = P k ∈ Z + e Γ k , w h ere e Γ k consists of F -linear maps e γ : A ⊗ k → F [ λ 1 , . . . , λ k ] ⊗ M , satisfying certain sesquilinearit y and sk ew-sym metry pr op erties, and endows this complex with a differen tial δ : e Γ k → e Γ k +1 , suc h that δ 2 = 0. This complex is isomorphic to the Lie al- gebra cohomology complex for the annihilation Lie algebra g − of A with co efficien ts in the g − -mo dule M [BKV, Theorem 6.1]. Next, one endo ws e Γ • ( A, M ) with a stru cture of a F [ ∂ ]-mo dule, su c h th at ∂ comm utes with δ , whic h allo ws on e to d efine the redu ced cohomology complex Γ • ( A, M ) = e Γ • ( A, M ) / ∂ e Γ • ( A, M ), and this is the Lie conformal algebra cohomology complex, introd u ced in [BKV]. Our firs t cont ribution to this theory is a m ore explicit constru ction of the reduced cohomol- ogy complex. Namely , w e introd uce a n ew cohomology complex C • ( A, M ) = ⊕ k ∈ Z + C k , where C 0 = M /∂ M , C 1 = Hom F [ ∂ ] ( A, M ), and for k ≥ 2, C k consists of p oly λ -brac k ets, namely of F -linear maps c : A ⊗ k → F [ λ 1 , · · · , λ k − 1 ] ⊗ M , satisfying certain sesquilinearit y and skew- symmetry conditions, and w e end ow C • ( A, M ) with a square zero d ifferen tial d . W e construct em b eddings of complexes: (0.4) Γ • ( A, M ) ⊂ ¯ C • ( A, M ) ⊂ C • ( A, M ) , where ¯ C • ( A, M ) consists of cocycles whic h v anish if one of the argumen ts is a torsion elemen t of A . In fact, ¯ C k = C k , unless k = 1. W e sho w that Γ • ( A, M ) = ¯ C • ( A, M ), pro vid ed that, as an F [ ∂ ]-mo dule, A is isomorphic to a direct sum of it s torsion and a free F [ ∂ ]-mo dule (whic h is alw a ys th e case if A is a finitely generated F [ ∂ ]-mo dule). Our o pinion is that the sligh tly larger complex C • ( A, M ) is a more correct Lie conformal algebra cohomology complex than the complex Γ • ( A, M ) of [BKV]. This is illustrated b y our Theorem 3.1 (c), which says th at the F [ ∂ ]-split ab elian extensions of A by M are parameterized by H 2 ( A, M ) for the complex C • ( A, M ). This holds for the cohomology theory of [BKV] only if A is a free F [ ∂ ]-mo dule. F ollo wing [BKV], we also consider the sup erspace of b asic chains e Γ • ( A, M ) and its sub - space of reduced c h ains Γ • ( A, M ) (they a re n ot complexes in general) . Corresp ondin g to the em b eddings of c omplexes (0.4), we in tro duce the v ector sup erspaces of chains C • ( A, M ) and ¯ C • ( A, M ), and the maps: (0.5) C • ( A, M ) ։ ¯ C • ( A, M ) → Γ • ( A, M ) . W e devel op the theory furth er in the imp ortant for the calculus of v ariations case, wh en the A -mo dule M is endo we d with a co mm utativ e asso ciativ e pr o d uct, suc h that ∂ and a λ for all a ∈ A are d eriv atio ns of th is pro du ct. In this case one can endo w the sup erspace e Γ • ( A, M ) w ith a comm utativ e asso ciativ e pro duct [BKV]. F urthermore, w e introdu ce a Lie algebra brac k et on the sp ace g := Π e Γ 1 ( A, M ) (Π, as usual, stands for rev ers in g of the p arit y). Let b g = η g ⊕ g ⊕ F ∂ η b e a Z -graded Lie s u p eralgebra extension of g , where η is an o dd in determinate, η 2 = 0. W e end ow e Γ • ( A, M ) with a str u cture of a g - c omp lex , whic h is a Z -grading preserving Lie sup eralgebra h omomorphism ϕ : b g → End F e Γ • ( A, M ), such that ϕ ( ∂ η ) = δ . W e also sho w that ϕ ( b g ) lies in the subalgebra of deriv ations of the sup eralgebra e Γ • ( A, M ). F or eac h X ∈ g w e th u s h a v e the Lie deriv ativ e L X = ϕ ( X ) and the con tr actio n o p erator ι X = ϕ ( η X ), satisfying all usual relations, in particular, the Cartan formula L X = ι X δ + δ ι X . Denoting by g ∂ the cen tr alize r o f ∂ in g , w e obtain the indu ced stru cture of a g ∂ -complex for Γ • ( A, M ), w hic h we, furthermore, extend to the larger complex C • ( A, M ). Namely , we 2 in tro d uce a canonical Lie algebra brac k et on all spaces of 1-c hains w ith rev ersed parity (see (0.5)), so that all th e maps Π C 1 ։ Π ¯ C 1 → ΠΓ 1 ֒ → Π e Γ 1 are Lie algebra homomorphisms, and the em b edd ings (0.4) are morp hisms of complexes, en do wed with a corresp ond ing Lie algebra structure. What do es it all ha ve to d o with the calculus of v ariations? In order to explain this, int ro duce the notion of an algebr a of differ entiable functions (in ℓ v ariables). This is a different ial algebra, i.e., a unital comm u tativ e associativ e algebra V with a d er iv ation ∂ , endow ed with comm uting deriv ations ∂ ∂ u ( n ) i , i ∈ I = { 1 , . . . , ℓ } , n ∈ Z + , su c h that only a finite num b er of ∂ f ∂ u ( n ) i are non-zero for eac h f ∈ V , and th e follo wing comm utation ru les with ∂ hold: (0.6) " ∂ ∂ u ( n ) i , ∂ # = ∂ ∂ u ( n − 1) i (the RHS is 0 if n = 0) . The most imp ortan t example is the algebra of d ifferen tial p olynomials F [ u ( n ) i | i ∈ I , n ∈ Z + ] with ∂ ( u ( n ) i ) = u ( n +1) i , n ∈ Z + , i ∈ I . Other examples include any lo calizatio n by a m u ltiplicati v e sub set or an y algebraic extension of this algebra. The b asic de Rham complex e Ω • = e Ω • ( V ) o ver V is defined as an exterior su p eralgebra o ve r the free V -mo dule e Ω 1 = P i ∈ I , n ∈ Z + V δu ( n ) i on generators δ u ( n ) i with o dd parity . W e ha ve : e Ω • = L k ∈ Z + e Ω k , where e Ω 0 = V , e Ω k = Λ k V e Ω 1 . T his Z -graded sup eralgebra is endo wed b y an o dd deriv ation δ of d egree 1, su c h th at δ f = P i ∈ I , n ∈ Z + ∂ f ∂ u ( n ) i δ u ( n ) i for f ∈ e Ω 0 and δ ( δ u ( n ) i ) = 0. One easily c h ec ks that δ 2 = 0, so that e Ω • is a cohomology complex. Let g b e the Lie algebra of deriv ations of the algebra V of the form (0.7) X = X i ∈ I , n ∈ Z + P i,n ∂ ∂ u ( n ) i , where P i,n ∈ V . T o any suc h deriv ation X w e asso ciate an ev en deriv ation L X (Lie deriv ativ e) and an o dd deriv a- tion ι X (con traction) of the sup eralgebra e Ω • b y letting L X | V = X, L X ( δ u ( n ) i ) = δ P i,n , ι X | V = 0 , ι X ( δ u ( n ) i ) = P i,n . This pr o vides e Ω • with a structure of a g -complex, b y letting ϕ ( X ) = L X and ϕ ( η X ) = ι X . Also, the deriv ation ∂ extends to an (ev en) deriv ation of e Ω • b y letting ∂ ( δ u ( n ) i ) = δ u ( n +1) i . It is easy to c heck, using (0.6), that ∂ and δ commute, h ence w e can consider the r educed complex Ω • ( V ) = e Ω • ( V ) /∂ e Ω • ( V ) , whic h is called the variational c omplex . Th is is, of course, a g ∂ -complex. Our m ain observ ation is the in terp retation of the v ariational complex Ω • ( V ) in terms of Lie conformal algebra cohomology , giv en by Theorem 0.1 b elo w. Let R = L i ∈ I F [ ∂ ] u i b e a free F [ ∂ ]-mo du le of rank ℓ , endo wed with the trivia l λ -brac ke t [ a λ b ] = 0 for all a, b ∈ R . Let V b e an algebra of differenti able functions. W e endo w V with th e structure of an R -mo dule b y letting u i λ f = X n ∈ Z + λ n ∂ f ∂ u ( n ) i , i ∈ I , 3 and extending to R by sesquilinearit y . Let g b e the Lie algebra of d eriv atio ns of V of the form (0.7), and let g ∂ b e the subalgebra of g , consisting of deriv ations comm uting with ∂ . Theorem 0.1. The g ∂ -c omplexes C • ( R, V ) and Ω • ( V ) ar e isomorphic. As a result, we obtain the follo wing interpretatio n of the complex Ω • ( V ), w hic h explains the name “calculus of v ariations”. W e hav e: Ω 0 = V /∂ V , Ω 1 = Hom F [ ∂ ] ( R, V ) = V ⊕ ℓ . Elemen ts of Ω 0 are called lo cal function- als and the image of f ∈ V in Ω 0 is d enoted by R f . Elemen ts of Ω 1 are called lo cal 1-forms. Th e differen tial δ : Ω 0 → Ω 1 is iden tified w ith the v ariat ional deriv ativ e: δ R f =  δ R f δu i  i ∈ I = δf δu , where (0.8) δ f δ u i = X n ∈ Z + ( − ∂ ) n ∂ f ∂ u ( n ) i . F urthermore, the space of 2-co c hains C 2 is id entified with the space of sk ew-adjoint dif- feren tial op er ators b y asso ciating to th e λ -b r ac k et {· λ ·} : R ⊗ 2 → F [ λ ] ⊗ V the ℓ × ℓ matrix S ij ( ∂ ) = { u j ∂ u i } → , where th e arr o w means that ∂ is mov ed to the right. The d ifferen tial δ : Ω 1 → Ω 2 is expressed in terms of the F rec het deriv ativ e (0.9) D F ( ∂ ) ij = X n ∈ Z + ∂ F i ∂ u ( n ) j ∂ n , i, j ∈ I , whic h defines an F -linear map: V ℓ → V ⊕ ℓ . Namely: