Essential variational Poisson cohomology

ESSENTIAL V ARIA TIONAL POISSON COHO MO LOGY ALBER TO DE SOLE, VICTOR G. KA C Abstra ct. In our recen t paper [DSK 11] w e computed the dimension of the v ariational Poi sson cohomology H • K ( V ) for any quasiconstan t co- efficien t ℓ × ℓ matrix differential op erator K of order N with inv ertible leading co efficient, p ro vided that V is a normal algebra of differential functions o ver a linearly closed differential field. In the p resen t pap er w e show th at, for K skew adjoin t , th e Z -graded Lie sup eralgebra H • K ( V ) is isomorphic to the finite dimensional Lie sup eralgebra e H ( N ℓ, S ). W e also pro v e that t he subalgebra of “essentia l” v ariational P oiss on coho- mology , consisting of classes v anishing on the Casimirs of K , is zero. This vanis hing result has applications to the theory of bi-Hamiltonian structures and their deformations. At th e en d of t h e paper we consider also the translation inv ariant case. 1. Introduction The Z -graded L ie sup eralgebra W v ar (Π V ) = L ∞ k = − 1 W v ar k of variational p olyve ctor fields is a v ery con venien t framework for the theory of in tegrable Hamiltonian PDE’s. This Lie s u p eralgebra is asso ciated to an algebra of differen tial f u nctions V , which is an extension of the algebra of different ial p olynomials R ℓ = F [ u ( n ) i | i = 1 , . . . , ℓ ; n ∈ Z + ] o v er a differen tial field F with th e deriv ation ∂ extended to R ℓ b y ∂ u ( n ) i = u ( n +1) i . The fi rst three p ieces, W v ar k for k = − 1 , 0 , 1, are iden tified with the most imp orta n t ob jects in the theory of integrable systems: First, W v ar − 1 = Π( V /∂ V ), w h ere V /∂ V is the s pace of Hamiltonian functions (or local fu nc- tionals), and wh ere Π is ju st to remin d that it should b e considered as an o dd subs pace of W v ar (Π V ). Second, W v ar 0 is the Lie algebra of evolutionary ve ctor fields X P = ℓ X i =1 ∞ X n =0 ( ∂ n P i ) ∂ ∂ u ( n ) i , P ∈ V ℓ , whic h w e identify with V ℓ . Third, W v ar 1 is iden tified with the space of sk ew adj oin t ℓ × ℓ m atrix differen tial op erators o ver V end o w ed with odd parit y . Dipartimento di Matematica, Universi t` a di Roma “La Sapienza”, 00185 Roma, Italy . desole@mat.uniroma1.it Supp orted by a PR IN grant and F ondi Ateneo, from the Univer- sit y of Rome. Department of Mathematics, M.I.T., Cambridge, MA 02139, U S A. k ac@math.mit.edu Supp orted in part b y an NSF gran t. 1 2 ALBER TO DE SOLE, VICTOR G. KAC F or R f , R g ∈ W v ar − 1 , X , Y ∈ W v ar 0 , an d H = H ( ∂ ) ∈ W v ar 1 , the com- m utators are defined as f ollo ws (as u sual, R denotes the canonical map V → V /∂ V ): [ R f , R g ] = 0 , (1.1) [ X, R f ] = R X ( f ) , (1.2) [ X, Y ] = X Y − Y X , (1.3) [ H , R f ] = H ( ∂ ) δ f δ u , (1.4) [ X P , H ] = X P ( H ( ∂ )) − D P ( ∂ ) ◦ H ( ∂ ) − H ( ∂ ) ◦ D ∗ P ( ∂ ) . (1.5) Here δ δu is the v ariational deriv ativ e (see (3.4)), D P is the F r ec h et der iv ativ e (see (3.7)), and D ∗ ( ∂ ) d enotes th e matrix differential op erato r adjoint to D ( ∂ ). The form ula for the comm utator of t w o elemen ts K, H of W v ar 1 (the so called Sc h outen br ac ket) is more complicated (see (3.17), bu t one needs only to know that cond itions [ K, K ] = 0 , [ H , H ] = 0 means that th ese matrix differen tial operators are H amiltonian , and the condition [ K , H ] = 0 means that th ey are c omp atible . There h a v e b een v arious v arious versions of the notion of v ariatio nal p olyv ector fields, but [Kup80] is probably the earliest reference. The basic n otions of the theory of integrable Hamiltonian equations can b e easily describ ed in terms of the Lie sup eralg ebra W v ar (Π V ). Giv en a Hamiltonian op erator H and a Hamiltonian function R h ∈ V /∂ V , th e cor- resp ond in g H amiltonian e q u ation is (1.6) du dt = [ H, R h ] , u = ( u 1 , . . . , u ℓ ) . One s a ys that t wo Hamiltonian fun ctions R h 1 and R h 2 are in involution if (1.7) [[ H , R h 1 ] , R h 2 ] = 0 . (Note th at the LHS of (1.7) is skew asymmetric in R h 1 and R h 2 , since b oth are o d d elemen ts of the Lie sup eralgebra W v ar (Π V )). Any R h 1 whic h is in in volutio n with R h is called an inte gr al of motion of the Ha miltonian equation (1.6), and this equation is call ed inte gr able if there exists an infinite dimensional su bspace Ω of V /∂ V cont aining R h suc h that all elemen ts of Ω are in in v olution. I n this case w e obtain a hierarch y of compatible in tegrable Hamiltonian equations, lab eled b y elements ω ∈ Ω: du dt ω = [ H, ω ] . The basic device f or p ro ving integrabilit y of a Hamiltonian equation is the so calle d L enar d- M agri scheme , prop osed by Lenard in early 1970’s (unpub lished), w ith an imp ortan t imput by Magri [Mag 78]. A surve y of related results u p to early 1990’s can b e found in [Dor93], and a discuss ion in terms of Poisson v ertex algebras can b e found in [BDSK09]. ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 3 The Lenard-Magri sc h eme requires t w o compatible Hamiltonian op erators H and K and a sequence of Hamiltonian functions R h n , n ∈ Z + , suc h that (1.8) [ H , R h n ] = [ K , R h n +1 ] , n ∈ Z + . Then it is a trivial exercise in Lie sup eralge bra to sho w that all Hamiltonian functions R h n are in inv olution (hin t: use the parenthetic al r emark after (1.7)). Note to solv e exercise one only uses the fact that K, H lie in W v ar 1 , but in order to construct th e sequence R h n , n ∈ Z + , one needs the Hamiltonian prop erty of H and K and their compatibilit y . The appropr iate language here is the cohomolog ical one. S ince [ K, K ] = 0 and K is an (o dd) elemen t of W v ar 1 , it follo ws that we ha v e a cohomology complex  W v ar (Π V ) = M k ≥− 1 W v ar k , ad K  , called the v ariational P oisson cohomol ogy complex. As usu al, let Z • K ( V ) = L k ≥− 1 Z − K b e the subalgebra of closed elemen ts (= Ker(ad K )), and let B • K ( V ) = L k ≥− 1 B − K b e its ideal of exact elemen ts (= Im(ad K )). Then the variational Poisson c oho molo gy H • K ( V ) = Z • K ( V )  B • K ( V ) = M k ≥− 1 H k K , is a Z -graded Lie sup eralg ebra. (F or usual p olyve ctor fields the corresp ond- ing Poisso n cohomolog y was in trod uced in [Lic77]; cf. [DSK11]). No w we can try to fin d a solution to (1.8) by induction on n as follo ws (see [Kra88] and [Olv87 ]). Since [ K, H ] = 0, w e h a ve, b y the Jacobi iden tit y: (1.9) [ K, [ H , R h n ]] = − [ H, [ K, R h n ]] , hence, b y the inductive assumption, the RHS of (1.9) is − [ H , [ H , R h n +1 ]], whic h is zero since [ H , H ] = 0 and H is o dd. Thus, [ H , R h n ] ∈ Z 0 K . T o complete the n -th step of induction w e need that this elemen t is exact, i.e. it equ als [ H , R h n +1 ] for some R h n +1 . But in general w e h av e (1.10) [ H , R h n ] = [ K, R h n +1 ] + z n +1 , where z n +1 ∈ Z 0 K only dep ends on the cohomology class in H 0 K . The b est place to start the Lenard-Magri scheme is to tak e R h 0 = C 0 Z − 1 K , a c entr al element for K . Then the fir s t step of the Lenard-Magri sc heme requires the existence of R h 1 suc h that (1.11) [ H , C 0 ] = [ K , R h 1 ] . T aking brac k et of b oth sides of (1.11) with arbitrary C 1 ∈ Z − 1 K , w e obtain (1.12) [[ H , C 0 ] , C 1 ] = 0 . Th us, if w e wish the Lenard-Magri sc heme to w ork starting with an arbitrary cen tral elemen t C 0 for K , the Hamiltonian op erator H (whic h lies in Z 1 K ), m ust satisfy (1 .12) for an y C 0 , C 1 ∈ Z − 1 K . In other w ords, H m ust b e “essen tial ly closed”. 