δ F = D F ( ∂ ) − D F ( ∂ ) ∗ . Th e su b space of closed 2-cochains in C 2 is identified with the space of symplectic d ifferen tial op er ators. A 2-cochai n, whic h is a sk ew-adjoint differen tial op er ator S ij ( ∂ ), can b e id entified w ith the corresp onding F -linear map ( V ℓ ) 2 → V /∂ V , of “differen tial t yp e”, giv en b y S ( P , Q ) = Z X i,j ∈ I Q i S ij ( ∂ ) P j . Sk ew-adjoin tness of S translat es to the s k ew-symmetry condition S ( P , Q ) = − S ( Q, P ). More generally , th e s pace of k -co c h ains C k for k ≥ 2 is id en tified w ith the sp ace of all sk ew-symm etric F -linear maps S : ( V ℓ ) k → V /∂ V , of “differen tial t yp e”: S ( P 1 , · · · , P k ) = Z X i 1 , ··· ,i k ∈ I n 1 , ··· ,n k ∈ Z + f n 1 , ··· ,n k i 1 , ··· ,i k ( ∂ n 1 P 1 i 1 ) · · · ( ∂ n k P k i k ) , where f n 1 , ··· ,n k i 1 , ··· ,i k ∈ V . The sk ew-symmetry condition is simply S ( P 1 , · · · , P k ) = sign( σ ) S ( P σ (1) , · · · , P σ ( k ) ), for eve ry σ ∈ S k . The subspace of c losed k -cochains for k ≥ 2 is the sub s pace of “symplectic” k − 1- differen tial op erators. W e pro v e in [BDK] that the cohomolo gy H j of the complex Ω • ( V ) is zero for j ≥ 1 and H 0 = C / ( C ∩ ∂ V ), wh ere C := { f ∈ V | ∂ f ∂ u ( n ) i ∀ i ∈ I , n ∈ Z + } , pr ovided that V is normal, as defined in Section 4.6. (An y algebra of differentiable functions can b e includ ed in a normal one.) As a corollary , w e obtain (cf. [D]) that Ker δ δu = ∂ V + C , and F ∈ Im δ δu iff D F ( ∂ ) is a 4 self-adjoin t d ifferen tial op erator, p r o vided that V is n ormal. The first result can b e found in [D] (see also [Di] and [Vi], where it is pro ved under stronger conditions on V ), but it is certa inly m u c h older. The second result, at least under stronger conditions on V , goes bac k to [H], [V]. W e also obtai n the classification of symplectic differen tial op erators (cf. [D ]) and of symplectic p oly-differen tial op erators for normal V , whic h seems to b e a new result. Th us, the int eraction b et wee n the Lie conform al algebra cohomology and the v ariational calculus has led to progress in b oth theories. On the one hand, the v ariational calculus moti- v ated some of our constructions in the Lie conformal algebra cohomolo gy . O n the other h and, the Lie conformal algebra cohomology inte rpretation of the v ariatio nal complex h as led to a b etter und erstanding of this complex an d to a classification of symplectic and p oly-symplectic differen tial op erators. The ground field is an arbitrary field F of c h aracteristic 0. 1 Lie conformal algebra cohomology complexes. 1.1 The basic cohomology complex e Γ • and t he reduced cohomology complex Γ • . Let us review, follo wing [BKV ], the defin ition of the basic and redu ced cohomolo gy complexes asso ciated to a Lie conformal algebra A and an A -mo dule M . A k - c o chain of A with co efficien ts in M is an F -linear map e γ : A ⊗ k → F [ λ 1 , . . . , λ k ] ⊗ M , a 1 ⊗ · · · ⊗ a k 7→ e γ λ 1 , ··· ,λ k ( a 1 , · · · , a k ) , satisfying the follo wing t w o conditions: A1. e γ λ 1 , ··· ,λ k ( a 1 , · · · , ∂ a i , · · · , a k ) = − λ i e γ λ 1 , ··· ,λ k ( a 1 , · · · , a k ) for all i , A2. e γ is sk ew-symmetric with resp ect to sim u ltaneous p ermutat ions of the a i ’s and the λ i ’s. R emark 1.1 . Note that condition A1. i mplies th at e γ λ 1 , ··· ,λ k ( a 1 , · · · , a k ) is zero if one of the elemen ts a i is a torsion element of the F [ ∂ ]-mo dule A . W e let e Γ k = e Γ k ( A, M ) b e the space of all k -co c hains, and e Γ • = e Γ • ( A, M ) = L k ≥ 0 e Γ k . The differen tial δ of a k -co c h ain e γ is defined b y th e follo wing form ula: ( δ e γ ) λ 1 , ··· ,λ k +1 ( a 1 , · · · , a k +1 ) = k +1 X i =1 ( − 1) i +1 a i λ i  e γ λ 1 , i ˇ ··· ,λ k +1 ( a 1 , i ˇ · · · , a k +1 )  (1.1) + k +1 X i,j =1 i σ ( j ). Hence, for σ ( i ) < σ ( j ) w e ha v e  a σ (1) λ σ (1) i ˇ · · · j ˇ · · · a σ ( k +1) λ σ ( k +1) [ a σ ( i ) λ σ ( i ) a σ ( j ) ]  c    λ k +1 = λ † k +1 (1.8) = sign( σ )( − 1) i + j + σ ( i )+ σ ( j )  a 1 λ 1 σ ( i ) ˇ · · · σ ( j ) ˇ · · · a k +1 λ k +1 [ a σ ( i ) λ σ ( i ) a σ ( j ) ]  c    λ k +1 = λ † k +1 , while for σ ( i ) > σ ( j ) w e ha v e, by the sk ew-symmetry of the λ -brac ket in A ,  a σ (1) λ σ (1) i ˇ · · · j ˇ · · · a σ ( k +1) λ σ ( k +1) [ a σ ( i ) λ σ ( i ) a σ ( j ) ]  c    λ k +1 = λ † k +1 = sign( σ )( − 1) i + j + σ ( i )+ σ ( j )  a 1 λ 1 σ ( j ) ˇ · · · σ ( i ) ˇ · · · a σ ( k +1) λ k +1 [ a σ ( j ) − λ σ ( i ) − ∂ a σ ( i ) ]  c    λ k +1 = λ † k +1 (1.9) = sign( σ )( − 1) i + j + σ ( i )+ σ ( j )  a 1 λ 1 σ ( j ) ˇ · · · σ ( i ) ˇ · · · a σ ( k +1) λ k +1 [ a σ ( j ) λ σ ( j ) a σ ( i ) ]  c    λ k +1 = λ † k +1 . In the last iden tit y we used th e assumption that c sat isfies condition B2. Clearly , equ ations (1.7), (1.8) and (1.9), together with the definition (1.6) of dc , imply that dc satisfies condition B3. W e are left to pro v e that d 2 c = 0. W e h a v e, by (1.6), { a 1 λ 1 · · · a k +1 λ k +1 a k +2 } d 2 c =  k +2 X i =1 ( − 1) i +1 a i λ i  a 1 λ 1 i ˇ · · · a k +1 λ k +1 a k +2  dc + k +2 X i,j =1 i j . Similarly , b y (1.4) the second term in the RHS of (1.10) is  k +2 X i,j,h =1 i t . Hence, the second term in the RHS of (1.32) b ecomes h +1 X i,j =1 i h + j , the pro of is similar to that of Prop osition 3.4. Thus we only hav e to consider the case k = h + j . R ecalling (3.40) and (3.41 ), w e ha ve ι y ( ι x c ) = R ψ µ  ( χ h φ ) µ  { a 1 λ 1 · · · a h − 1 λ h − 1 a h λ h b 1 µ 1 · · · b j − 1 µ j − 1 b j } c  . Applying the ske w -symmetry condition B3. for c and using the defi nition (3.34) of χ h , w e get, after int egration by parts, that the RHS is ( − 1) hj R ( χ j ψ ) µ  φ µ  { b 1 µ 1 · · · b j − 1 µ j − 1 b j µ j a 1 λ 1 · · · a h − 1 λ h − 1 a h } c  , whic h is the same as ( − 1) hj ι x ( ι y c ). 35 F or example, for m ∈ C 0 = { m ′ ∈ M | ∂ m ′ = 0 } , we hav e { a 1 λ 1 · · · a k − 1 λ k − 1 a k } ι m c = m { a 1 λ 1 · · · a k − 1 λ k − 1 a k } c . Recall also that C 1 = A ⊗ M /∂ ( A ⊗ M ). The con traction operators asso ciated to 1-c hains are giv en b y the follo wing form u las: if c ∈ C 1 = Hom F [ ∂ ] ( A, M ), then (3.46) ι a ⊗ m c = Z mc ( a ) , while if c ∈ C k , with k ≥ 2, then (3.47) { a 2 λ 2 · · · a k − 1 λ k − 1 a k } ι a 1 ⊗ m c = { a 1 ∂ M a 2 λ 2 · · · a k − 1 λ k − 1 a k } c → m , where the arro w in the RHS m eans, as usual, that ∂ M should b e mov ed to the righ t. Also w e ha ve the follo wing form u las for the Lie deriv ativ e L x = [ d, ι x ] by a 1-c hain x ∈ C 1 acting on C 0 = M /∂ M M and C 1 = Hom F [ ∂ ] ( A, M ): L a ⊗ m R n = R ( a ∂ M n ) → m , (3.48) ( L a ⊗ m c )( b ) =  a ∂ M c ( b )  → m + ←  ( b − ∂ M m ) c ( a )  − c  [ a ∂ M b ]  → m , where the left arr o w in the RHS means, as usual, that ∂ M should b e mov ed to the left. The definitions of the cont raction op erators asso ciated to elemen ts of Γ • and C • are “com- patible”. Th is is stated in the follo w in g: Theorem 3.15. F or x ∈ C h and γ ∈ Γ k , with k ≥ h , we have ι x ( ψ k ( γ )) = ψ k − h ( ι χ h ( x ) ( γ )) , wher e ψ k : Γ k ֒ → C k , denotes the inje ctive line ar map define d in The or e m 1.5, and χ h : C h → Γ h , denotes the line ar map define d in Pr op osition 3.11 . In other wor ds, ther e is a c omm utative diagr am of line ar maps: (3.49) C k ι x / / C k − h Γ k ?  ψ k O O ι ξ / / Γ k − h ?  ψ k − h O O , pr ovide d tha t ξ ∈ Γ h and x ∈ C h ar e r elate d by ξ = χ h ( x ) . Pr o of. Let e γ ∈ e Γ k b e a r epresen tativ e of γ ∈ Γ k , and let a 1 ⊗ · · · ⊗ a h ⊗ φ ∈ A ⊗ h ⊗ Hom( F [ λ 1 , . . . , λ h − 1 ] , M ) b e a representat iv e of x ∈ C h . Recalli ng the definition (1.13) of ψ k and the d efi nition (3.40) of ι x , we hav e (3.50) { a h +1 λ h +1 · · · a k − 1 λ k − 1 a k } ι x ψ k ( e γ ) = ( χ h φ ) µ  e γ λ 1 , ··· ,λ k − 1 ,λ † k ( a 1 , · · · , a k )  , where, in the RHS , λ † k stands for − P k − 1 j =1 λ j − ∂ M , with ∂ M acting on the argum en t of ( χ h φ ) µ . By Lemmas 3.6 and (3.10)(c), we can replace λ † k b y − P k − 1 j = h +1 λ j − ∂ M , wh ere no w ∂ M is mo v ed to the left of ( χ h φ ) µ . Hence, the RHS of (3.50) is the same as ( χ h φ ) µ  e γ λ 1 ,λ 2 , ··· ,λ k ( a 1 , · · · , a k )    λ k 7→ λ † k = { a h +1 λ h +1 · · · a k − 1 λ k − 1 a k } ψ k − h ( ι χ h ( x ) ( e γ )) , th u s completing the pro of of the theorem. 36 3.7 Lie conformal algeroids. A Lie conformal algebroid is an analogue of a Lie algebroid. Definition 3.16 . A Lie c onformal algebr oid is a pair ( A, M ), where A is a L ie conformal algebra, M is a comm utativ e asso ciativ e different ial algebra with deriv ativ e ∂ M , suc h that A is a left M -mo d ule an d M is a left A -mo dule, satisfying the follo w ing compatibilit y conditions ( a, b ∈ A, m, n ∈ M ): L1. ∂ ( ma ) = ( ∂ M m ) a + m ( ∂ a ), L2. a λ ( mn ) = ( a λ m ) n + m ( a λ n ), L3. [ a λ mb ] = ( a λ m ) b + m [ a λ b ]. It follo ws from condition L3. an d sk ew-symmetry (0.2) of the λ -br ac k et, that L3’. [ ma λ b ] =  e ∂ M ∂ λ m  [ a λ b ] + ( a λ + ∂ m ) → b , where the first term in the RHS is P ∞ i =0 1 i !  ( λ + ∂ M ) i m  ( a ( i ) b ), and in the second term the arro w means, as usual, that ∂ should b e mo ved to the right, acting on b . W e next giv e t w o examples analogous to th ose in the Lie algebroid case. Let M b e, as ab o ve, a comm utativ e a sso ciativ e d ifferen tial algebra. Recall from Section 2 that a conformal endomorphism on M is an F -linear map ϕ (= ϕ λ ) : M → F [ λ ] ⊗ M satisfying ϕ λ ( ∂ M m ) = ( ∂ M + λ ) ϕ λ ( m ). The space Cen d( M ) of conformal endomorphism is then a Lie conformal algebra with the F [ ∂ ]-mo du le structure giv en b y ( ∂ ϕ ) λ = − λϕ λ , and the λ -brac ket giv en by [ ϕ λ ψ ] µ = ϕ λ ◦ ψ µ − λ − ψ µ − λ ◦ ϕ λ . Example 3.17 . Let Cder( M ) b e the su balgebra of the Lie conformal algebra C en d( M ) consisting of all conformal deriv ations on M , n amely of the the conformal endomorph isms satisfying the Leibniz rule: ϕ λ ( mn ) = ϕ λ ( m ) n + mϕ λ ( n ). Then the pair (Cder( M ) , M ) is a Lie conformal algebroid, where M carries the tautologic al Cder( M )-mo du le structure, and Cd er( M ) carr ies the follo win g M -mo du le structure: (3.51) ( mϕ ) λ =  e ∂ M ∂ λ m ) ϕ λ . This is ind eed an M -mo d ule, since e x∂ M ( mn ) = ( e x∂ M m )( e x∂ M n ). F ur thermore, condition L1. h olds thanks to the obvious identit y e ∂ M ∂ λ λ = ( λ + ∂ M ) e ∂ M ∂ λ . C ondition L2. holds b y definition. Finally , for condition L3. we ha v e [ ϕ λ mψ ] µ ( n ) = ϕ λ  ( mψ ) µ − λ ( n )  − ( mψ ) µ − λ  ϕ λ ( n )  = ϕ λ  e ∂ M ∂ µ m  ψ µ − λ ( n )  −  e ∂ M ∂ µ m  ψ µ − λ  ϕ λ ( n )  =  e ( λ + ∂ M ) ∂ µ ϕ λ ( m )  ψ µ − λ ( n ) +  e ∂ M ∂ µ m   ϕ λ  ψ µ − λ ( n )  − ψ µ − λ  ϕ λ ( n )   =  e ∂ M ∂ µ ϕ λ ( m )  ψ µ ( n ) +  e ∂ M ∂ µ m  [ ϕ λ ψ ] µ ( n ) =  ϕ λ ( m ) ψ + m [ ϕ λ ψ ]  µ ( n ) . Example 3.18 . Assume, as in S ection 3.3, th at A is a Lie conformal algebra and M is an A -mo dule endo wed with a commutat iv e, asso ciativ e pr o d uct, such that ∂ M : M → M , and 37 a λ : M → C [ λ ] ⊗ M , f or a ∈ A , s atisfy the Leibniz rule. The sp ace M ⊗ A has a natur al structure of F [ ∂ ]-mo dule, w h ere ∂ a cts as (3.52) e ∂ ( m ⊗ a ) = ( ∂ M m ) ⊗ a + m ⊗ ( ∂ a ) . Clearly , M ⊗ A is a left M -mo d ule via multiplic ation on th e first factor. W e define a left λ -action of M ⊗ A on M b y (3.53) ( m ⊗ a ) λ n =  e ∂ M ∂ λ m  ( a λ n ) , and a λ -brack et on M ⊗ A b y (3.54)  ( m ⊗ a ) λ ( n ⊗ b )  =  e ∂ M ∂ λ m  n  ⊗ [ a λ b ] +  ( m ⊗ a ) λ n  ⊗ b − e e ∂ ∂ λ  ( n ⊗ b ) − λ m ⊗ a  . W e claim that (3.52) and (3.54) make M ⊗ A a Lie conformal algebra, (3.53) make s M an M ⊗ A -mo dule, and the pair ( M ⊗ A, M ) is a Lie conformal alge broid. This will b e prov ed in Prop osition 3.21, us in g Lemm as 3.19 and 3.20. Lemma 3.19. (a) The fol lowing λ -br acket defines a Lie c onfo rmal al gebr a st ructur e on the C [ ∂ ] -mo dule M ⊗ A : (3.55)  ( m ⊗ a ) λ ( n ⊗ b )  0 =  e ∂ M ∂ λ m  n  ⊗ [ a λ b ] . (b) F or x, y ∈ M ⊗ A and m ∈ M , we have (3.56) [ mx λ y ] 0 =  e ∂ M ∂ λ m  [ x λ y ] 0 , [ x λ my ] 0 = m [ x λ y ] 0 . Pr o of. F or the first sesquilinearit y condition, we ha v e  e ∂ ( m ⊗ a ) λ ( n ⊗ b )  0 =  e ∂ M ∂ λ ∂ M m  n  ⊗ [ a λ b ] −  e ∂ M ∂ λ m  n  ⊗ λ [ a λ b ] = − λ  ( m ⊗ a ) λ ( n ⊗ b )  0 . The second sesquilinearity condition and sk ew-symm etry ca n b e pro ved in a similar w a y , and they are left to the reader. Let us c h ec k the Jacobi identit y . W e hav e,  ( m ⊗ a ) λ  ( n ⊗ b ) µ ( p ⊗ c )  0  0 =  e ∂ M ∂ λ m  e ∂ M ∂ µ n  p ⊗ [ a λ [ b µ c ]] . Exc hanging a ⊗ m with b ⊗ n and λ with µ , we get  ( n ⊗ b ) µ  ( m ⊗ a ) λ ( p ⊗ c )  0  0 =  e ∂ M ∂ λ m  e ∂ M ∂ µ n  p ⊗ [ b µ [ a λ c ]] . F urthermore, we ha v e [[ m ⊗ a λ n ⊗ b ] 0 ν p ⊗ c ] 0 =  e ∂ M ∂ ν  e ∂ M ∂ λ m  n  p ⊗ [[ a λ b ] ν c ] . Putting ν = λ + µ , the RHS b ecomes  e ∂ M ∂ λ m  e ∂ M ∂ µ n  p ⊗ [[ a λ b ] λ + µ c ] . Hence, the Jacobi identit y for the λ -brac k et (3.55) follo ws immediately from the Jacobi id en tit y for the λ -brack et on A . This prov es part (a). P art (b) is immed iate. 38 W e defi ne another λ -pro duct on M ⊗ A : (3.57) ( m ⊗ a ) λ ( n ⊗ b ) =  ( m ⊗ a ) λ n  ⊗ b . Notice that the λ -brac k et (3.54) can b e nicely written in terms of the λ -brac k et (3.55 ) and the λ -pro duct (3.57): (3.58) [ x λ y ] = [ x λ y ] 0 + x λ y − y − λ − e ∂ x . Lemma 3.20. (a) The λ -pr o duct (3.57) satisfies b oth sesquiline arity c onditions ( x, y ∈ M ⊗ A ): (3.59) ( e ∂ x ) λ y = − λ x λ y , x λ ( e ∂ y ) = ( λ + e ∂ )( x λ y ) . (b) F or x ∈ M ⊗ A, m ∈ M and y either in M ⊗ A or in M , we have (3.60) ( mx ) λ y =  e ∂ M ∂ λ m  x λ y , x λ ( my ) = ( x λ m ) y + m ( x λ y ) . (c) We have the fol lowing identity for x, y , z ∈ M ⊗ A : (3.61) x λ [ y µ z ] 0 = [( x λ y ) λ + µ z ] 0 + [ y µ ( x λ z )] 0 . (d) We have the fol lowing identity for x, y ∈ M ⊗ A and z either in M or in M ⊗ A : (3.62) x λ ( y µ z ) − y µ ( x λ z ) = [ x λ y ] λ + µ z . Pr o of. W e h a ve ( e ∂ ( m ⊗ a )) λ ( n ⊗ b ) =  e ∂ M ∂ λ ( ∂ M − λ ) m  ( a λ n ) ⊗ b . The first sesqu ilinearit y condition follo w s from the obvio us iden tit y e ∂ M ∂ λ ( ∂ M − λ ) = − λe ∂ M ∂ λ . The second sesqu ilinearit y cond ition can b e p ro ve d in a s im ilar w ay . T his p ro ves p art (a). Part (b) is immediate. F or part (c) and (d), let x = a ⊗ m, y = b ⊗ n, z = c ⊗ p ∈ A ⊗ M . W e ha v e (3.63) x λ [ y µ z ] 0 =  e ∂ M ∂ λ m  a λ  e ∂ M ∂ µ n  p  ⊗ [ b µ c ] , Similarly , (3.64) [( x λ y ) ν z ] 0 =  e ∂ M ∂ ν  e ∂ M ∂ λ m  ( a λ n )  p ⊗ [ b ν c ] . Hence, if we put ν = λ + µ , the RHS b ecomes (3.65)  e ∂ M ∂ λ m  e ∂ M ∂ µ ( a λ n )  p ⊗ [ b λ + µ c ] =  e ∂ M ∂ λ m  a λ  e ∂ M ∂ µ n  p ⊗ [ b µ c ] , where we used the sesquilinearit y of the λ -brac ket on A . F ur thermore, we hav e (3.66) [ y µ ( x λ z )] 0 =  e ∂ M ∂ µ n  e ∂ M ∂ λ m  ( a λ p ) ⊗ [ b µ c ] . 39 Com b ining equations (3.63), (3.65) and (3.66), we immediately get (3.61), thanks to the as- sumption that the λ -action of A on M is a der iv ation of the comm utativ e a sso ciativ e pro du ct on M . W e are left to prov e part (d). W e ha v e x λ ( y µ p ) =  e ∂ M ∂ λ m  a λ   e ∂ M ∂ µ n  ( b µ p )  =  e ∂ M ∂ λ m  e ∂ M ∂ µ n  a λ ( b µ p ) +  e ∂ M ∂ λ m  e ∂ M ∂ µ ( a λ n )  ( b λ + µ p ) . (3.67) F or the second equalit y , w e used the Leibniz rule and the sesquilinearit y condition for the λ -action of A on M . Exchanging x with y and λ w ith µ , w e hav e (3.68) y µ ( x λ p ) =  e ∂ M ∂ λ m  e ∂ M ∂ µ n  b µ ( a λ p ) +  e ∂ M ∂ µ n  e ∂ M ∂ λ ( b µ m )  ( a λ + µ p ) . By similar computations, w e get (3.69) ( x λ y ) λ + µ z =  e ∂ M ∂ λ m  e ∂ M ∂ µ ( a λ n )  ( b λ + µ p ) , and (3.70) ( y − λ − ∂ x ) λ + µ p =  e ∂ M ∂ µ n  e ∂ M ∂ λ ( b µ m )  ( a λ + µ p ) . Finally , it f ollo ws by a straigh tforwa rd computation that (3.71) [ x λ y ] 0 λ + µ z =  e ∂ M ∂ λ m  e ∂ M ∂ µ n  [ a λ b ] λ + µ p . Equation (3.62) is obtained combining equations (3.67), (3.68) , (3.69), (3.70) and (3.71). Prop osition 3.21. (a) The λ -br acket (3.54) defines a Li e c onformal algebr a structur e on the F [ ∂ ] -mo dule M ⊗ A . (b) The λ -action (3.53) defines a structur e of a M ⊗ A -mo dule on M . (c) The p air ( M ⊗ A, M ) is a Lie c onformal algebr oid. (d) We have a homomorp hism of Li e c onformal algebr oids ( M ⊗ A, M ) → (Cder( M ) , M ) , given by the identity map on M and th e fol lowing Lie c onformal a lgebr a homomorp hism fr om M ⊗ A to Cd er( M ) : m ⊗ a 7→  e ∂ M ∂ λ m  a λ . Pr o of. It immediately follo ws from Lemma 3.19 and L emma 3.20(a) that the λ -br ac k et (3.58 ) satisfies sesquilinearity and s k ew-symmetry . F u rthermore, the Jacobi id entit y for the λ -br ack et (3.54) follo ws f rom Lemma 3.19 and equations (3.61) and (3.62). This p ro ve s part (a). P art (b ) is Lemm a 3.5 9 (c), in the case z ∈ M . F or p art (c) we need to c hec k conditions L1., L2. and L3. The first t wo conditions are immediate. The last one follo w s from equations (3.56) and (3.60). Finally , part (d) is straigh tforwa rd and is left to the reader. 