4 ALBER TO DE SOLE, VICTOR G. KAC It wa s remark ed in [DMS05] that condition (1.12) is an obstru ction to trivialit y of deformations of th e Hamiltonian op erator K , whic h is, of course, another imp ortan t reason to b e in terested in “essen tial” v ariational P oisson cohomology . W e d efine the sub algebra E Z • K ( V ) = L k ≥− 1 E Z k K ⊂ Z • K ( V ) of essential ly close d elements , by induction on k ≥ − 1, as follo w s : E Z − 1 K = 0 , E Z k K =  z ∈ Z k K   [ z , Z − 1 K ] ⊂ E Z k − 1  , k ∈ Z + . It is immediate to see that exa ct ele men ts are essent ially closed, and we define the essential variational Poisson c ohomolo gy as E H • K ( V ) = E Z • K ( V )  B • K ( V ) . The fi rst main result of the present pap er is T heorem 4.3, whic h asserts that E H • K ( V ) = 0, p ro vided that K is an ℓ × ℓ matrix differen tial op erator of order N with co efficien ts in Mat ℓ × ℓ ( F ) and in v ertible leading co efficien t, that th e d ifferen tial fi eld F is linearly closed, and th at the algebra of differen- tial fun ctions V is normal. Recall that a differen tial fi eld F is called line arly close d [DSK11] if an y linear in homogenous (resp ectiv ely homogenous) dif- feren tial equation of order greater than or equal to 1 w ith coefficient s in F has a solution (resp . n on zero solution) in F . The pro of of Theorem 4.3 relies on our previous pap er [DSK 11], wh ere, under the same assump tions on K , F and V , we p ro v e that d im C ( H k K ) =  N ℓ k +2  , w here C ⊂ F is the sub field of constan ts, and we constru cted exp licit represent ativ es of cohomolog y classes. In turn, Theorem 4.3 allo ws us to compute the Lie sup eralgebra structure of H • K ( V ), whic h is our second main result. Namely , Th eorem 3.6 asserts that the Z -graded Lie s u p eralgebra H • K ( V ) is isomorphic to the finite di- mensional Z -graded Lie sup eralgebra e H ( N ℓ, S ), of Hamiltonian vecto r fields o v er the Grassman su p eralgebra in N ℓ indetermin ates { ξ i } N ℓ i =1 , with P oisson brac k et { ξ i , ξ j } = s ij , d ivided by the cen tral ideal C 1, wh ere S = ( s ij ) is a nondegenerate s ymmetric N ℓ × N ℓ matrix o ver C . W e hop e that Theorem 4.3 will allo w further p rogress in the study of the L enard-Magri sc heme (wo rk in progress). First, it leads to classificatio n of Hamilto nian op erat ors H compatible to K , us in g tec hniques an d results from [DSKW10]. Second, it sho ws that if the elemen ts z n +1 in (1.10) are essen tially closed, then th ey can b e remov ed. Also, of course, Th eorem 4.3 sho ws that, if (1.12) h olds for a Hamil- tonian op er ator obtained b y a formal d eformation of K , then this formal deformation is trivial. In conclusion of the pap er w e discuss the other “extreme” – the translation in v arian t case – when F = C . In this case, we giv e an upp er b ound for the dimension of H k K , f or an arbitrary Hamiltonian op erator K w ith co efficien ts in Mat ℓ × ℓ ( C ) and inv ertible leading co efficient, and w e sho w that this b oun d is sharp if and on ly if K = K 1 ∂ , wh ere K 1 is a symmetric nondegenerate matrix o ver C . Since an y Hamiltonian op erator of h ydro dymanic type can ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 5 b e b rough t, by a change of v ariables, to this form , our result generalizes the results of [LZ11, LZ11pr] on K of hydro dynamic t yp e. F urthermore, for su c h op erators K w e also pro v e that the essen tial v ariational P oisson cohomolo gy is tr ivial, and we fi nd a nice description of the Z -graded Lie sup eralgebra H • K . W e are grateful to Tsinghua Universit y and the Mathematica l Sciences Cen ter (MSC), Beijing, where this pap er w as written, for their hospitalit y , and esp ecially Y oujin Zhang and Si-Qi L iu for enligh tening lectures and discussions. W e also thank th e Cente r of Mathematics and Th eoretica l Ph ysics (CMTP), Rome, for con tin uing encouragemen t and supp ort. 2. Transitive Z -graded Lie su peralgebr as and p rolonga tions Recall [GS64, Kac77] that a Z -graded L ie su p eralgebra g = L k ≥− 1 g k is called tr ansitive if any a ∈ g k , k ≥ 0, suc h that [ a, g − 1 ] = 0, is zero. Tw o equiv alen t d efi nitions are as follo ws: (i) There are no nonzero ideals of g con tained in L k ≥ 0 g k . (ii) I f a ∈ g k is such that [ . . . [[ a, C 0 ] , C 1 ] , . . . , C k ] = 0 f or all C 0 , . . . , C k ∈ g − 1 , then a = 0. If a Z -graded Lie su p eralgebra g = L k ≥− 1 g k is transitiv e, the Lie subalge- bra g 0 acts faithfu lly on g − 1 , hence w e ha ve an em b edding g 0 → gl ( g − 1 ). Giv en a Lie algebra g acting faithfully on a pu rely o dd v ector sup erspace U , one calls a pr olongation of the pair ( U, g ) any transitiv e Z -graded Lie sup eralgebra g = L k ≥− 1 g k suc h that g − 1 = U , g 0 = g , and the Lie brac k et b et w een g 0 and g − 1 is giv en by the action of g on U . Th e ful l pr olon gation of the pair ( U, g ) is a prolongation con taining any other prolongation of ( U, g ). It alwa ys exists and is unique. 2.1. T he Z -graded Lie sup eralgebra W ( n ) . Let Λ( n ) b e the Grass- man sup eralg ebra o ver the fi eld C on o dd generators ξ 1 , . . . , ξ n . Let W ( n ) b e the Lie sup eralg ebra of all deriv ations of the su p eralgebra Λ( n ), with the follo wing Z -grading: for k ≥ − 1, W k ( n ) is span n ed by deriv ati ons of the form ξ i 1 . . . ξ i k +1 ∂ ∂ ξ j . In particular, W − 1 ( n ) = h ∂ ∂ ξ i i n i =1 = Π C n , and W 0 ( n ) = h ξ i ∂ ∂ ξ j i n i,j =1 ≃ g l ( n ). It is easy to see that W ( n ) is the full prolon- gation of (Π C n , g l ( n )) [Kac77]. C onsequen tly , any transitive Z -graded Lie sup eralgebra g = L k ≥− 1 g k , with dim C g − 1 = n , embeds in W ( n ). 2.2. T he Z -graded L ie sup eralgebra e H ( n, S ) . Let S = ( s ij ) n i,j =1 b e a symmetric n × n matrix ov er C . C onsider the f ollo wing subalgebra of the Lie algebra g l ( n ): (2.1) so ( n, S ) =  A ∈ Mat n × n ( C )   A T S + S A = 0 , T r( A ) = 0  . W e endow the Grassman sup eralg ebra Λ( n ) with a structure of a P oisson sup eralgebra by letting { ξ i , ξ j } S = s ij . A closed formula for the P oisson 6 ALBER TO DE SOLE, VICTOR G. KAC brac k et on Λ( n ) is { f , g } S = ( − 1) p ( f )+1 n X i,j =1 s ij ∂ f ∂ ξ i ∂ g ∂ ξ j . W e in trod u ce a Z -gradin g of the sup ersp ace Λ( n ) b y letting deg( ξ i 1 . . . ξ i s ) = s − 2. Note that this is a Lie s up eralgebra Z gradin g Λ( n ) = L n − 2 k = − 2 Λ k ( n ) (but it is not an asso ciativ e su p eralgebra grading). Note also that Λ − 2 ( n ) = C 1 ⊂ Λ( n ) is a cen tral ideal of this Lie su p eralgebra. Hence Λ( n ) / C 1 inherits the structure of a Z -graded Lie sup eralgebra of dim en sion 2 n − 1, whic h w e denote b y e H ( n, S ) = L n − 2 k = − 1 e H k ( n, S ). The − 1-st degree su bspace is e H − 1 ( n, S ) = h ξ i i n i =1 ≃ Π C n , and the 0-th degree subspace e H 0 ( n, S ) = h ξ i ξ j i n i,j =1 is a Lie su balgebra of dimension  n 2  . Iden tifying e H − 1 ( n, S ) w ith Π C n (using the basis ξ i , i = 1 , . . . , n ) and e H 0 ( n, S ) with the sp ace of s kewsymmetric n × n matrices o v er C (via ξ i ξ j 7→ ( E ij − E j i ) / 2), the action of e H 0 ( n, S ) on e H − 1 ( n, S ) b ecomes: { A, v } S = AS v . Note that, if A is sk ewsymmetric, then AS lies in so ( n, S ). Hence, we ha v e a homomorph ism of Lie su p eralgebras: (2.2) e H − 1 ( n, S ) ⊕ e H 0 ( n, S ) → Π C n ⊕ so ( n, S ) , ( v , A ) 7→ ( v , AS ) . Lemma 2.1. The map (2.2) is bije ctive if and only if S has r ank n or n − 1 . Pr o of. Clearly , if S is nondegenerate, the map (2.2) is b ijectiv e. Moreo ver, if S has rank less th an n − 1, the map (2.2 ) is clearly not injectiv e. In the remaining case when S has rank n − 1, we can assu me it has the form (2.3) S =  0 0 0 T  , where T is a nondegenerate symmetric ( n − 1) × ( n − 1) matrix. In this case, one immediately c hec ks that th e map (2.2) is injectiv e. Moreo v er, so ( n, S ) = n  0 B T 0 A     B ∈ C ℓ , A ∈ s o ( n − 1 , T ) o . Hence, d im C so ( n, S ) = n − 1 +  n − 1 2  =  n 2  = d im C e H 0 ( n, S ).  Prop osition 2.2. If S has r ank n or n − 1 , then e H ( n, S ) is the ful l pr olon- gation of the p air ( C n , so ( n, S )) . Pr o of. F or S nondegenerate, the p ro of is can b e found in [Kac77]. W e reduce b elo w the case r k ( S ) = n − 1 to th e case of nond egenerate S . If rk( S ) = ℓ = n − 1, we can choose a basis h η , ξ 1 , . . . , ξ ℓ i , such that the matrix S is of the form (2.3 ). Define the map ϕ S : e H ( n, S ) → W ( n ), giv en b y (2.4) ϕ S ( f ( ξ 1 , . . . , ξ ℓ )) = { f , ·} S = ( − 1) p ( f )+1 ℓ X i,j =1 t ij ∂ f ∂ ξ i ∂ ∂ ξ j , ϕ S ( f ( ξ 1 , . . . , ξ ℓ ) η ) = f ( ξ 1 , . . . , ξ ℓ ) ∂ ∂ η . ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 7 It is easy to chec k that ϕ S is an inj ectiv e homomorp hism of Z -graded Lie sup eralgebras. Hence, w e can identi fy e H ( n, S ) with its image in W ( n ). Since ϕ S ( e H − 1 ( n, S )) = Π C n = W − 1 ( n ), the Z -graded Lie sup eralgebra ϕ S ( e H ( n, S )) (hence e H ( n, S )) is transitiv e. It remains to pro v e that it is the full prolongation of the pair ( e H − 1 ( n, S ) , e H 0 ( n, S )). F or this, we will pr ov e that, if X = f 0 ∂ ∂ η + ℓ X i =1 f i ∂ ∂ ξ i ∈ W k ( n ) , with f i ∈ Λ( n ), homogenous p olynomials of degree k + 1 ≥ 2, is suc h that (2.5)  ∂ ∂ η , X  ,  ∂ ∂ ξ i , X  ∈ ϕ S ( e H k − 1 ( n, S )) ∀ i = 1 , . . . ℓ , then X ∈ ϕ S ( e H k ( n, S )). C onditions (2.5) imp ly that all f 0 , . . . , f ℓ are p oly- nomials in ξ 1 , . . . , ξ ℓ only , an d there exist g 1 , . . . , g ℓ , p olynomials in ξ 1 , . . . , ξ ℓ , suc h that (2.6) ∂ f j ∂ ξ i = ( − 1) p ( g i )+1 ℓ X k =1 t j k ∂ g i ∂ ξ k , for ev ery i, j ∈ { 1 , . . . ℓ } . O n the other hand, the condition that X ∈ ϕ S ( e H k ( n, S )) means that there exists h , a p olynomial in ξ 1 , . . . , ξ ℓ , suc h that (2.7) f i = ( − 1) p ( h )+1 ℓ X k =1 t ik ∂ h ∂ g k . T o conclude, w e observe that conditions (2.6 ) imply the existence of h solving equation (2.7) , since e H ( ℓ, T ) is a full pr olongatio n.  R emark 2.3 . T h e notation e H ( n, S ) comes from the fact that, if S is nonde- generate, then the derive d Lie sup eralgebra H ( n, S ) = { e H ( n, S ) , e H ( n, S ) } = L n − 3 k = − 1 e H k ( n, S ) h as co dimen s ion 1 in e H ( n, S ), and it is simple for n ≥ 4. 3. V aria tional Poisso n cohomolo gy In this section we recall our r esults from [DSK11] on the v ariational P ois- son cohomology , in the notation of the p r esen t pap er. 3.1. Algebras of differen tial functions. An algebr a of differ ential func- tions V in one indep end ent v ariable x and ℓ dep endent v ariables u i , indexed b y the set I = { 1 , . . . , ℓ } , is, by definition, a differential alge bra (i.e. a unital comm utativ e asso ciativ e algebra w ith a deriv ation ∂ ), endo w ed with comm uting deriv ations ∂ ∂ u ( n ) i : V → V , for all i ∈ I and n ∈ Z + , suc h that, 8 ALBER TO DE SOLE, VICTOR G. KAC giv en f ∈ V , ∂ ∂ u ( n ) i f = 0 f or all but finitely man y i ∈ I and n ∈ Z + , and the follo win g commutatio n rules with ∂ hold: (3.1) h ∂ ∂ u ( n ) i , ∂ i = ∂ ∂ u ( n − 1) i , where the RHS is considered to b e zero if n = 0. An equiv alen t wa y to wr ite the identitie s (3.1) is in terms of generating series: (3.2) X n ∈ Z + z n ∂ ∂ u ( n ) i ◦ ∂ = ( z + ∂ ) ◦ X n ∈ Z + z n ∂ ∂ u ( n ) i . As usu al we shall denote by f 7→ R f the canonical quotien t map V → V /∂ V . W e call C = Ker( ∂ ) ⊂ V the subalgebra of c onstant functions , and w e denote b y F ⊂ V the subalgebra of quasic onstant functions , defined by (3.3) F =  f ∈ V   ∂ f ∂ u ( n ) i = 0 ∀ i ∈ I , n ∈ Z +  . It is n ot hard to sh o w [DSK11] that C ⊂ F , ∂ F ⊂ F , and F ∩ ∂ V = ∂ F . Throughout the pap er w e w ill assum e th at F is a field of c haracteristic zero, hence so is C ⊂ F . Unless otherwise sp eci fied, all vecto r spaces, as well as tensor pro du cts, direct sums, and Hom’s, will b e considered o v er the field C . One sa ys that f ∈ V has differ ential or der n in the v ariable u i if ∂ f ∂ u ( n ) i 6 = 0 and ∂ f ∂ u ( m ) i = 0 for all m > n . The main example of an algebra of differential functions is the ring of dif- feren tial p olynomials o v er a differen tial field F , R ℓ = F [ u ( n ) i | i ∈ I , n ∈ Z + ], where ∂ ( u ( n ) i ) = u ( n +1) i . Other examples can b e constructed s tarting from R ℓ b y taking a lo calization by some multiplica tiv e subs et S , or an alge- braic extension obtained by add in g solutions of some p olynomial equ ations, or a differen tia l extension obtained by addin g solutions of some differen tial equations. The v ariationa l derivative δ δu : V → V ℓ is defi ned by (3.4) δ f δ u i := X n ∈ Z + ( − ∂ ) n ∂ f ∂ u ( n ) i . It follo ws immediately f rom (3.2 ) that ∂ V ⊂ Ker δ δu . A ve ctor field is, b y defin ition, a deriv ation of V of the form (3.5) X = X i ∈ I ,n ∈ Z + P i,n ∂ ∂ u ( n ) i , P i,n ∈ V . W e den ote b y V ect( V ) th e Lie algebra of all v ector fields. A vect or fi eld X is called evolutionary if [ ∂ , X ] = 0, and w e denote by V ect ∂ ( V ) ⊂ V ect ( V ) ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 9 the Lie sub algebra of all ev olutionary vec tor fi elds. By (3.1), a v ecto r field X is ev olutionary if and only if it h as the form (3.6) X P = X i ∈ I ,n ∈ Z + ( ∂ n P i ) ∂ ∂ u ( n ) i , where P = ( P i ) i ∈ I ∈ V ℓ , is called the char acteristic of X P . Giv en P ∈ V ℓ , we denote by D P =  ( D P ) ij ( ∂ )  i,j ∈ I its F r e chet derivative , giv en by (3.7) ( D P ) ij ( ∂ ) = X n ∈ Z + ∂ P i ∂ u ( n ) j ∂ n . Recall fr om [BDSK09 ] that an algebra of different ial functions V is called normal if w e ha v e ∂ ∂ u ( m ) i  V m,i  = V m,i for all i ∈ I , m ∈ Z + , where we let (3.8) V m,i := n f ∈ V    ∂ f ∂ u ( n ) j = 0 if ( n, j ) > ( m, i ) in lexicographic order o . W e also denote V m, 0 = V m − 1 ,ℓ , and V 0 , 0 = F . The algebra R ℓ is ob viously normal. Moreo v er, any its extension V can b e further extended to a n ormal algebra. Conv ersely , it is prov ed in [DSK09] that any n ormal algebra of differentia l functions V is automatically a differ- en tial algebra extension of R ℓ . Throughout the pap er w e shall assume that V is an extension of R ℓ . Recall also f rom [DSK11] th at a differential field F is called line arly close d if any linear differen tial equation, a n u ( n ) + · · · + a 1 u ′ + a 0 u = b , with n ≥ 0, a 0 , . . . , a n ∈ F , a n 6 = 0, has a solution in F for ev ery b ∈ F , and it h as a nonzero solution for b = 0, pro vided that n ≥ 1. 