40 3.8 The Lie algera structure on Π C 1 and t he Π C 1 -structure on the complex ( C • , d ) . Recall that the space of 1-c hains of the complex ( C • , d ) is C 1 = ( A ⊗ M ) /∂ ( A ⊗ M ) with od d parit y . W e w ant to define a Lie alge b ra structure on Π C 1 , where, as usual, Π denotes parit y rev ersin g, making C • in to a Π C 1 -complex. By Prop osition 3. 21(a), we ha v e a Lie conformal algebra structur e on M ⊗ A . Hence, if w e iden tify M ⊗ A with A ⊗ M by exc hanging the tw o factors, w e get a stru ctur e of a Lie algebra on the quotien t s pace ( A ⊗ M ) /∂ ( A ⊗ M ), induced b y the λ -brac ket at λ = 0 [K]. Explicitly , we get the follo wing w ell-defin ed Lie algebra brac ket on Π C 1 = ( A ⊗ M ) /∂ ( A ⊗ M ): (3.72) [ a ⊗ m, b ⊗ n ] = [ a ∂ M 1 b ] → ⊗ mn + b ⊗  a ∂ M n  → m − a ⊗  b ∂ M m  → n , where in the RHS, as usual, the righ t arrow means that ∂ M should b e mo v ed to the right, and in the fir st s u mmand ∂ M 1 denotes ∂ M acting only on the fi rst factor m . Recall from Sectio n 3.4 that e Γ 1 = ( A ⊗ M [[ x ]])  ( ∂ ⊗ 1 + 1 ⊗ ∂ x )( A ⊗ M [[ x ]]), and Γ 1 =  ξ ∈ e Γ 1 | ∂ ξ = 0  , where th e acti on of ∂ on e Γ 1 is giv en b y (3.15). Under this id en tification, th e map χ 1 : C 1 → Γ 1 defined by (3.3 4) and (3.38) is giv en b y (3.73) χ 1 ( a ⊗ m ) = a ⊗ e x∂ M m . Prop osition 3.22. The map χ 1 : C 1 → Γ 1 is a Lie algebr a homomorph ism, which f actors thr ough a Lie algebr a isomorph ism χ 1 : ¯ C 1 → Γ 1 , pr ovide d tha t A de c omp oses as in (1.14) . Pr o of. W e h a ve , by (3.72) and (3.73) that χ 1  [ a ⊗ m, b ⊗ n ]  (3.74) = [ a ∂ M 1 b ] → ⊗  e x∂ M m  e x∂ M n  + b ⊗ e x∂ M  ( a ∂ M n ) → m  − a ⊗ e x∂ M  ( b ∂ M m ) → n  . Recalling formula (3.2 1 ) for th e Lie brac k et on e Γ 1 , we ha v e [ χ 1 ( a ⊗ m ) , χ 1 ( b ⊗ n )]  (3.75) = [ a ∂ x 1 b ] ⊗  e x 1 ∂ M m  e x∂ M n     x 1 = x + b ⊗  m ( x 1 ) , a λ 1 n ( x )  − a ⊗  n ( x 1 ) , b λ 1 m ( x )  . Clearly , the first term in th e RHS of (3.74) is the same as the first term in the RHS of (3.75). Recalling the definition (3.18) of the pairing h , i , and u sing the sesqu ilinearit y of th e λ -action of A on M , we hav e that the second term in the RHS of (3. 74) is the same as the second term in the RHS of (3.75), and sim ilarly for the third terms. The last statemen t follo ws from Prop osition 3.12. Prop osition 3.23. The c ohomoloy c omplex ( C • , d ) h as a Π C 1 -structur e ϕ : d Π C 1 → End C • , given by ϕ ( ∂ η ) = d, ϕ ( η x ) = ι x , ϕ ( x ) = L x = [ d, ι x ] . Mor e over, ( ¯ C • , d ) is a Π C 1 -sub c omplex. Pr o of. Due to Remark 3.1 and Prop osition 3.1 4, w e only need to c h eck t hat, for x, y ∈ Π C 1 , w e ha ve (3.76) [ L x , ι y ] = ι [ x,y ] . 41 This follo ws f rom a long b ut str aightforw ard computation, using the explicit form ulas (1.4) and (3.47) for the differentia l and the con traction op erators. It is left to th e reader. Notice though that, in the sp ecial case when A decomp oses as in (1.14) , equation (3.76) is a corollary of Prop osition 3.8, Theorem 1.5 and T heorem 3.15 for h = 1. Indeed, d ue to these results, it suffices to c heck that b oth sides of (3.76 ) coincide w hen acting on C 1 = Hom F [ ∂ ] ( A, M ). In the latter case, using equations (1.3), (1.5), (3.46), (3.47), (3.48) and (3.72), w e ha ve , for c ∈ C 1 , L a ⊗ m ( ι b ⊗ n c ) = R c ( b )  a ∂ M n  → m + R n  a ∂ M c ( b )  → m , ι b ⊗ n ( L a ⊗ m c ) = R n  a ∂ M c ( b )  → m + R c ( a )  b ∂ M m  → n − R nc  [ a ∂ M b ]  → m , ι [ a ⊗ m,b ⊗ n ] c = R nc  [ a ∂ M b ]  → m + R c ( b )  a ∂ M n  → m − R c ( a )  b ∂ M m  → n . It follo ws that (3.76) holds when app lied to elemen ts of C 1 . The ab o ve results imply the follo wing Theorem 3.24. The maps ψ • : Γ • → ¯ C • ⊂ C • and χ 1 : C 1 → Γ 1 define a homomorp hism of g - c omplexes. Pr ovide d that A de c om p oses as in (1.14) , we obtain an isomorphism of Π C 1 ≃ ΠΓ 1 - c omplexes ψ • : Γ • ∼ → ¯ C • . Pr o of. It follo w s fr om Theorem 1.5, Prop osition 3.12, Theorem 3.15 an d Prop osition 3.22. 3.9 P airings b et ween 1-c hains and 1-co c ha ins. Recall that e Γ 0 = M . Hence, the con- traction op erators of 1-c hains, restricted to the sp ace of 1-co c hains, d efi ne a natural p airin g e Γ 1 × e Γ 1 → M , whic h, to ξ ∈ e Γ 1 and e γ ∈ e Γ 1 , asso ciates (3.77) ι ξ e γ = φ µ ( e γ λ ( a )) ∈ M , where a ⊗ φ ∈ A ⊗ Hom( F [ λ ] , M ) is a represen tativ e of ξ . When w e consider the reduced spaces, w e hav e Γ 0 = M /∂ M , and th e ab o v e map induces a natural pairing Γ 1 × Γ 1 → M /∂ M , whic h, to ξ ∈ e Γ 1 and γ ∈ Γ 1 , asso ciate s (3.78) ι ξ γ = R φ µ ( e γ λ ( a )) ∈ M /∂ M , where again a ⊗ φ ∈ A ⊗ Hom( F [ λ ] , M ) is a representat iv e of ξ , and e γ ∈ e Γ 1 is a rep resen tativ e of γ . A similar pairing can b e defined for 1-c hains in C 1 and 1-coc hains in C 1 . Recall that C 0 = M /∂ M , C 1 is th e space of F [ ∂ ]-mo dule homomorphisms c : A → M , and C 1 = A ⊗ M /∂ ( A ⊗ M ). The corresp onding p airin g C 1 × C 1 → M /∂ M , is obtained as follo ws. T o x ∈ C 1 and c ∈ C 1 , w e associate, r ecalli ng (3.41), (3.79) ι x ( c ) = R m · c ( a ) ∈ M /∂ M , where a ⊗ m ∈ A ⊗ M is a representati v e of x . Recalling Theorems 1.5 and 3.15, the ab ov e p airings (3.78) an d (3 .79) are compatible in the sense that ι x ( c ) = ι ξ ( γ ), pro vided that γ ∈ Γ 1 and c ∈ C 1 are related by c = ψ 1 ( γ ), and ξ ∈ Γ 1 and x ∈ C 1 are related by ξ = χ 1 ( x ). 42 3.10 Con traction by a 1 -c hain as an o dd deriv at ion of e Γ • . Recall that, if the A -mo dule M h as a commutativ e asso ciativ e p ro duct, and ∂ M and a M λ are ev en deriv ations of it, then the basic cohomology complex e Γ • is a Z -graded comm utativ e asso ciativ e sup eralgebra with r esp ect to the exterior p r o duct (1.2 6 ), and th e differen tial δ is an o dd deriv ation of degree + 1. Prop osition 3.25. The c ontr action op er ator ι ξ , asso ciate d to a 1-chain ξ ∈ e Γ 1 , is an o dd derivation of the sup er algebr a e Γ • of de gr e e -1. Pr o of. Let a 1 ⊗ φ , with a 1 ∈ A and φ ∈ Hom ( F [ λ 1 ] , M ), b e a represen tativ e of ξ ∈ e Γ 1 . By the definition (1.26) of the exterior pro duct, we hav e ( ι ξ ( e α ∧ e β )) λ 2 , ··· ,λ h + k ( a 2 , · · · , a h + k ) = X σ ∈ S h + k sign( σ ) h ! k ! φ µ  e α λ σ (1) , ··· ,λ σ ( h ) ( a σ (1) , · · · , a σ ( h ) ) × e β λ σ ( h +1) , ··· ,λ σ ( h + k ) ( a σ ( h +1) , · · · , a σ ( h + k ) )  . (3.80) By the ske w -symmetry condition A2. for e α and e β , w e can r ewr ite the RHS of (3.80) as h X i =1 X σ | σ ( i )=1 sign( σ ) h ! k ! ( − 1) i +1 φ µ  e α λ 1 ,λ σ (1) , i ˇ ··· ,λ σ ( h ) ( a 1 , a σ (1) , i ˇ · · · , a σ ( h ) )  × × e β λ σ ( h +1) , ··· ,λ σ ( h + k ) ( a σ ( h +1) , · · · , a σ ( h + k ) ) (3.81) + h + k X i = h +1 X σ | σ ( i )=1 sign( σ ) h ! k ! ( − i ) i − h +1 e α λ σ (1) , ··· ,λ σ ( h ) ( a σ (1) , · · · , a σ ( h ) ) × × φ µ  e β λ 1 ,λ σ ( h +1) , i ˇ ··· ,λ σ ( h + k ) ( a 1 , a σ ( h +1) , i ˇ · · · , a σ ( h + k ) )  . By Lemma 1.7, the set of all p erm u tations σ ∈ S h + k suc h that σ ( i ) = 1, is naturally in bijection with the set of all p erm utations τ of { 2 , . . . , h + k } , and the corresp ondence b et ween the signs is sign( τ ) = ( − 1) i +1 sign( σ ). Hence, (3.8 1) can b e rewritten as X τ sign( τ ) h ! k !  