3.2. T he univ ersal Lie sup era lgebra W v ar (Π V ) of v ariat ional p oly– v ector fields. Recall the definition of the universal Lie sup eralg ebra of v ariational p olyv ector fields W v ar (Π V ), asso ciated to the algebra of differ- en tial fu n tions V [DSK 11]. W e let W v ar (Π V ) = ∞ M k = − 1 W v ar k , where W v ar k is the sup ersp ace of parity k mo d 2 consisting of all skewsym- metric arr ays , i.e. arr a ys of p olynomials (3.9) P =  P i 0 ,...,i k ( λ 0 , . . . , λ k )  i 1 ,...,i k ∈ I , where P i 0 ,...,i k ( λ 0 , . . . , λ k ) ∈ V [ λ 0 , . . . , λ k ] / ( ∂ + λ 0 + · · · + λ k ) are sk ewsymmet- ric with resp ect to sim ultaneous p ermutat ions of the v ariables λ 0 , . . . , λ k and the indices i 0 , . . . , i k . By V [ λ 0 , . . . , λ k ] / ( ∂ + λ 0 + · · · + λ k ) we mean the quo- tien t of the space V [ λ 0 , . . . , λ k ] by the image of th e op erato r ∂ + λ 0 + · · · + λ k . 10 ALBER TO DE SOLE, VICTOR G. KAC Clearly , for k = − 1 this space is V /∂ V and, for k ≥ 0, w e can id en tify it with th e algebra of p olynomials V [ λ 0 , . . . , λ k − 1 ] by letting λ k = − λ 0 − · · · − λ k − 1 − ∂ , with ∂ acting from the left. W e then define the follo wing Z -graded Lie sup eralgebra brac k et on W v ar (Π V ). F or P ∈ W v ar h and Q ∈ W v ar k − h , with − 1 ≤ h ≤ k +1, w e let [ P , Q ] := P  Q − ( − 1) h ( k − h ) Q  P , w here P  Q ∈ W v ar k is zero if h = k − h = − 1, and otherwise it is giv en b y (3.10)  P  Q  i 0 ,...,i k ( λ 0 , . . . , λ k ) = X σ ∈S h,k sign( σ ) X j ∈ I ,n ∈ Z + P j,i σ ( k − h +1) ,...,i σ ( k ) ( λ σ (0) + · · · + λ σ ( k − h ) + ∂ , λ σ ( k − h +1) , . . . , λ σ ( k ) ) → ( − λ σ (0) − · · · − λ σ ( k − h ) − ∂ ) n ∂ ∂ u ( n ) j Q i σ (0) ,...,i σ ( k − h ) ( λ σ (0) , . . . , λ σ ( k − h ) ) , where S h,k denotes the set of h - shuffles in the group S k +1 = P er m { 0 , . . . , k } , i.e. the p ermutatio ns σ satisfying σ (0) < · · · < σ ( k − h ) , σ ( k − h + 1) < · · · < σ ( k ) . The arro w in (3.10) means that ∂ should b e mo v ed to the right. Note that, b y the sk ewsymmetry conditions on P and Q , we can replace the sum o v er shuffles b y the sum o v er the whole p erm u tation group S k +1 , p ro vided that w e divide by h !( k − h + 1)!. It follo ws fr om Prop ositio n 9.1 and the iden tification (9.22) in [DSK11], that the b o x pro duct (3.10) is well defined and the corresp ondin g commutato r mak es W v ar (Π V ) into a Z -graded Lie sup eralgebra. R emark 3.1 . In [DSK 11] we identified W var (Π V ) with the quotient space Ω • ( V ) = e Ω • ( V ) /∂ e Ω • ( V ), where e Ω • ( V ) is the comm utativ e asso ciativ e u nital sup eralgebra freely generated o v er V by o d d generators θ ( m ) i = δ u ( m ) i , i ∈ I , m ∈ Z + , and w here ∂ : e Ω • ( V ) → e Ω • ( V ) extends ∂ : V → V to an even deriv ation su c h that ∂ θ ( m ) i = θ ( m +1) i . Th is iden tification is giv en by mapping the array P =  X m 0 ,...,m k ∈ Z + f m 0 ,...,m k i 0 ,...,i k λ m 0 0 . . . λ m k k  i 0 ,...,i k ∈ I ∈ W var k to the elemen t Z X i 0 ,...,i k ∈ I X m 0 ,...,m k ∈ Z + f m 0 ,...,m k i 0 ,...,i k θ ( m 0 ) i 0 . . . θ ( m k ) i k ∈ Ω k +1 ( V ) . (It is easy to see that this m ap is well defined and b ijectiv e.) Here R denotes, as u sual, the quotien t map e Ω • ( V ) → e Ω • ( V ) /∂ e Ω • ( V ) = Ω • ( V ). W e extend the v ariational deriv ativ e to a map δ δ u i = X n ∈ Z + ( − ∂ ) n ◦ ∂ ∂ u ( n ) i : Ω k ( V ) → Ω k +1 ( V ) , ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 11 b y letting ∂ ∂ u ( n ) i acts on co efficien ts ( ∈ V ). F ur thermore, w e introd uce the o dd v ariati onal deriv ativ es δ δ θ i = X n ∈ Z + ( − ∂ ) n ◦ ∂ ∂ θ ( n ) i : Ω k ( V ) → Ω k +1 ( V ) . Then the b o x pro du ct (3.10) tak es, under the identificat ion W var (Π V ) ≃ Ω • ( V ), the follo wing simple form [Get02]: P  Q = X i ∈ I δ P δ θ i δ Q δ u i . W e describ e explicitly the s p aces W v ar k for k = − 1 , 0 , 1. Clearly , W v ar − 1 = V /∂ V . Also W v ar 0 = V ℓ thanks to the ob vious identificatio n of V [ λ ] / ( ∂ + λ ) with V . Finally , the space V [ λ, µ ] / ( ∂ + λ + µ ) is iden tified with V [ λ ] ≃ V [ ∂ ], by letting µ = − ∂ mov ed to the left and λ = ∂ mo v ed to th e righ t. Hence elemen ts in W v ar 1 corresp ond to ℓ × ℓ matrix d ifferential op erators o v er V , and the sk ewsymmetry condition for an elemen t of W v ar 1 translates in to the sk ew adj oin tness of the corresp onding matrix differen tial op er ator (i.e. to the condition H ∗ j i ( ∂ ) = − H ij ( ∂ ), wher e, as usual, for a differentia l op erator L ( ∂ ) = P n l n ∂ n , its adjoin t is L ∗ ( ∂ ) = P n ( − ∂ ) n ◦ l n ). In order to k eep the same ident ification as in [DSK 11], we asso ciate to the arra y P =  P ij ( λ, µ )  i,j ∈ I ∈ W v ar 1 , the f ollo wing skew adjoint ℓ × ℓ matrix differentia l op erator H =  H ij ( ∂ )  i,j ∈ I , wh ere (3.11) H ij ( λ ) = P j i ( λ, − λ − ∂ ) , and ∂ acts f rom the left. Next, we write some explicit formulas for the Lie brac k ets in W v ar (Π V ). Since S − 1 ,k = ∅ an d S k +1 ,k = { 1 } , we ha v e, for R h ∈ V /∂ V = W v ar − 1 and Q ∈ W v ar k +1 : (3.12) [ R h, Q ] i 0 ,...,i k ( λ 0 , . . . , λ k ) = ( − 1) k [ Q, R h ] i 0 ,...,i k ( λ 0 , . . . , λ k ) = ( − 1) k X j ∈ I Q j,i 0 ,...,i k ( ∂ , λ 0 , . . . , λ k ) → δ h δ u j , In particular, [ R h, R f ] = 0 for R f ∈ V /∂ V . F or Q ∈ V ℓ = W v ar 0 w e h av e (3.13) [ Q, R h ] = − [ R h, Q ] = X j ∈ I Z Q j δ h δ u j = R X Q ( h ) , where X Q is the ev olutionary v ector field with c h aracteristics Q , defin ed in (3.6). F urth er m ore, for H =  H ij ( ∂ )  i,j ∈ I ∈ W v ar 1 (via the identi fication (3.11)), we hav e (3.14) [ H , R h ] = H ( ∂ ) δ h δ u ∈ V ℓ . 12 ALBER TO DE SOLE, VICTOR G. KAC Since S 0 ,k = { 1 } and S k ,k = { ( α, 0 , α ˇ . . . , k ) } k α =0 , w e ha v e, for P ∈ V ℓ = W v ar 0 and Q ∈ W v ar k , [ P , Q ] i 0 ,...,i k ( λ 0 , . . . , λ k ) = X P  Q i 0 ,...,i k ( λ 0 , . . . , λ k )  − k X α =0 X j ∈ I ,n ∈ Z + Q i 0 ,..., α ˇ j ,. ..,i k ( λ 0 , . . . , λ α + ∂ , . . . , λ k ) → ( − λ α − ∂ ) n ∂ P i α ∂ u ( n ) j . In p articular, for Q ∈ V ℓ = W v ar 0 , we get the usu al commutat or of ev olu- tionary v ector fields: [ P , Q ] i = X P ( Q i ) − X Q ( P i ) , while, for a ske w adjoin t ℓ × ℓ matrix differen tial op erator H ( ∂ ) ∈ W v ar 1 , w e get (3.15) [ P , H ]( ∂ ) = X P  H ( ∂ )  − D P ( ∂ ) ◦ H ( ∂ ) − H ( ∂ ) ◦ D ∗ P ( ∂ ) , where, in the first term of the RHS, X P ( H ( ∂ )) denotes the ℓ × ℓ matrix differen tial op erator whose ( i, j ) en try is obtained by applying X P to the co efficien ts of th e d ifferen tial op erator H ij ( ∂ ). In the last t w o terms of th e RHS of (3.15), D P denotes the F r ec h et deriv ativ e of P , defined in (3.7), and D ∗ P is its adjoin t m atrix d ifferen tial op erato r. Finally , w e write equation (3.10) in the case when h = 1. Since S 1 ,k = { (0 , α ˇ . . . , k , α ) } k α =0 and S k − 1 ,k = { ( α, β , 0 , α ˇ . . . β ˇ . . ., k ) } k 0 ≤ α<β ≤ k , w e ha v e, for a sk ew adj oin t matrix differen tial op erator H =  H ij ( ∂ )  i,j ∈ I ∈ W v ar 1 (via the iden tificatino (3.11 )) and for P ∈ W v ar k − 1 : (3.16) [ H , P ] i 0 ,...,i k ( λ 0 , . . . , λ k ) = ( − 1) k +1 X j ∈ I ,n ∈ Z + k X α =0 ( − 1) α ∂ P i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . ., λ k ) ∂ u ( n ) j ( λ α + ∂ ) n H j,i α ( λ α ) + k X β = α +1 ( − 1) β × P j,i 0 , α ˇ ... β ˇ ...,i k ( λ α + λ β + ∂ , λ 0 , α ˇ . . . β ˇ . . ., λ k ) → ( − λ α − λ β − ∂ ) n ∂ H i β ,i α ( λ α ) ∂ u ( n ) j ! . In p articular, if K =  K ij ( ∂ )  i,j ∈ I ∈ W v ar 1 , we h a v e [ K, H ] = [ H , K ] = K  H + H  K , where (3.17) ( K  H ) i 0 ,i 1 ,i 2 ( λ 0 , λ 1 , λ 2 ) = X j ∈ I ,n ∈ Z +  ∂ H i 0 ,i 1 ( λ 1 ) ∂ u ( n ) j ( λ 2 + ∂ ) n K j,i 2 ( λ 2 ) + ∂ H i 1 ,i 2 ( λ 2 ) ∂ u ( n ) j ( λ 0 + ∂ ) n K j,i 0 ( λ 0 ) + ∂ H i 2 ,i 0 ( λ 0 ) ∂ u ( n ) j ( λ 1 + ∂ ) n K j,i 1 ( λ 1 )  . ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 13 R emark 3.2 . Give n a sk ew adjoint matrix d ifferen tial op erato r H =  H ij ( ∂ )  , w e can define the corresp onding “v ariational” λ -br ac kets {· λ ·} H : V × V → V [ λ ], giv en by the follo wing formula (cf. [DSK06]): (3.18) { f λ g } = X i,j ∈ I , m,n ∈ Z + ∂ g ∂ u ( n ) j ( λ + ∂ ) n H j i ( λ + ∂ )( − λ − ∂ ) m ∂ f ∂ u ( m ) i . One can write the ab o v e form ulas in this language (cf. [DSK 11]). Prop osition 3.3. The Z -gr ade d Lie sup er algebr a W v ar (Π V ) is tr ansitive, henc e it i s a pr olongation of the p air (Π V /∂ V , V ect ∂ ( V )) . Pr o of. First note that, if H ( ∂ ) is an ℓ × ℓ matrix differenti al op erator such that H ( ∂ ) δf δu = 0 for ev ery f ∈ V , then H ( ∂ ) = 0 (cf. [BDSK09]). In deed, if H ( ∂ ) has order N and H ij ( ∂ ) = P N n =0 h ij ; n ∂ n with h ij ; N 6 = 0, then letting f = ( − 1) M 2 ( u ( M ) j ) 2 , w e ha v e δf δu k = δ k ,j u (2 M ) j and, for M sufficien tly large, ∂ ∂ u (2 M + N ) j  H ( ∂ ) δf δu  i = h ij ; N 6 = 0 (here w e are using the assu mption that V con tains R ℓ ). The claim follo ws immediately b y this observ ation and equation (3.12 ).  3.3. T he cohomology complex ( W v ar (Π V ) , δ K ) . L et K =  K ij ( ∂ )  i,j ∈ I ∈ W v ar 1 b e a Hamiltonian op er ator , i.e. K is sk ew adjoin t and [ K, K ] = 0. Then (ad K ) 2 = 0, and we can consider the asso ciated variational Poisson c oh o- molo gy c omplex ( W v ar (Π V ) , ad K ). Let Z • K ( V ) = L ∞ k = − 1 Z k K , where Z k K = Ker  ad K   W v ar k  , an d B • K ( V ) = L ∞ k = − 1 B k K , w here B k K = (ad K )  W v ar k − 1  . Then Z • K ( V ) is a Z -graded subalgebra of th e Lie sup eralge bra W v ar (Π V ), and B • K ( V ) is a Z -graded id eal of Z • K ( V ). Hence, the corresp onding varia- tional Poisson c oho molo gy H • K ( V ) = ∞ M k = − 1 H k K , H k K = Z k K  B k K , is a Z -graded Lie sup eralgebra. In the sp eci al case w h en K =  K ij ( ∂ )  i,j ∈ I has co efficien ts in F , whic h, as in [DSK 11], we shall call a quasic onstant ℓ × ℓ matrix d ifferential op erator, form ula (3.16) for the different ial δ K = ad K b ec omes for P ∈ W v ar k − 1 , k ≥ 0, (3.19) ( δ K P ) i 0 ,...,i k ( λ 0 , . . . , λ k ) = ( − 1) k +1 X j ∈ I ,n ∈ Z + k X α =0 ( − 1) α ∂ P i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . ., λ k ) ∂ u ( n ) j ( λ α + ∂ ) n K j,i α ( λ α ) . In fact, as sh o wn in [DSK 11, Prop.9.9], if K =  K ij ( ∂ )  i,j ∈ I is an arbitrary quasiconstan t ℓ × ℓ matrix differen tial op erator (not necessarily ske w ad- join t), then th e same form ula (3.19) still giv es a well defin ed linear map δ K : W v ar k − 1 → W v ar k , k ≥ 0, such that δ 2 K = 0. Hence, w e get a cohomolog y 14 ALBER TO DE SOLE, VICTOR G. KAC complex ( W v ar (Π V ) , δ K ). As b efore, we denote Z k K = Ker  δ K   W v ar k  , B k K = δ K  W v ar k − 1  and H k K = Z k K  B k K . F or example, H − 1 K = Z − 1 K = R f ∈ V /∂ V    K ∗ ( ∂ ) δf δu = 0 o , wh ic h is called the set of c entr al elements (or Casimir elements ) of K ∗ . Next, w e hav e (see [DSK11]): B 0 K = n K ∗ ( ∂ ) δ f δ u o f ∈V , Z 0 K = n P ∈ V ℓ    D P ( ∂ ) ◦ K ( ∂ ) = K ∗ ( ∂ ) ◦ D ∗ P ( ∂ ) o . F urther m ore, given P ∈ V ℓ = W v ar 0 , the elemen t δ K P ∈ W v ar 1 , under the iden tification (3.11) of W v ar 1 with the space of ℓ × ℓ sk ew adjoint matrix differen tial op erators, coincides with (3.20) δ K P = D P ( ∂ ) ◦ K ( ∂ ) − K ∗ ( ∂ ) ◦ D ∗ P ( ∂ ) . Hence, B 1 K =  D P ( ∂ ) ◦ K ( ∂ ) − K ∗ ( ∂ ) ◦ D ∗ P ( ∂ )  P ∈V ℓ . Finally , Z 1 K consists, under the same iden tificatio n, of the ℓ × ℓ skew adjoint matrix differentia l op erators H ( ∂ ) for which the RHS of (3.17) is zero. R emark 3.4 . If R f , R g ∈ V /∂ V , w e h a ve [ R f , R g ] = 0 and [ δ K R f , R g ] − [ R f , δ K R g ] = R  − δ g δ u K ∗ ( ∂ ) δ f δ u − δ f δ u K ∗ ( ∂ ) δ g δ u  . Hence, the different ial δ K in (3.19) is not an o d d d eriv ation unless K ( ∂ ) is sk ew adj oin t. In particular, th e corresp onding cohomolo gy H • K ( V ) do es not ha v e a n atural structur e of a Lie sup eralg ebra unless K ( ∂ ) is a sk ew adj oin t op erator. 3.4. T he v ariational P oisson cohomo logy H ( W v ar (Π V ) , δ K ) for a qua- siconstan t matrix differen tial op erator K ( ∂ ) . Let V b e an algebra of differen tial functions extension of R ℓ , the algebra of differen tial p olynomi- als in th e differen tial v ariables u 1 , . . . , u ℓ o v er a differential fi eld F . Let K =  K ij ( ∂ )  i,j ∈ I b e a quasiconstan t ℓ × ℓ matrix differen tial op erato r of or- der N (not necessarily sk ew adjoint). F or k ≥ − 1, w e d enote b y A k K ⊂ W v ar k the su b set consisting of arra ys of the form (3.21)  X j ∈ I  P j,i 0 ,...,i k ( λ 0 , . . . , λ k ) u j   i 0 ,...,i k ∈ I , where [ x ] denotes the coset of x ∈ V [ λ 0 , . . . , λ k ] mo d ulo ( λ 0 + · · · + λ k + ∂ ) V [ λ 0 , . . . , λ k ], s atisfying the f ollo wing p rop erties. F or j, i 0 , . . . , i k ∈ I , P j,i 0 ,...,i k ( λ 0 , . . . , λ k ) are p olynomials in λ 0 , . . . , λ k with co efficien ts in F of degree at most N − 1 in eac h v ariable λ i , skewsymmetric with resp ect to simultaneous p erm utations of the indices i 0 , . . . , i k , and the v ariables λ 0 , . . . , λ k , and satisfying the follo wing condition: (3.22) k +1 X α =0 ( − 1) α X j ∈ I P j,i 0 , α ˇ ...,i k +1 ( λ 0 , α ˇ . . ., λ k +1 ) K j i α ( λ α ) ≡ 0 mo d ( λ 0 + · · · + λ k +1 + ∂ ) F [ λ 0 , . . . , λ k +1 ] . ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 15 F or example, A − 1 K consists of elemen ts of the form P j ∈ I R P j u j ∈ V /∂ V , where P ∈ F ℓ solv es th e equation K ∗ ( ∂ ) P = 0 . In fact it is not hard to sho w that A − 1 K coincides with the set Z − 1 K of cen tral elemen ts of K ∗ (see Lemma 4.4 b elo w). Next, A 0 K consists of elements of the form  P j ∈ I P ∗ ij ( ∂ ) u j  i ∈ I ∈ V ℓ = W v ar 0 , where P =  P ij ( ∂ )  i,j ∈ I is a quasiconstan t ℓ × ℓ matrix differen tial op erator of order at most N − 1, solving the follo wing equation: (3.23) K ∗ ( ∂ ) ◦ P ( ∂ ) = P ∗ ( ∂ ) ◦ K ( ∂ ) . The description of the set A 1 K is more complicated. Giv en a p olynomial in t w o v ariables P ( λ, µ ) = P N m,n =0 c mn λ m µ n ∈ F [ λ, µ ], w e denote P ∗ 1 ( λ, µ ) = P N m,n =0 ( − λ − ∂ ) m c mn µ n , and P ∗ 2 ( λ, µ ) = P N m,n =0 ( − µ − ∂ ) n c mn λ m . Then, under the iden tification of W v ar 1 with the space of skew adjoint ℓ × ℓ ma- trix differentia l op erat ors giv en by (3.11), A 1 K consists of operators H =  H ij ( ∂ )  i,j ∈ I of the form H ij ( λ ) = − X k ∈ I P ∗ k ij ( λ + ∂ , λ ) u k , where, f or i, j, k ∈ I , P k ij ( λ, µ ) ∈ F [ λ, µ ] are p olynomials of degree at most N − 1 in eac h v ariable, suc h that P k ij ( λ, µ ) = − P k j i ( µ, λ ), and such that X h ∈ I  K ∗ ih ( λ + µ + ∂ ) P hj k ( λ, µ ) + P ∗ 2 hk i ( µ, λ + µ + ∂ ) K hj ( λ ) + P ∗ 1 hij ( λ + µ + ∂ , λ ) K hk ( µ )  = 0 . Theorem 11.9 from [DSK11] can b e stated as follo ws: Theorem 3.5. L et V b e a normal algebr a of differ ential functions in ℓ dif- fer ential variables over a line arly close