h ( ι ξ e α ) λ τ (2) , ··· ,λ τ ( h ) ( a τ (2) , · · · , a τ ( h ) ) e β λ τ ( h +1) , ··· ,λ τ ( h + k ) ( a τ ( h +1) , · · · , a τ ( h + k ) ) + k ( − 1) h e α λ τ (2) , ··· ,λ τ ( h +1) ( a τ (2) , · · · , a τ ( h +1) )( ι ξ e β ) λ τ ( h +2) , ··· ,λ τ ( h + k ) ( a τ ( h +2) , · · · , a τ ( h + k ) )  = ( ι ξ ( e α ) ∧ e β ) λ 2 , ··· ,λ h + k ( a 2 , · · · , a h + k ) + ( − 1) h ( e α ∧ ι ξ ( e β )) λ 2 , ··· ,λ h + k ( a 2 , · · · , a h + k ) . R emark 3.26 . On e can sho w that the g -structur e of all our complexes e Γ • , Γ • and C • can b e extended to a structure of a calculus algebra, as defined in [DTT]. Namely , one can extend the Lie alge bra brac k et from the space of 1-c hains to the whole space of c hains (with rev erse parit y), and define there a commutati v e sup eralgebra structure, whic h extends our g -structure and satisfies all the p r op erties of a calculus algebra. 43 4 The complex of v ariational calculus as a Lie conformal algebra cohomology complex 4.1 Algebras of differen tiable functions. An algebr a of differ entiable functions V in the v ariables u i , indexed by a finite set I = { 1 , . . . , ℓ } , is, by d efinition, a different ial algebra (i.e. a unital comm utative associativ e algebra with a d eriv ation ∂ ), endo w ed with comm uting deriv ations ∂ ∂ u ( n ) i : V → V , for all i ∈ I and n ∈ Z + , such that, giv en f ∈ V , ∂ ∂ u ( n ) i f = 0 for all but finitely many i ∈ I and n ∈ Z + , and the follo w ing comm u tatio n rules with ∂ hold: (4.1) h ∂ ∂ u ( n ) i , ∂ i = ∂ ∂ u ( n − 1) i , where the R HS is considered to b e ze ro if n = 0. As in the previous sec tions, we denote by f 7→ R f the canonical qu otien t m ap V → V /∂ V . Denote by C ⊂ V the s ubspace of constan t functions, i.e. (4.2) C =  f ∈ V   ∂ f ∂ u ( n ) i = 0 ∀ i ∈ I , n ∈ Z +  . It follo ws from (4.1) by d o wnw ard indu ction that (4.3) Ker ( ∂ ) ⊂ C . Also, clearly , ∂ C ⊂ C . T yp ical examples of algebras of differen tiable fun ctions are: the ring of p olynomials (4.4) R ℓ = F [ u ( n ) i | i ∈ I , n ∈ Z + ] , where ∂ ( u ( n ) i ) = u ( n +1) i , an y lo calizat ion of it b y some m u ltiplicativ e sub set S ⊂ R , suc h as the whole fi eld of fractions Q = F ( u ( n ) i | i ∈ I , n ∈ Z + ), or any algebraic ext ension of the alge bra R or of the field Q obtained by adding a solution of certain p olynomial equation. In all these examples the action of ∂ : V → V is given by ∂ = X i ∈ I ,n ∈ Z + u ( n +1) i ∂ ∂ u ( n ) i . Another example of an algebra of differenti able fun ctions is the r in g R ℓ [ x ] = F [ x, u ( n ) i | i ∈ I , n ∈ Z + ], where ∂ = ∂ ∂ x + X i ∈ I ,n ∈ Z + u ( n +1) i ∂ ∂ u ( n ) i . The variational derivative δ δu : V → V ⊕ ℓ is defined by (4.5) δ f δ u i := X n ∈ Z + ( − ∂ ) n ∂ f ∂ u ( n ) i . It follo ws immediately from (4.1) th at (4.6) δ δ u i ( ∂ f ) = 0 , for ev ery i ∈ I and f ∈ V , namely , ∂ V ⊂ Ker δ δu . 44 A ve ctor field is, b y d efinition, a deriv ation of V of th e form (4.7) X = X i ∈ I ,n ∈ Z + P i,n ∂ ∂ u ( n ) i , P i,n ∈ V . W e let g b e the Lie algebra of all v ector fields. The su b algebra of evolutionary ve ctor fields is g ∂ ⊂ g , consisting of th e vec tor fields comm u ting w ith ∂ . By (4.1), a vec tor fi eld X is ev olutionary if and only if it h as the form (4.8) X P = X i ∈ I ,n ∈ Z + ( ∂ n P i ) ∂ ∂ u ( n ) i , wh ere P = ( P i ) i ∈ I ∈ V ℓ . 4.2 Normal algebras of differen t iable functions. Let V b e an a lgebra o f differen tiable functions in the v aria bles u i , i ∈ I = { 1 , . . . , ℓ } . F or i ∈ I and n ∈ Z + w e let (4.9) V n,i := n f ∈ V    ∂ f ∂ u ( m ) j = 0 if ( m, j ) > ( n, i ) in lexic ographic ord er o . W e also let V n, 0 = V n − 1 ,ℓ . A natural assump tion on V is to con tain elemen ts u ( n ) i , for i ∈ I , n ∈ Z + , such that (4.10) ∂ u ( n ) i ∂ u ( m ) j = δ ij δ mn . Clearly , such elemen ts are u niquely defin ed up to adding constan t fun ctions. Moreo v er, c ho osing these constan ts appr opriately , we can assume that ∂ u ( n ) i = u ( n +1) i . Thus, und er this assumption V is an algebra of differen tiable functions extension of the algebra R ℓ in (4.4). Lemma 4.1. L et V b e an algebr a of differ entiable fu nctions extension of the algebr a R ℓ . Then: (a) We have ∂ = ∂ R + ∂ ′ , wher e (4.11) ∂ R = X i ∈ I ,n ∈ Z + u ( n +1) i ∂ ∂ u ( n ) i , and ∂ ′ is a derivation of V which c ommutes with al l ∂ ∂ u ( n ) i and which vanishes on R ℓ ⊂ V . In p art icular, ∂ ′ V n,i ⊂ V n,i . (b) If f ∈ V n,i \V n,i − 1 , then ∂ f ∈ V n +1 ,i \V n +1 ,i − 1 , and i t has the form (4.12) ∂ f = X j ≤ i h j u ( n +1) j + r , wher e h j ∈ V n,i for al l j ≤ i, r ∈ V n,i , and h i 6 = 0 . (c) F or f ∈ V , R f g = 0 for every g ∈ V if and only if f = 0 . 45 Pr o of. Part (a) is clear. By part (a), we ha ve that ∂ f is as in (4.12), where h j = ∂ f ∂ u ( n ) j ∈ V n,i , and r = P j ∈ I ,m ≤ n u ( m ) j ∂ f ∂ u ( m − 1) j + ∂ ′ f ∈ V n,i . W e are left to pr o v e part (c). S upp ose f 6 = 0 is suc h t hat R f g = 0 f or ev ery g ∈ V . By taking g = 1, w e ha v e th at f ∈ ∂ V . Hence f has the form (4.12) for some i ∈ I and n ∈ Z + . But then u ( n +1) i f do es not ha ve this form, so that R u ( n +1) i f 6 = 0. Definition 4.2. T he algebra of d ifferen tiable functions V is called normal if ∂ ∂ u ( n ) i  V n,i  = V n,i for all i ∈ I , n ∈ Z + . Gi v en f ∈ V n,i , we d en ote by R du ( n ) i f ∈ V n,i a preimage of f un d er th e map ∂ ∂ u ( n ) i . Th is in tegral is d efined up to addin g element s from V n,i − 1 . Prop osition 4.3. Any normal algebr a of differ entiable functions V is an extension o f R ℓ . Pr o of. As pointed out a b o ve, w e need to fi n d elemen ts u ( n ) i ∈ V , for i ∈ I , n ∈ Z + , su c h that (4.10) holds. By the normalit y assump tion, there exists v n i ∈ V n,i suc h that ∂ v n i ∂ u ( n ) i = 1. Note that ∂ ∂ u ( n ) i ∂ v n i ∂ u ( n ) i − 1 = ∂ 1 ∂ u ( n ) i − 1 = 0, hence ∂ v n i ∂ u ( n ) i − 1 ∈ V n,i − 1 . If w e then replace v n i b y w n i = v n i − R du n i − 1 ∂ v n i ∂ u ( n ) i − 1 , w e ha ve that ∂ w n i ∂ u ( n ) i = 1 and ∂ w n i ∂ u ( n ) i − 1 = 0. Pro ceeding by do wnw ard induction, w e obtained the desired elemen t u ( n ) i . Clearly , the algebra R ℓ is norm al. Moreo ver, any extension V of R ℓ can b e fur ther ex- tended to a norm al algebra, by adding missing in tegrals. F or example, the lo calizatio n of R 1 = F [ u ( n ) | n ∈ Z + ] b y u is not a normal algebra, since it do esn’t con tain R du u . Note that an y differen tial algebra ( A, ∂ ) can b e view ed as a trivial algebra of d ifferen tiable functions with ∂ ∂ u ( n ) i = 0. S uc h an alg ebra do es not con tain R ℓ , hence it is not normal. 4.3 The complex of v ariational calculus. Let V b e an algebra of different iable fu nctions. The b asic de R ham c omplex e Ω • = e Ω • ( V ) is defined as the free comm utativ e sup eralgebra ov er V with o dd ge nerators δu ( n ) i , i ∈ I , n ∈ Z + . In other w ords e Ω • consists of fin ite sums of th e form (4.13) e ω = X i r ∈ I ,m r ∈ Z + f m 1 ··· m k i 1 ··· i k δ u ( m 1 ) i 1 ∧ · · · ∧ δ u ( m k ) i k , f m 1 ··· m k i 1 ··· i k ∈ V , and it has a (sup er)comm utativ e pr o duct giv en by the w edge pro duct ∧ . W e hav e a natur al Z + -grading e Ω • = L k ∈ Z + e Ω k defined b y saying th at elements in V ha v e degree 0, while the generators δ u ( n ) i ha ve degree 1. Hence e Ω k is a free mo d ule o ve r V with basis giv en by th e elemen ts δ u ( m 1 ) i 1 ∧ · · · ∧ δ u ( m k ) i k , with ( m 1 , i 1 ) > · · · > ( m k , i k ) (with resp ect to the lexico graphic order). In particular e Ω 0 = V and e Ω 1 = L i ∈ I ,n ∈ Z + V δu ( n ) i . Not ice that there is a natural V - linear pairing e Ω 1 × g → V defined on generators by  δ u ( m ) i , ∂ ∂ u ( n ) j  = δ i,j δ m,n , and extended to e Ω 1 × g by V -bilinearity . 