d differ ential field F , and let C ⊂ F b e the su bfield of c onstan ts. L et K ( ∂ ) b e a quasic onsta nt ℓ × ℓ matr ix differ en- tial op er ator of or der N with invertible le ading c o efficient K N ∈ Mat ℓ × ℓ ( F ) . Then we have the fol lowing de c omp osition of Z k K in a dir e ct sum of ve ctor sp ac es over C : Z k K = A k K ⊕ B k K . Henc e, we have a c anonic al isomorph ism H k K ≃ A k K . M or e over, A k K (henc e H k K ) is a ve ctor sp ac e over C of dimension  N ℓ k +2  . Recall th at, if K is a ske w adjoin t op erator, then H • K ( V ) = L k ≥− 1 H k K is a Lie su p eralgebra with consisten t Z -grading. I n Section 5 w e will prov e th e follo win g Theorem 3.6. L et V b e a normal algebr a of differ ential func tions, over a line arly close d differ ential field F . L et K ( ∂ ) b e a quasic onstant skewadjoint ℓ × ℓ matrix diffe r ential op er ator of or der N with invertible le ading c o efficient 16 ALBER TO DE SOLE, VICTOR G. KAC K N ∈ Mat ℓ × ℓ ( F ) . Then the Z -g r ade d Lie sup er algebr a H • K ( V ) is isomorphic to the Z -gr ade d Lie sup er algebr a e H ( N ℓ, S ) c onstructe d in Se ction 2.2 , wher e S is the matrix, in some b asis, of the nonde gener ate symmetric biline ar f orm h ·|· i 0 K c onstructe d in Se ction 5.1. R emark 3.7 . The su bspace A • K ( V ) = L ∞ k = − 1 A k K is NOT, in general, a sub- algebra of the Lie sup eralgebra Z • ( V ). W e can enlarge it to b e a su balgebra b y letting e A k K ⊂ Z k K b e the sub set consisting of arra ys of the form (3.21) where P j,i 0 ,...,i k ( λ 0 , . . . , λ k ) are p olynomia ls in λ 0 , . . . , λ k with co efficient s in F of arbitrary degree, sk ews y m metric w ith resp ect to s imultaneous p ermu- tations of the indices i 0 , . . . , i k , and the v ariables λ 0 , . . . , λ k , and satisfying condition (3.22). Then, clearly , A • K ( V ) ≃ e A • K ( V )  e A • K ( V ) ∩ B • K ( V )  . F or example, it is not hard to show that e A 0 K ∩ B 0 K =  S ( ∂ ) K ( ∂ )   S ∗ ( ∂ ) = S ( ∂ )  , so that, A 0 K is a L ie algebra,  S ( ∂ ) K ( ∂ )   S ∗ ( ∂ ) = S ( ∂ )  is its ideal, and, b y Th eorem 3.6, th e quotient is isomorphic to the Lie algebra so ( N ℓ ). R emark 3.8 . If N ≤ 1, then A • K ( V ) is a subalgebra of the Lie sup eralge- bra Z • K ( V ), i.e. in this case the complex ( W var (Π V ) , ad K ) is form al (cf. [Get02]). Ho w ev er, this is not the case for N > 1. 4. Essent ial v aria tional Poisson cohom ology In this section we in tro d uce the sub algebra of essen tia l v ariatio nal P oisson cohomology and w e p ro v e a v anishing theorem for this cohomology . 4.1. T he Casimir subalgebra Z − 1 K ⊂ V /∂ V and the essen t ial sub- complex E W v ar (Π V ) . T hroughout th is section w e let V b e an algebra of differen tial functions in the v ariables u i , i ∈ I , and w e denote, as usual, by F the s ubalgebra of quasiconstan t, and by C ⊂ F the subalgebra of constan ts. Let K =  K ij ( ∂ )  i,j ∈ I b e a Hamiltonian ℓ × ℓ m atrix different ial op erator with co efficien ts in V . I n other words, we can view K as an element of W v ar 1 suc h that [ K , K ] = 0, hence, w e can consider the corresp on d ing cohomology complex ( W v ar (Π V ) = L k ≥− 1 W v ar k , ad K ). Reca ll from Section 3.3 that w e ha v e the Z -graded subalgebra Z • K ( V ) = L k ≥− 1 Z k K of closed elemen ts in W v ar (Π V ), and, inside it, the ideal of exact elements B • ( V ) = L k ≥− 1 B k K . The space Z − 1 K of cen tral elemen ts is, in this case, (4.1) Z − 1 K = n C ∈ V /∂ V    [ K, C ]  = K ( ∂ ) δ C δ u  = 0 o . W e call an elemen t P ∈ W v ar k essential if the f ollo wing condition holds: (4.2)  . . .  [ P , C 0 ] , C 1  , . . . , C k  = 0 , ∀ C 0 , . . . , C k ∈ Z − 1 K . W e denote by E W v ar k ⊂ W v ar k the su b space of essentia l elemen ts. F or exam- ple, E W v ar − 1 = 0 and E W v ar 0 consists of elemen ts P ∈ V ℓ suc h that R P δC δu = 0 for all central elemen ts C ∈ Z − 1 K . F urthermore, E W v ar 1 consists, u n der the ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 17 iden tification (3.11), of sk ew adj oint ℓ × ℓ matrix differen tial op erators H ( ∂ ), suc h that Z δ C 1 δ u H ( ∂ ) δ C 2 δ u = 0 , ∀ C 1 , C 2 ∈ Z − 1 K . Let E W v ar = L k ≥− 1 E W v ar k . This is a Z -graded su bspace of W v ar (Π V ), dep end in g on the op erator K ( ∂ ). Finally , denote by E Z • K ( V ) = L k ≥− 1 E Z k K the Z -graded subspace of essential ly close d elements , i.e. E Z k K = Z k K ∩ E W v ar k . Prop osition 4.1. (a) E W v ar is a Z -gr ade d sub algebr a of the Li e sup er alge- br a W v ar (Π V ) . Conse qu ently E Z • K ( V ) i s a Z -gr ade d sub algebr a of E W v ar . (b) E xact elements ar e essential ly close d, i.e. B • K ( V ) ⊂ E Z • K ( V ) , henc e they form a Z -gr ade d ide al of the Lie sup er algebr a E Z • K ( V ) . Pr o of. Let P ∈ E W v ar h and Q ∈ E W v ar k − h , with 0 ≤ h ≤ k , and let C 0 , . . . , C k ∈ Z − 1 K . Using iterativ ely th e J acobi identit y , we can express [ . . . [[[ P , Q ] , C 0 ] , C 1 ] , . . . , C k ] as a linear combinatio n of the comm utators of the p airs of elemen ts of the form [ . . . [[ P , C i 0 ] , C i 1 ] , . . . , C i s − 1 ] and [ . . . [[ Q, C i s ] , C i s +1 ] , . . . , C i k ] , where s is either h or h + 1. In the latter ca se the first eleme n t is zero since P is essentia l, w hile in the former case the second elemen t is zero since Q is essen tia l. Hence, [ P , Q ] is essenti al. The second claim of part (a) follo ws s in ce E Z • K ( V ) is the in tersectio n of E W v ar and Z • K ( V ), whic h are b oth Z -graded subalgebra of W v ar (Π V ). F or p art (b), giv en the exact elemen t [ K, P ], where P ∈ E W v ar k − 1 , an d give n C 0 , · · · , C k ∈ Z − 1 K , w e hav e, using again the J acobi iden tit y , [ . . . [[[ K, P ] , C 0 ] , C 1 ] , . . . , C k ] = [ K , [ . . . [[ P , C 0 ] , C 1 ] , . . . , C k ]] = 0 .  So, w e defin e the essential variational Poisson c oho molo gy as E H • K ( V ) = M k ≥− 1 E H k K , where E H k K = E Z k K / B k K . Clearly , this is a Z -graded subalgebra of the Lie sup eralgebra H • K ( V ) = H ( W v ar (Π V ) , ad K ). R emark 4.2 . Let H ( ∂ ) b e a Hamiltonian op erator compatible with K ( ∂ ), i.e. [ K, H ] = 0. Sup p ose that the first step of the Lenard-Magri sc heme alw a ys w orks, namely for ev ery cen tral elemen t C ∈ Z − 1 K there exists R h ∈ V /∂ V su ch that [ H, C ] = [ K, R h ]. Th en H is essential ly closed. Indeed, [[ H , C ] , C 1 ] = [[ K, R h ] , C 1 ] = [ R h, [ K, C 1 ]] = 0 for ev ery C, C 1 ∈ Z − 1 K . This is one of the reasons for the name ”essen tial”, since only for the essentia lly 18 ALBER TO DE SOLE, VICTOR G. KAC closed op er ators H the Lenard -Magri scheme ma y w ork. Con v er s ely , sup- p ose H ( ∂ ) is an essen tially closed Hamiltonian op erato r, i.e. H ( ∂ ) ∈ E Z 1 K . Then, for ev ery cen tral elemen t C ∈ Z − 1 K , it is immediate to see that there exists R h ∈ V /∂ V and A ∈ E Z 0 K suc h that [ H , C ] = [ K, R h ] + A . If the first essenti al v ariational Poisson cohomology is zero, we can choose A to b e zero, wh ic h means that the first step in the L en ard-Magri scheme works. 4.2. V a nishing of the essen tial v ariational Po isson cohomology. In this section w e p ro v e the follo wing Theorem 4.3. If V b e a normal algebr a of differ ential functions in ℓ dif- fer ential variables over a line arly close d differ ential field F , and i f K ( ∂ ) is a quasic onstant ℓ × ℓ matrix differ ential op er ator of or der N with i nv e rtible le ading c o e ffic i ent K N ∈ Mat ℓ × ℓ ( F ) , then E H • K ( V ) = 0 . In order to pro v e Theorem 4.3 we will need some preliminary lemmas. Lemma 4.4. L et V b e an arbitr ary algebr a of differ ential functions. L et K ( ∂ ) : V ℓ → V ℓ b e a quasic onstant ℓ × ℓ matrix differ ential op er ator with invertible le ading c o efficient K N ∈ Mat ℓ × ℓ ( F ) Then: (a) Ker( K ( ∂ )) = Ker  K ( ∂ )   F ℓ  . (b) The map δ δu : V /∂ V → V ℓ r estricts to a surje ctive map δ δu : Z − 1 K → Ker  K ( ∂ )   F ℓ  . (c) If, mor e over, V is a normal algebr a of differ ential functions and ∂ : F → F is surje ctive, then we have a bije c tion δ δu : Z − 1 K ∼ − → Ker  K ( ∂ )   F ℓ  . Pr o of. F or p art (a), w e need to sh ow that, if F ∈ V ℓ solv es K ( ∂ ) F = 0, then F ∈ F ℓ . S upp ose , by con tradictio n, that F / ∈ F ℓ . W e ma y assum e, w ithout loss of generalit y , that K N = 1 I, and that the fir s t co ord inate F 1 has maximal differen tial order, i.e. F 1 , . . . , F ℓ ∈ V n,i and F 1 / ∈ V n,i − 1 , for some i ∈ I , n ∈ Z + . Then ∂ ∂ u ( n + N ) i  K ( ∂ ) F  1 = ∂ F 1 ∂ u ( n ) i 6 = 0, a con tradictio n. Next, w e prov e part (b). The inclusion δ δu ( Z − 1 K ) ⊂ Ker  K ( ∂ )   F ℓ  immediately follo ws from part (a). F urthermore, if P ∈ K er  K ( ∂ )   F ℓ  , then C = R P i P i u i ∈ Z − 1 K is suc h that δC δu = P . He nce, δ δu  Z − 1 K  = Ker  K ( ∂ )   F ℓ  , as desired. Finally , for part (c), if V is normal, we ha v e by [BDSK09, Prop.1.5] that Ker  δ δu : V /∂ V → V ℓ ) = F /∂ F , hence, if ∂ F = F , w e conclude that δ δu : V /∂ V → V ℓ is inj ective .  T o simplify notation, let Z := Ker  K ( ∂ )  . Un d er the assu mptions of Theorem 4.3, by part (a) in Lemma 4.4, we ha ve Z ⊂ F ℓ , an d by part (c) w e ha v e a bijection (4.3) δ δ u : Z − 1 K ∼ − → Z , ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 19 the inv erse map b eing Z ∋ F =    f 1 . . . f ℓ    7→ X i R f i u i ∈ Z − 1 K . Lemma 4.5. If F 1 , . . . , F N ℓ ar e elements of F ℓ , line arly indep endent over C , and satisfying a differ ential e quation (4.4) F ( N ) = A 0 F + A 1 F ′ + · · · + A N − 1 F ( N − 1) , for some A 0 , . . . , A N − 1 ∈ Mat ℓ × ℓ ( F ) , then the ve ctors (4.5) G 1 :=      F 1 F ′ 1 . . . F ( N − 1) 1      , . . . , G N ℓ :=      F N ℓ F ′ N ℓ . . . F ( N − 1) N ℓ      ∈ F N ℓ ar e line arly indep endent over F . Pr o of. Sup p ose b y cont radiction that (4.6) a 1 G 1 + a 2 G 2 + · · · + a N ℓ G N ℓ = 0 , is a nontrivial relation of linear dep endence o v er F . W e can assume, without loss of generalit y , th at suc h relation has minimal n um b er of nonzero co effi- cien ts a 1 , . . . , a N ℓ ∈ F , and that a 1 = 1. Note that equation (4.6) can b e equiv alen tly rewritten as the follo wing system of equations in F ℓ : (4.7) a 1 F 1 + a 2 F 2 + · · · + a N ℓ F N ℓ = 0 a 1 F ′ 1 + a 2 F ′ 2 + · · · + a N ℓ F ′ N ℓ = 0 . . . a 1 F ( N − 1) 1 + a 2 F ( N − 1) 2 + · · · + a N ℓ F ( N − 1) N ℓ = 0 Applying ∂ to b oth sides of equation (4.6), we get (4.8) a 1 G ′ 1 + a 2 G ′ 2 + · · · + a N ℓ G ′ N ℓ + a ′ 1 G 1 + a ′ 2 G 2 + · · · + a ′ N ℓ G N ℓ = 0 . The ve ctor a 1 G ′ 1 + a 2 G ′ 2 + · · · + a N ℓ G ′ N ℓ is an elemen t of F N ℓ whose fi rst ℓ co ordinates are a 1 F ′ 1 + a 2 F ′ 2 + · · · + a N ℓ F ′ N ℓ , w hic h are zero by the second equation in (4.7 ), the s econd ℓ co ordin ates are a 1 F (2) 1 + a 2 F (2) 2 + · · · + a N ℓ F (2) N ℓ , whic h are zero by the third equation in (4.7), and so on, up to th e last set of ℓ coord inates, whic h are, b y the equation (4.4) , a 1 F ( N ) 1 + a 2 F ( N ) 2 + · · · + a N ℓ F ( N ) N ℓ = A 0  a 1 F 1 + a 2 F 2 + · · · + a N ℓ F N ℓ  + A 1  a 1 F ′ 1 + a 2 F ′ 2 + · · · + a N ℓ F ′ N ℓ  + · · · + A N − 1  a 1 F ( N − 1) 1 + a 2 F ( N − 1) 2 + · · · + a N ℓ F ( N − 1) N ℓ  , whic h is zero again b y the equations (4.7). Hence, equation (4.8) reduces to a ′ 1 G 1 + a ′ 2 G 2 + · · · + a ′ N ℓ G N ℓ = 0 , 20 ALBER TO DE SOLE, VICTOR G. KAC whic h, by the assu mption that a 1 = 1 and the minimalit y assumption on the co efficien ts of linear dep endence (4.6) , implies that all co efficien ts a 1 , . . . , a N ℓ are constan t. This, b y the fi rst equation in (4.7), con tradicts th e assumption that F 1 , . . . , F N ℓ are linearly indep enden t o ver C .  Lemma 4.6. If P ( ∂ ) i s a quasic onsta nt m × ℓ ( m ≥ 1 ) matrix diffe r ential op er ator of or der at most N − 1 such that P ( ∂ ) F = 0 f or ev ery F ∈ Z = Ker  K ( ∂ )  , then P ( ∂ ) = 0 . Pr o of. Recall from [DSK11, C or.A.3.7] th at, if K ( ∂ ) = K 0 + K 1 ∂ + · · · + K N ∂ N , with K i ∈ Mat ℓ × ℓ ( F ) , i = 0 , . . . , N and K N in v ertible, then the set of solutions in F ℓ of the homogeneous system K ( ∂ ) F = 0 is a v ect or space o v er C of dimension N ℓ . Let F 1 , . . . , F N ℓ ∈ F ℓ b e a b asis of this space. Note that the equation K ( ∂ ) F = 0 has the form (4.4) with A i = − K − 1 N K i , i = 0 , . . . , N − 1. Hence, by Lemma 4.5, all the v ecto rs G 1 , . . . , G N ℓ in (4.5) are linearly indep endent ov er F , i.e. the W ronskian matrix W =     F 1 F 2 . . . F N ℓ F ′ 1 F ′ 2 . . . F ′ N ℓ . . . F ( N − 1) 1 F ( N − 1) 2 . . . F ( N − 1) N ℓ     is n on d egenerate. By assumption P ( ∂ ) F 1 = · · · = P ( ∂ ) F N ℓ = 0. Hence, letting P ( ∂ ) = P 0 + P 1 ∂ + · · · + P N − 1 ∂ N − 1 , wh ere P i ∈ Mat m × ℓ ( F ), we get  P 0 , P 1 , . . . , P N − 1  W = 0 , whic h, by the nondegeneracy of W , implies that P 0 = , · · · = P N − 1 = 0.  Pr o of of The or em 4.3. Let Q ∈ A k K . Recalling Theorem 3.5 and Pr op osition 4.1(b), it suffices to sho w that, if Q is essen tial, then it is zero. By the definition of A k K , w e hav e, in particular, that Q is an array with en tries Q i 0 ,...,i k ( λ 0 , . . . , λ k ) = X j ∈ I P j,i 0 ,...,i k ( λ 0 , . . . , λ k ) u j ∈ V [ λ 0 , . . . , λ k ] / ( ∂ + λ 0 + · · · + λ k ) V [ λ 0 , . . . , λ k ] , for some p olynomials P j,i 0 ,...,i k ( λ 0 , . . . , λ k ) ∈ F [ λ 0 , . . . , λ k ] of degree at most N − 1 in eac h v ariable λ i . Recalling formula (3.12), we hav e, for arbitrary C 0 , . . . , C k ∈ V /∂ V , (4.9) [ . . . [[ Q, C 0 ] , C 1 ] , . . . , C k ] = X j,i 0 ,...,i k ∈ I Z u j P j,i 0 ,...,i k ( ∂ 0 , . . . , ∂ k ) δ C 0 δ u i 0 . . . δ C k δ u i k , where ∂ s means ∂ acting on δC s δu i s . Hence, if Q is essential , (4.9 ) is zero for all C 0 , . . . , C k ∈ Z − 1 K . By Lemma 4.4, we th us h a v e X j,i 0 ,...,i k ∈ I Z u j P j,i 0 ,...,i k ( ∂ 0 , . . . , ∂ k ) F 0 . . . F k = 0 , ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 21 for all F 0 , . . . , F k ∈ Ker  K ( ∂ )   F ℓ  . Sin ce all co efficien ts of the P j,i 0 ,...,i k ’s and all en tries of the F i ’s are quasiconstan t, the ab o ve equation is equiv alent to X i 0 ,...,i k ∈ I P j,i 0 ,...,i k ( ∂ 0 , . . . , ∂ k ) F 0 . . . F k = 0 , ∀ j ∈ I . Applying Lemma 4.6 iterativ ely to eac h factor, we conclude that the p oly- nomials P j,i 0 ,...,i k ( λ 0 , . . . , λ k ) are zero.  