46 W e le t δ b e an o dd deriv ation of degree 1 of e Ω • , su c h that δ f = P i ∈ I , n ∈ Z + ∂ f ∂ u ( n ) i δ u ( n ) i for f ∈ V , and δ ( δ u ( n ) i ) = 0. It is immediate to c hec k that δ 2 = 0 and that, for e ω ∈ e Ω k as in (4.13) , w e ha ve (4.14) δ ( e ω ) = X i r ∈ I ,m r ∈ Z + j ∈ I ,n ∈ Z + ∂ f m 1 ··· m k i 1 ··· i k ∂ u ( n ) i j δ u ( n ) j ∧ δ u ( m 1 ) i 1 ∧ · · · ∧ δ u ( m k ) i k . F or X ∈ g we define the c ontr action op er ator ι X : e Ω • → e Ω • , as an od d deriv ation of e Ω • of degree -1, suc h that ι X ( f ) = 0 for f ∈ V , and ι X ( δ u ( n ) i ) = X ( u ( n ) i ). If X ∈ g is as in (4.7) and e ω ∈ e Ω k is as in (4.13), we ha v e (4.15) ι X ( e ω ) = X i r ∈ I ,m r ∈ Z + k X q =1 ( − 1) q +1 f m 1 ··· m k i 1 ··· i k P i q ,m q δ u ( m 1 ) i 1 ∧ q ˇ · · · ∧ δ u ( m k ) i k . In particular, for f ∈ V we ha v e (4.16) ι X ( δ f ) = X ( f ) . It is easy to c hec k that the op er ators ι X , X ∈ g , form an a b elian (p u rely o dd) su balgebra of the Lie sup er algebra Der e Ω • , namely (4.17) [ ι X , ι Y ] = ι X ◦ ι Y + ι Y ◦ ι X = 0 . The Lie derivative L X along X ∈ g is defined as a d egree 0 deriv ation of the su p eralgebra e Ω • , comm u ting with δ , and suc h that (4.18) L X ( f ) = X ( f ) for f ∈ e Ω 0 . One easily c h ec k (on generators) Cartan’s formula (cf. (3.1)): (4.19) L X = [ δ, ι X ] = δ ◦ ι X + ι X ◦ δ . W e next pr o v e the follo wing: (4.20) [ ι X , L Y ] = ι X ◦ L Y − L Y ◦ ι X = ι [ X,Y ] . It is clear by d egree considerations that b oth sides of (4.20) act as zero on e Ω 0 = V . Moreo v er, it follo ws b y (4.16) that [ ι X , L Y ]( δ f ) = ι X δ ι Y δ f − ι Y δ ι X δ f = X ( Y ( f )) − Y ( X ( f )) = [ X, Y ]( f ) = ι [ X,Y ] ( δ f ) for ev ery f ∈ V . Equation (4.20) then follo ws b y the fact t hat b oth sides are ev en deriv ations of the wedge pro duct in e Ω. Finally , as immediate consequence of equation (4.20), w e get that (4.21) [ L X , L Y ] = L X ◦ L Y − L Y ◦ L X = L [ X,Y ] . Th us, e Ω • is a g -complex, b g acting on e Ω • b y deriv atio ns. 47 Note that the action of ∂ on V extends to a degree 0 deriv ation of e Ω • , suc h that (4.22) ∂ ( δ u ( n ) i ) = δ u ( n +1) i , i ∈ I , n ∈ Z + . This d eriv atio n comm utes with δ , hence w e can consider the corresp ond ing r e duc e d de R ham c omplex Ω • = Ω • ( V ), usually called the c omplex of variational c alculus : Ω • = M k ∈ Z + Ω k , Ω k = e Ω k /∂ e Ω k , with the in duced action of δ . With an ab u se of notation, we denote by δ and , for X ∈ g ∂ , b y ι X , L X , the maps indu ced on the quotien t space Ω k b y the corresp ondin g m aps on e Ω k . Ob viously , Ω • is a g ∂ -complex. 4.4 Isomorphism of the cohomology g ∂ -complexes Ω • and Γ • . Prop osition 4.4. L et V b e an algebr a of differ entiable fu nctions. Consider the Lie c onformal algebr a A = ⊕ i ∈ I F [ ∂ ] u i with the zer o λ - br acket. Th en V is a mo dule over the Lie c onformal algebr a A , with the λ -action given by (4.23) u i λ f = X n ∈ Z + λ n ∂ f ∂ u ( n ) i . Mor e over, the λ -action of A on V is by derivations of the asso ciative pr o duct in V . Pr o of. Th e fact th at V is an A -mo dule follo ws from the d efinition of an algebra of differentia ble functions. The second statemen t is clear as w ell. Let e Γ • = e Γ • ( A, V ) and Γ • = Γ • ( A, V ) b e the basic and redu ced Lie conformal algebra cohomology complexes for the A -mod ule V , defined in Prop osition 4.4. Thus, to ev ery algebra of differenti able fu n ctions V we can associate t wo app aren tly u nrelated t yp es of cohomology complexes: the basic and reduced de Rham cohomology complexes, e Ω • ( V ) and Ω • ( V ), defi n ed in Section 4.3, and the basic an d reduced Lie cofo rmal alge bra cohomolo gy complexes e Γ • ( A, V ) and Γ • ( A, V ), defined in Section 1.1, for the Lie conform al algebra A = L i ∈ I F [ ∂ ] u i , with the zero λ -brack et, acting on V , with the λ -action giv en by (4.23). W e are going to pr o v e that, in fact, these complexes are isomorphic, and all the related structur es (su c h as exterior pr o ducts, con traction op erators, Lie deriv ativ es,...) corresp ond via th is isomorphism. W e d enote, as in S ection 3.2, by e Γ • = e Γ • ( A, V ) (r esp . Γ • = Γ • ( A, V )) the b asic (resp. reduced) sp ace of c hains of A with co efficien ts in V . Recall from Secton 3.4 that Π e Γ 1 is iden tifi ed with the space ( A ⊗ V [[ x ]])  ( ∂ ⊗ 1 + 1 ⊗ ∂ x )( A ⊗ V [[ x ]]), and it carries a Lie algebra structure giv en by th e Lie brac k et (3.21) , whic h in this ca se tak es th e form, for i, j ∈ I and P ( x ) = P m ∈ Z + 1 m ! P m x m , Q ( x ) = P n ∈ Z + 1 n ! Q n x n ∈ V [[ x ]]: (4.24) [ u i ⊗ P ( x ) , u j ⊗ Q ( x )] = − u i ⊗ X n ∈ Z + Q n ∂ P ( x ) ∂ u ( n ) j + u j ⊗ X m ∈ Z + P m ∂ Q ( x ) ∂ u ( m ) i . Moreo v er , ∂ acts on e Γ 1 b y (3.15). Its kernel ΠΓ 1 consists of elemen ts of the form (4.25) X i ∈ I u i ⊗ e x∂ P i , wh ere P i ∈ V , 48 and it is a Lie subalgebra of Π e Γ 1 . W e also denote, as in S ection 4.1, by g the Lie algebra of all v ector fields (4.7) acting on V , and by g ∂ ⊂ g the Lie subalgebra of ev olutionary v ector fields (4.8). Prop osition 4.5. The map Φ 1 : Π e Γ 1 → g , which maps (4.26) ξ = X i ∈ I u i ⊗ P i ( x ) = X i ∈ I ,n ∈ Z + 1 n ! u i ⊗ P i,n x n ∈ e Γ 1 , to (4.27) Φ 1 ( ξ ) = X i ∈ I , n ∈ Z + P i,n ∂ ∂ u ( n ) i , is a Lie algebr a isomorphism. Mor e over, the image of the sp ac e of r e duc e d 1-chains via Φ 1 is the sp ac e of evolutionary ve ctor fields. Henc e we ha ve t he induc e d Lie algebr a isom orphism Φ 1 : ΠΓ 1 ∼ → g ∂ . Pr o of. Clearly , Φ 1 is a bijectiv e map, and, by (4.25), Φ 1 (Γ 1 ) = g ∂ . Hence w e only need to c hec k Φ 1 is a Lie algebra homomorphism. This is immediate from equation (4.24). Theorem 4.6. The m ap Φ • : e Γ • → e Ω • , su c h that Φ 0 = 1 I | V and, for k ≥ 1 , Φ k : e Γ k → e Ω k is given by (4.28) Φ k ( e γ ) = 1 k ! X i r ∈ I ,m r ∈ Z + f m 1 ··· m k i 1 ··· i k δ u ( m 1 ) i 1 ∧ · · · ∧ δ u ( m k ) i k , wher e f m 1 ··· m k i 1 ··· i k ∈ V i s the c o efficient of λ m 1 1 · · · λ m k k in e γ λ 1 ,...,λ k ( u i 1 , . . . , u i k ) , i s an isomorph ism of sup er algebr as, an d an isomorp hism of g -c omplexes, (onc e we identify the Lie algebr as g and Π e Γ 1 via Φ 1 , as in Pr op osition 4.5). Mor e over, Φ • c ommutes with the action of ∂ , henc e it induc es an i somorp hism of the c or- r esp on ding r e duc e d g ∂ -c omplexes: Φ • : Γ • ∼ → Ω • . Pr o of. Notice that since, b y assumption, I is a finite in dex set, the RHS of (4.28) is a fi n ite sum, so that Φ k ( e Γ k ) ⊂ e Ω k . By the sesquilinearit y and skew-symmetry conditions A1. and A2. in S ectio n 1.1, elements e γ ∈ e Γ k are uniquely determined by the collection of p olynomials e γ λ 1 , ··· ,λ k ( u i 1 , · · · , u i k ) = P m r ∈ Z + f m 1 ··· m k i 1 ··· i k λ m 1 1 · · · λ m k k , wh ic h are skew-symmetric with resp ect to simultaneous p erm utation of the v ariables λ r and the indices i r . W e wan t to c h eck that Φ k is a bijectiv e linear map from e Γ k to e Ω k . In fact, d en ote by Ψ k : e Ω k → e Γ k the linear map whic h to e ω as in (4.13) asso ciates the k -co c hain Ψ k ( e ω ), such that Ψ k ( e ω ) λ 1 ,...,λ k ( u i 1 , . . . , u i k ) = X m r ∈ Z + h f i m 1 ··· m k i 1 ··· i k λ m 1 1 · · · λ m k k , where h f i denotes the skew-symmetrizat ion of f : h f i m 1 ··· m k i 1 ··· i k = X σ sign( σ ) f m σ (1) ··· m σ ( k ) i σ (1) ··· i σ ( k ) , 49 and Ψ k ( e ω ) is extend ed to A ⊗ k b y the sesquilinearit y condition A1. It is straigh tforw ard to c heck that Ψ k ( e ω ) is indeed a k -co c h ain, and that the m aps Φ k and Ψ k are inv erse to eac h other. This prov es that Φ • is a bijectiv e map. Next, let us p ro ve that Φ • is an asso ciativ e sup eralgebra homomorphism. Let e α ∈ e Γ h , e β ∈ e Γ k and let α m 1 , ··· ,m h i 1 , ··· ,i h b e th e co efficien t of λ m 1 1 · · · λ m h h in e α λ 1 , ··· ,λ h ( u i 1 , · · · , u i h ), and let β n 1 , ··· ,n k j 1 , ··· ,j k b e the co efficien t of λ n 1 1 · · · λ n k k in e β λ 1 , ··· ,λ k ( u j 1 , · · · , u j k ). By (1.2 6), the co efficient of λ m 1 1 · · · λ m h + k h + k in ( e α ∧ e β ) λ 1 , ··· ,λ h + k ( u i 1 , · · · , u i h + k ) is X σ ∈ S h + k sign( σ ) h ! k ! α m σ (1) , ··· ,m σ ( h ) i σ (1) , ··· ,i σ ( h ) β m σ ( h +1) , ··· ,m σ ( h + k ) i σ ( h +1) , ··· ,i σ ( h + k ) . The ident it y Φ h + k ( e α ∧ e β ) = Φ h ( e α ) ∧ Φ k ( e β ) follo ws b y the defin ition (4.2 8 ) of Φ k . Let e γ ∈ e Γ k , and denote by f m 1 ··· m k i 1 ··· i k ∈ V the co efficien t of λ m 1 1 · · · λ m k k in e γ λ 1 ,...,λ k ( u i 1 , . . . , u i k ). W e wan t to pro v e that Φ k +1 ( δ e γ ) = δ Φ k ( e γ ). By assum ption, the λ -br ac k et on A is ze ro, and the λ -action of A on V is giv en b y (4.23). Hence, recalling (1.1), the co efficien t of λ m 1 1 · · · λ m k +1 k +1 in ( δ e γ ) λ 1 , ··· ,λ k +1 ( u i 1 , · · · , u i k +1 ) is k +1 X r =1 ( − 1) r +1 ∂ f m 1 r ˇ ··· m k +1 i 1 r ˇ ··· i k +1 ∂ u ( m r ) i r . It follo ws that Φ k +1 ( δ e γ ) = 1 ( k + 1)! X i r ∈ I ,m r ∈ Z + k +1 X q =1 ( − 1) q +1 ∂ f m 1 q ˇ ··· m k +1 i 1 q ˇ ··· i k +1 ∂ u ( m q ) i q δ u ( m 1 ) i 1 ∧ · · · ∧ δ u ( m k +1 ) i k +1 = 1 k ! X i r ∈ I ,m r ∈ Z + ∂ f m 1 ··· m k i 1 ··· i k ∂ u ( m 0 ) i 0 δ u ( m 0 ) i 0 ∧ · · · ∧ δ u ( m k ) i k = δ Φ k ( e γ ) , th u s pro vin g the claim. Similarly , the co efficien t of λ m 1 1 · · · λ m k k in ( ∂ e γ ) λ 1 , ··· ,λ k ( u i 1 , · · · , u i k ) is ∂ M f m 1 ··· m k i 1 ··· i k + P k r =1 f m 1 ··· m r − 1 ··· m k i 1 ··· i k , so that Φ k ( ∂ e γ ) = 1 k ! X i r ∈ I ,m r ∈ Z +  ∂ M f m 1 ··· m k i 1 ··· i k δ u ( m 1 ) i 1 ∧ · · · ∧ δ u ( m k ) i k + f m 1 ··· m k i 1 ··· i k k X q =1 δ u ( m 1 ) i 1 ∧ · · · ∧ δ u ( m q +1) i q ∧ · · · ∧ δ u ( m k ) i k  = ∂ Φ k ( e γ ) . This prov es that Φ • is compatible with the actio n of ∂ . Finally , w e pro ve that Φ • is compatible with the con traction op erators. Let e γ ∈ e Γ k b e as in the s tatement of the theorem, and let ξ ∈ e Γ 1 b e as in (4.26). By equation (3.19), w e ha ve the follo wing form u la for the con traction op erator ι ξ , ( ι ξ e γ ) λ 2 , ··· ,λ k ( u i 2 , · · · , u i k ) = X i 1 ∈ I  P i 1 ( x 1 ) , e γ λ 1 ,λ 2 , ··· ,λ k ( u i 1 , u i 2 , · · · , u i k )  , 50 where h , i denotes the con traction of x 1 with λ 1 defined in (3.18). Hence, th e coefficient of λ m 2 2 · · · λ m k k in ( ι ξ e γ ) λ 2 , ··· ,λ k ( u i 2 , · · · , u i k ) is X i 1 ∈ I ,m 1 ∈ Z + P i 1 ,m 1 f m 1 m 2 ··· m k i 1 i 2 ··· i k . It follo ws that Φ k − 1 ( ι ξ ( e γ )) = 1 ( k − 1)! X i r ∈ I ,m r ∈ Z + P i 1 ,m 1 f m 1 m 2 ··· m k i 1 i 2 ··· i k δ u ( m 2 ) i 2 ∧ · · · ∧ δ u ( m k ) i k , whic h , recalling (4.15) and (4.27), is th e same as ι Φ 1 ( ξ ) (Φ k ( e γ )). T his completes the pro of of th e theorem. 4.5 An explicit construction of the g ∂ -complex of v ariational calculus. Let V b e an algebra of differentia b le fun ctions in th e v ariables { u i } i ∈ I , let A = L i ∈ I F [ ∂ ] u i b e the free F [ ∂ ]-mo du le of rank ℓ , consid ered as a Lie conformal algebra with the zero λ -br ack et, and consider the A -mo dule structure on V , with the λ -action giv en by (4.23). By Theorem 4.6, the g ∂ -complex of v ariational calculus Ω • ( V ) is isomorphic to the ΠΓ 1 -complex Γ • ( A, V ). F urthermore, du e to T heorems 1.5 and 3.15, the ΠΓ 1 -complex Γ( A, V ) is isomorph ic to the Π C 1 -complex C • ( A, V ) = L k ∈ Z + C k , whic h is explicitly describ ed in S ectio ns 1.3 and 3.6. In this section we use this isomorphism to describ e explicitly the Π C 1 ≃ g ∂ -complex of v ariational calc ulus C • ( A, V ) ≃ Ω • ( V ), b oth in terms of “p oly-symb ols”, and in terms of skew- symmetric “p oly-differen tial op erators”. W e shall iden tify these tw o complexes via this isomor- phism. W e start b y d escrib ing all v ector spaces Ω k and the maps d : Ω k → Ω k +1 , k ∈ Z + . First, w e ha ve (4.29) Ω 0 = V /∂ V . Next, Ω 1 = Hom F [ ∂ ] ( A, V ), hence we ha ve a canonical iden tification (4.30) Ω 1 = V ⊕ ℓ . Comparing (1.3 ) and (4.23), we see that d : Ω 0 → Ω 1 is giv en b y the v ariatio nal d eriv ativ e: (4.31) d R f = δ f δ u . F or arbitrary k ≥ 1, the space Ω k can b e identi fied with the space of k - symb ols in u i , i ∈ I . By definition, a k -symb ol is a collection of expressions of the form (4.32)  u i 1 λ 1 u i 2 λ 2 · · · u i k − 1 λ k − 1 u i k  ∈ F [ λ 1 , . . . , λ k − 1 ] ⊗ V , where i 1 , . . . , i k ∈ I , satisfying the f ollo wing sk ew-symmetry prop ert y: (4.33)  u i 1 λ 1 u i 2 λ 2 · · · u i k − 1 λ k − 1 u i k  = sign( σ )  u i σ (1) λ σ (1) · · · u i σ ( k − 1) λ σ ( k − 1) u i σ ( k )  , 51 for eve ry p er mutation σ ∈ S k , where λ k is replaced, if it o ccurs in the RHS, by λ † k = − P k − 1 j =1 λ j − ∂ , with ∂ acting from the left. Clearly , by sesquilinearit y , f or k ≥ 1, the space Ω k = C k of k - λ -brac ke ts is one-to-o ne corresp ondence with the space of k -symb ols. F or example, the sp ace of 1-symbols is the same as V ⊕ ℓ . A 2-sym b ol is a collectio n of elemen ts  u i λ u j  ∈ F [ λ ] ⊗ V , f or i, j ∈ I , such that  u i λ u j  = −  u j − λ − ∂ u i  . A 3-sym b ol is a collectio n of elemen ts  u i λ u j µ u k  ∈ F [ λ, µ ] ⊗ V , for i, j, k ∈ I , suc h that  u i λ u j µ u k  = −  u j µ u i λ u k  = −  u i λ u k − λ − µ − ∂ u j  , and similarly for k > 3. Comparing (1.4) and (4.23) w e see that, if F ∈ V ⊕ ℓ , its differen tial dF corresp onds to t he follo wing 2-sym b ol: (4.34)  u i λ u j  = X n ∈ Z +  λ n ∂ F j ∂ u ( n ) i − ( − λ − ∂ ) n ∂ F i ∂ u ( n ) j  = ( D F ) j i ( λ ) − ( D ∗ F ) j i ( λ ) , where D F is the F rec het d eriv ativ e defin ed b y (0.9). More generally , the d ifferen tial of a k - sym b ol for k ≥ 1 is giv en b y the follo wing form ula: d  { u i 1 λ 1 · · · u i k − 1 λ k − 1 u i k }  i 1 ,...,i k ∈ I =  X n ∈ Z + k X s =1 ( − 1) s +1 λ n s ∂ ∂ u ( n ) i s  u i 1 λ 1 s ˇ · · · u i k λ k u i k +1  +( − 1) k X n ∈ Z +  − k X j =1 λ j − ∂  n ∂ ∂ u ( n ) i k +1  u i 1 λ 1 · · · u i k − 1 λ k − 1 u i k   i 1 ,...,i k +1 ∈ I . (4.35) Pro vid ed that V is an algebra of differen tiable fu n ctions extension of R ℓ , an equiv alen t language is that of sk ew-symmetric p oly-differen tial op erators. By defin ition, a k - differ ential op er ator is an F -linear map S : ( V ℓ ) k → V /∂ V , of the form (4.36) S ( P 1 , · · · , P k ) = Z X n 1 , ··· ,n k ∈ Z + i 1 , ··· ,i k ∈ I f n 1 , ··· ,n k i 1 , ··· ,i k ( ∂ n 1 P 1 i 1 ) · · · ( ∂ n k P k i k ) . The op erator S is calle d sk ew-symmetric if Z S ( P 1 , · · · , P k ) = sign( σ ) Z S ( P σ (1) , · · · , P σ ( k ) ) , for ev ery P 1 , · · · , P k ∈ V ℓ and ev ery p ermuta tion σ ∈ S k . Giv en a k -symb ol (4.37)  u i 1 λ 1 · · · u i k − 1 λ k − 1 u i k  = X n 1 ,...,n k − 1 ∈ Z + f n 1 , ··· ,n k − 1 i 1 , ··· ,i k − 1 ,i k λ n 1 1 · · · λ n k − 1 k − 1 , i 1 , . . . , i k ∈ I , where f n 1 , ··· ,n k − 1 i 1 , ··· ,i k ∈ V , w e asso ciate to it the follo wing p oly-differen tial op erator: S : ( V ℓ ) k → V /∂ V , is (4.38) S ( P 1 , · · · , P k ) = Z X n 1 , ··· ,n k − 1 ∈ Z + i 1 , ··· ,i k ∈ I f n 1 , ··· ,n k − 1 i 1 , ··· ,i k − 1 ,i k ( ∂ n 1 P 1 i 1 ) · · · ( ∂ n k − 1 P k − 1 i k − 1 ) P k i k . 52 Clearly , the ske w-symmetry p r op ert y of the k -sym b ol is translated to th e sk ew-sym m etry of the p oly-differentia l op erator. Conv ersely , in tegrating b y p arts, any k -differen tial op erator can b e written in the form (4.38). Thus w e hav e a surjectiv e m ap Ξ form the sp ace of k -sym b ols to the s pace of sk ew-symmetric k -differential op erators. Pro vided that V is an algebra of differen tiable fun ctions extension of R ℓ , by Lemma 4.1(c), the k -differential op erator S can b e written uniquely in the form (4.38). Hence, the map Ξ is an isomorphism. Note that th e space of 1-differentia l op erators S : V ℓ → V /∂ V can b e canonical ly identified with the sp ace Ω 1 = V ⊕ ℓ . Exp licitly , to th e 1-differen tial op erator S ( P ) = R P i ∈ I ,n ∈ Z + f n i ∂ n P i , w e associate: (4.39)  X n ∈ Z + ( − ∂ ) n f n i  i ∈ I ∈ V ⊕ ℓ . W e can write down the expression of the differential d : Ω k → Ω k +1 in terms of p oly- differen tial op erators. First, if F ∈ Ω 1 = V ⊕ ℓ , the 2-different ial op erator corresp onding to dF ∈ Ω 2 is obtained by looking at equation (4.34): (4.40) dF ( P , Q ) = Z X i ∈ I  Q i X P ( F i ) − P i X Q ( F i )  = Z X i,j ∈ I  Q i D F ( ∂ ) ij P j − P i D F ( ∂ ) ij Q j  , where X P denotes the evol utionary ve ctor field asso ciated to P ∈ V ℓ , defined in (4.8), and D F ( ∂ ) is the F rec het deriv ativ e (0.9). Next, if S : ( V ℓ ) k → V /∂ V is a sk ew-symmetric k -d ifferen tial op erator, its different ial dS , obtained by looking at (4.35), is the follo wing k + 1-differen tial op erator: (4.41) dS ( P 1 , · · · , P k +1 ) = k +1 X s =1 ( − 1) s +1  X P s S  ( P 1 , s ˇ · · · , P k +1 ) . In the ab o ve formula, if S is as in (4.36), X P S denotes the k -differen tial op erator obtained fr om S by replacing the co efficien ts f n 1 , ··· ,n k i 1 , ··· ,i k b y X P ( f n 1 , ··· ,n k i 1 , ··· ,i k ). R emark 4 .7 . F o r k ≥ 2, a k -differen tial op erator can also b e u ndersto o d as a map S : ( V ℓ ) k − 1 → V ⊕ ℓ of the follo wing form: (4.42) S ( P 1 , · · · , P k − 1 ) i k = X n 1 , ··· ,n k − 1 ∈ Z + i 1 , ··· ,i k − 1 ∈ I f n 1 , ··· ,n k − 1 i 1 , ··· ,i k − 1 ,i k ( ∂ n 1 P 1 i 1 ) · · · ( ∂ n k − 1 P k − 1 i k − 1 ) . This corresp onds to the k -symbol (4.37) in the ob vious wa y . With th is notation, the differentia l dS is the follo wing map ( V ℓ ) k → V ⊕ ℓ : dS ( P 1 , · · · , P k ) i = k X s =1 ( − 1) s +1 ( X P s S )( P 1 , s ˇ · · · , P k ) i (4.43) +( − 1) k X j ∈ I ,n ∈ Z + ( − ∂ ) n  P k j ∂ S ∂ u ( n ) i ( P 1 , · · · , P k − 1 ) j  . 53 Recall that the Lie algebra g ∂ ≃ Π C 1 is iden tifi ed with the space V ℓ via the map P 7→ X P , defined in (4.8). Giv en P ∈ V ℓ , we wa n t to describ e explicitly the action of the corresp ond ing con traction op erator ι P and the Lie d eriv ativ e L P = [ d, ι P ]. First, for F ∈ V ⊕ ℓ = Ω 1 , w e ha ve (cf. (3.46)): (4.44) ι P ( F ) = R X i ∈ I P i F i ∈ V /∂ V = Ω 0 . Next, the con traction of a k -symb ol for k ≥ 2 is giv en b y the follo win g formula (cf. (3.47)): (4.45) ι P   u i 1 λ 1 · · · u i k − 1 λ k − 1 u i k   i 1 ,...,i k ∈ I =  X i 1 ∈ I  u i 1 ∂ u i 2 λ 2 · · · u i k − 1 λ k − 1 u i k  → P i 1  i 2 ,...,i k ∈ I , where, as us ual, th e arrow in the RHS means that ∂ is mo ved to the right. F or k = 2, the ab o ve form u la b ecomes (4.46) ι P   u i λ u j   i,j ∈ I =  X j ∈ I  u j ∂ u i  → P j  i ∈ I ∈ V ⊕ ℓ = Ω 1 . W e can write th e ab ov e formulas in the language of p oly-differentia l op erators. F or a k - differen tial op erator S , we ha ve (4.47) ( ι P 1 S )( P 2 , · · · , P k ) = S ( P 1 , P 2 , · · · , P k ) . F or k = 2 ι P 1 S is a 1-differentia l op erator which, by (4.39), is the same as an elemen t of V ⊕ ℓ = Ω 1 . R emark 4.8 . In the inte rpretation (4.42) of a k -differen tial operator, the action of the con trac- tion op erator is giv en by ( ι P 1 S )( P 2 , · · · , P k − 1 ) i k = S ( P 1 , P 2 , · · · , P k − 1 ) i k . Next, we write the formula for the Lie deriv ativ e L Q : Ω k → Ω k , asso ciated to Q ∈ V ℓ ≃ g ∂ , using Cartan’s f ormula L Q = [ ι Q , d ]. Recalling (4.31) and (4.44), after int egration by parts w e obtain, for R f ∈ Ω 0 = V /∂ V : (4.48) L Q  R f  = R X Q ( f ) , where X Q is the evo lutionary v ector field co rresp ondin g to Q (cf. (4.8)). Similarly , recalling (4.34) and (4.46), we obtain, for F ∈ Ω 1 = V ⊕ ℓ : dι Q ( F ) = D F ( ∂ ) ∗ Q + D Q ( ∂ ) ∗ F , ι Q d ( F ) = D F ( ∂ ) Q − D F ( ∂ ) ∗ Q , where D F ( ∂ ) denotes the F rec het deriv ativ e (0.9), and D F ( ∂ ) ∗ is the adjoin t differen tial op er- ator. Pu tting the ab o ve form ulas together, w e get: (4.49) L Q F = D F ( ∂ ) Q + D Q ( ∂ ) ∗ F . 54 F or k ≥ 2, L Q acts on a k -symb ol in Ω k b y th e f ollo wing f ormula, whic h can be derive d from (4.35) and (4.45): L Q { u i 1 λ 1 · · · u i k − 1 λ k − 1 u i k } = X Q { u i 1 λ 1 · · · u i k − 1 λ k − 1 u i k } + k − 1 X s =1 ( − 1) s +1 X j ∈ I { u j λ s + ∂ u i 1 λ 1 s ˇ · · · u i k − 1 λ k − 1 u i k } → D Q ( λ s ) j i s +( − 1) k +1 X j ∈ I { u j λ † k + ∂ u i 1 λ 1 · · · u i k − 2 λ k − 2 u i k − 1 } → D Q ( λ † k ) j i k . In the RHS the ev olutionary v ector field X Q is applied to the co efficien ts of the k -sym b ol, in the last tw o terms the arrow means, as us u al, that w e mo ve ∂ to the righ t, D Q ( λ ) den otes the F rec het deriv ativ e (0.9) considered as a p olynomial in λ , and, in the last term, λ † k = − λ 1 − · · · − λ k − 1 − ∂ , where ∂ is mov ed to the left. This f orm ula tak es a m uc h nicer form in the language of k - differen tial op erators. Namely we ha ve: (4.50) ( L Q S )( P 1 , · · · , P k ) = ( X Q S )( P 1 , · · · , P k ) + k X s =1 S ( P 1 , · · · , X Q P s , · · · , P k ) . Here X Q S has th e same meaning as in equation (4.41). Th is form ula can b e obtai ned from the previous one by in tegration by parts. 4.6 An application to the classification of symplectic differen t ial op e ra tors. Recall that C ⊂ V d enotes the subspace (4. 2) of constan t functions. In [BDK] w e pro ve the follo wing: Theorem 4.9. If V is normal, then H k (Ω • , d ) = δ k , 0 C / ( C ∩ ∂ V ) . Recall that a symple ctic diffe r ential op er ator (cf. [D] and [BDK ]) is a sk ew-adjoin t differen- tial op erator S ( ∂ ) =  S i,j ( ∂ )  i,j ∈ I : V ℓ → V ⊕ ℓ , which is closed, namely the follo wing condition holds (cf. (4.43)): (4.51) u i λ S k j ( µ ) − u j µ S k i ( λ ) − u k − λ − µ − ∂ S j i ( λ ) = 0 , where the λ -action of u i on V is defined b y (4.23). W e ha v e the follo wing corollary of Theorem 4.9. Corollary 4.10. If V is a no rmal algebr a of differ entiable functions, then any symple ctic d if- fer ential op er at or is of the form: S F ( ∂ ) = D F ( ∂ ) − D F ( ∂ ) ∗ , for some F ∈ V ⊕ ℓ . Mor e over, S F = S G if and only if F − G = δf δu for some f ∈ V . A skew-symmetric k -differen tial op erator S : ( V ℓ ) k → V /∂ V is called symple ctic if it is closed, i.e. k +1 X s =1 ( − 1) s +1  X P s S  ( P 1 , s ˇ · · · , P k +1 ) = 0 . The follo wing corollary of Th eorem 4.9 is a generalization of Corollary 4.10 and u ses Pr op osition 4.3 55 Corollary 4.11. If V is a normal algebr a of differ entiable functions, then any symple ctic k - differ ential op e r ator, for k ≥ 1 , is of the form: S ( P 1 , · · · , P k ) = k X s =1 ( − 1) s +1  X P s T  ( P 1 , s ˇ · · · , P k ) , for some skew-symmetric k − 1 -differ ential op er ator T . Mor e over, T is define d up to a dding a symple ctic k − 1 -differ ential op er at or. R emark 4.12 . It follo ws from the pro of of Theorem 4.9 that, Corollaries 4.10 and 4.11 h old in an y algebra of differentiable f u nctions V , provided that we are allo we d to tak e F and T resp ectiv ely in an extension of V , obtained by addin g fin itely many integ rals of elemen ts of V (an in tegral of an elemen t f ∈ V n,i is a preimage R du ( n ) i f of ∂ ∂ u ( n ) i indep endent on u ( m ) j with ( m, j ) > ( n, i )). R emark 4.13 . The map Ξ defin ed in Section 4.5 ma y ha ve a non-zero k ern el if V is not an extension of the alge bra R ℓ , but, of course, for any V the image of Ξ is a g ∂ -complex. The 0-th term of this complex is V /∂ V and the k -th term, for k ≥ 1, is the space of skew-symmetric k -differen tial op erators S : ( V ℓ ) k → V /∂ V . R emark 4.14 . Throughou t this section we assumed that th e num b er ℓ of v ariables u i is fi nite, but th is assumption is not essentia l, and our arguments go through with minor mo difications. This is the r eason for distinguishing V ℓ from V ⊕ ℓ , in order to accommo date the case ℓ = ∞ . References [BKV] B. Bak alo v, V.G. Kac, and A.A. V orono v, Cohomolo gy of c onformal algebr as , C omm un . Math. Phys. 200 (199 9), 561–598. [BDK] A. Barak at, A. De Sole, and V.G. Kac, Conformal algebr as in the the ory of Hamiltonian e quations , in pr eparation. [D] I. Dorfman, Di r ac structur es a nd inte gr ability of non-line ar evolution e quations , John Wiley and sons, 1993. [Di] L.A. Dic k ey , Soliton e quations and Hamilto nian systems , Ad v anced Ser. Math. Ph ys. 26 (second edition), W ord Sci., 2003. [DTT] V. Dolgushev, D. T ama rkin, and B. Tsygan, F or mality of the homotopy c alculus algebr a of Ho chschild (c o)chains , preprint arXiv:0807.51 17 . [H] H. Helmholtz, Ub er der physikalische Be dentung des Princi ps der Klinstein Wirkung , J. Reine Angen Math 100 (188 7), 137 –166. [K] V.G. K ac, V ertex algebr as for b e ginners , Univ. Lecture Ser., v ol 10, AMS, 1996. Second edition, 1998. [Vi] A.M. Vinogrado v, On the algebr a-ge ometric foundations of L agr angian field the ory , So v. Math. Dokl. 18 (1977), 1200–120 4. [V] V. V olterra, L e¸ cons sur les F onctions de Lignes , Gauthier-Villar, P aris, 191 3. 56