R emark 4.7 . By Remark 4.2, fr om the p oint of view of applicabilit y of th e Lenard-Magri scheme for a bi-Hamiltonian pair ( H , K ), we should consider only essen tia lly closed Hamiltonian op erators H ( ∂ ). Moreo v er, by Theorem 4.3, if K ( ∂ ) is a quasiconstan t matrix differentia l op erator with inv ertible leading co efficien t, an essentia lly closed H ( ∂ ) m ust b e exact, namely , r ecall- ing equation (3.20), it must ha v e the form H ( ∂ ) = D P ( ∂ ) ◦ K ( ∂ ) + K ( ∂ ) ◦ D ∗ P ( ∂ ) , for some P ∈ V ℓ , and tw o suc h P ’s d iffer b y an elemen t of the form K ( ∂ ) δf δu for some R f ∈ V /∂ V . Corollary 4.8. U nder the assumptions of The or em 4.3, the Z -gr ade d Lie sup er algebr a H • K ( V ) is tr ansitive. Pr o of. By T h eorem 4.3, if P ∈ H k K is such that [ . . . [[ P , C 0 ] , C 1 ] , . . . , C k ] = 0 for ev ery C 0 , . . . , C k ∈ Z − 1 K = H − 1 K , then P = 0. Th is, b y definition, m eans that H • K ( V ) is transitive.  5. Isomorphism of Z -graded Lie s uperalgeb ras H • K ( V ) ≃ e H ( N ℓ, S ) In this s ection we introdu ce an inner pro du ct h ·|· i K : F ℓ × F ℓ → F asso ciated to an ℓ × ℓ matrix differen tial op erator K =  K ij ( ∂ )  i,j ∈ I , wh ic h is us ed to pro v e Theorem 3.6. 5.1. T he inner pro duct asso ciated to K . Let F b e a different ial algebra with d eriv ation ∂ , and denote by C the su balgebra of constan ts. As u sual, w e denote b y · the standard inn er pr o duct on F ℓ , i.e. F · G = P i ∈ I F i G i ∈ V for F , G ∈ V ℓ , wh ere, as b efore, I = { 1 , . . . , ℓ } . Consider the algebra of p olynomia ls in t w o v ariables F [ λ, µ ]. Clearly , the map λ + µ + ∂ : F [ λ, µ ] → F [ λ, µ ] is injectiv e. Hence, giv en P ( λ, µ ) ∈ ( λ + µ + ∂ ) F [ λ, µ ], there is a u nique preimage of this map in F [ λ, µ ], that w e denote b y ( λ + µ + ∂ ) − 1 P ( λ, µ ) ∈ F [ λ, µ ]. Let no w K ( ∂ ) =  K ij ( ∂ )  i,j ∈ I b e an arbitrary ℓ × ℓ matrix d ifferen tial op erator ov er F . W e expand its matrix en tries as (5.1) K ij ( λ ) = N X n =0 K ij ; n λ n , K ij ; n ∈ F . 22 ALBER TO DE SOLE, VICTOR G. KAC The adjoint op erato r is K ∗ ( ∂ ), with en tr ies (5.2) K ∗ ij ( λ ) = K j i ( − λ − ∂ ) = N X n =0 ( − λ − ∂ ) n K j i ; n . It f ollo ws f rom the expans ions (5.1 ) and (5.2 ) that, for ev ery i, j ∈ I , the p olynomial K ij ( µ ) − K ∗ j i ( λ ) lies in the image of λ + µ + ∂ , so that w e can consider th e p olynomial (5.3) ( λ + µ + ∂ ) − 1  K ij ( µ ) − K ∗ j i ( λ )  ∈ F [ λ, µ ] . Next, for a p olynomia l P ( λ, µ ) = P N m,n =0 p mn λ m µ n ∈ F [ λ, µ ], we use the follo win g n otation (5.4) P ( λ, µ )  | λ = ∂ f  | µ = ∂ g  := N X m,n =0 p mn ( ∂ m f )( ∂ n g ) .  λ | = ∂ f  Based on the observ ation (5.3) , and using th e notation in (5.4), we define the follo win g in ner pro duct h ·|· i K : F ℓ × F ℓ → F , associated to K =  K ij ( ∂ )  i,j ∈ I ∈ Mat ℓ × ℓ ( F [ ∂ ]): (5.5) h F | G i K = X i,j ∈ I ( λ + µ + ∂ ) − 1  K ij ( µ ) − K ∗ j i ( λ )  | λ = ∂ F i  | µ = ∂ G j  . It is not hard to write an explicit formula for h F | G i K , using the expansion (5.1) f or K ij ( λ ): (5.6) h F | G i K = X i,j ∈ I N X n =0 n − 1 X m =0  n m  ( − ∂ ) n − 1 − m ( F i K ij ; n ∂ m G j ) . Lemma 5.1. F or every F , G ∈ V ℓ , we have ∂ h F | G i K = F · K ( ∂ ) G − G · K ∗ ( ∂ ) F . Pr o of. It immediately follo ws from the definition (5.5) of h F | G i K .  Lemma 5.2. F or every K ( ∂ ) ∈ Mat ℓ × ℓ ( F [ ∂ ]) and F , G ∈ F ℓ , we have h G | F i K ∗ = −h F | G i K . In p articular, the inner pr o duct h ·|· i K is symmetric (r esp e ctively skewsym- metric) if K is skewadjoint (r esp. selfadjoint). Pr o of. By equ ation (5.5 ) w e ha v e h G | F i K ∗ = X i,j ∈ I ( λ + µ + ∂ ) − 1  K ∗ ij ( µ ) − K j i ( λ )  | λ = ∂ G i  | µ = ∂ F j  = − X i,j ∈ I ( λ + µ + ∂ ) − 1  K ij ( µ ) − K ∗ j i ( λ )  | λ = ∂ F i  | µ = ∂ G j  = −h F | G i K .  ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 23 F ollo w ing the notation of the previous sections, we let Z = Ker  K ( ∂ )  ⊂ F ℓ . Clearly , Z is a submo dule of the C -mo du le F ℓ . Lemma 5.3. If K ( ∂ ) ∈ Mat ℓ × ℓ ( F [ ∂ ]) is skewadjoint, then h F | G i K ∈ C for every F , G ∈ Z Pr o of. It is an immediate consequence of Lemma 5.1.  According to Lemmas 5.2 and 5.3, if K ( ∂ ) ∈ Mat ℓ × ℓ ( F [ ∂ ]) is s k ew adjoint, the restriction of h ·|· i K to Z ⊂ F ℓ defines a symmetric bilinear form on Z with v alues in C , which w e d enote by h ·|· i 0 K := h ·|· i K   Z : Z × Z → C . Lemma 5.4. Assuming that K ( ∂ ) ∈ Mat ℓ × ℓ ( F [ ∂ ]) is a skewadjoint op er ator and P ( ∂ ) ∈ Mat ℓ × ℓ ( F [ ∂ ]) is such that K ( ∂ ) P ( ∂ ) + P ∗ ( ∂ ) K ( ∂ ) = 0 , we have h P ( ∂ ) F | G i K + h F | P ( ∂ ) G i K = 0 for every F , G ∈ F ℓ . Pr o of. By equ ation (5.5 ), we hav e h P ( ∂ ) F | G i K = X i,j,k ∈ I ( λ + µ + ∂ ) − 1  K k j ( µ ) + K j k ( λ )  | λ = ∂ P k i ( ∂ ) F i  | µ = ∂ G j  = X i,j,k ∈ I ( λ + µ + ∂ ) − 1  K k j ( µ ) + K j k ( λ + ∂ )  P k i ( λ )  | λ = ∂ F i  | µ = ∂ G j  = X i,j,k ∈ I ( λ + µ + ∂ ) − 1  P k i ( λ ) K k j ( µ ) − P ∗ j k ( λ + µ ) K k i ( λ )  | λ = ∂ F i  | µ = ∂ G j  . In the last identit y we used the assump tion that K ( ∂ ) P ( ∂ ) = − P ∗ ( ∂ ) K ( ∂ ). Similarly , h F | P ( ∂ ) G i K = X i,j,k ∈ I ( λ + µ + ∂ ) − 1 ×  − P ∗ ik ( µ + ∂ ) K k j ( µ ) + P k j ( µ ) K k i ( λ )  | λ = ∂ F i  | µ = ∂ G j  . Com bining these t w o equations, we get (5.7) h P ( ∂ ) F | G i K + h F | P ( ∂ ) G i K = X i,j,k ∈ I ( λ + µ + ∂ ) − 1   P k i ( λ ) − P ∗ ik ( µ + ∂ )  K k j ( µ ) +  P k j ( µ ) − P ∗ j k ( λ + µ )  K k i ( λ )   | λ = ∂ F i  | µ = ∂ G j  . W e next observe that the differen tial op erato r P k i ( λ ) − P ∗ ik ( µ + ∂ ) lies in ( λ + µ + ∂ ) ◦ ( F [ λ, µ ])[ ∂ ], i.e. it is of the form P k i ( λ ) − P ∗ ik ( µ + ∂ ) = ( λ + µ + ∂ ) ◦ Q k i ( λ, µ + ∂ ) , for some p olynomial Q k i . Hence, ( λ + µ + ∂ ) − 1  P k i ( λ ) − P ∗ ik ( µ + ∂ )  K k j ( µ )  | µ = ∂ G j  = Q ik ( λ, ∂ ) K k j ( ∂ ) G j , 24 ALBER TO DE SOLE, VICTOR G. KAC whic h, after su mming with resp ect to j ∈ I , b ecomes zero since, b y assump- tion, G ∈ Ker( K ( ∂ )). S imilarly , ( λ + µ + ∂ ) − 1  P k j ( µ ) − P ∗ j k ( λ + µ )  K k i ( λ )  | λ = ∂ F i  = Q k j ( µ, ∂ ) K k i ( ∂ ) F i , whic h is zero after summin g with resp ect to i ∈ I , sin ce F ∈ Ker( K ( ∂ )). Therefore th e RHS of (5.7) is zero, proving the claim.  Prop osition 5.5. Assuming that F is a line arly close d differ ential field, and that K ( ∂ ) ∈ Mat ℓ × ℓ ( F [ ∂ ]) is a skewad joint ℓ × ℓ matrix diffe r ential op er ator with invertible le ading c o efficient, the C -bilie anr form h ·|· i 0 K : Z × Z → C is nonde gener ate. Pr o of. Giv en F ∈ F ℓ , co nsider the map P F : F ℓ → F giv en by G 7→ P F ( G ) = h F | G i 0 K . Equ ation (5.6 ) can b e rewritten b y sa ying that P F is a 1 × ℓ m atrix differen tial op erator, of order less than or equal to N − 1, with en tries ( P F ) j ( ∂ ) = X i ∈ I N X n =0 n − 1 X m =0  n m  ( − ∂ ) n − 1 − m ◦ F i K ij ; n ∂ m . Supp ose no w that P F ( G ) = h P | G i 0 K = 0 for all G ∈ Z ⊂ F ℓ . By Lemma 4.6 we get that P F ( ∂ ) = 0. On the other hand, the (left) co efficien t of ∂ N − 1 in ( P F ) j ( ∂ ) is 0 = X i ∈ I N − 1 X m =0  N m  ( − 1) N − 1 − m F i ( K N ) ij = X i ∈ I F i ( K N ) ij . Since, by assumption, K N ∈ Mat ℓ × ℓ ( F ) is in vertible, we conclude that F = 0.  5.2. Pro of of Theorem 3.6 . Recal l from Lemma 4.4 that H − 1 K = Z − 1 K is isomorphic, as a C -v ector space, to Z = Ker  K ( ∂ )  , an d , from T heorem 3.5, that dim C Z = N ℓ . By Corollary 4.3, the Z -graded Lie sup eralge bra H • K ( V ) is transitiv e, i.e. if P ∈ H k K , k ≥ 0, is suc h that [ P , H − 1 K ] = 0, then P = 0. Hence, due to transitivit y , the repr esen tati on of H 0 on H − 1 K = Z − 1 K is faithful. Iden tifying Z − 1 K ≃ Z , w e can therefore view H 0 K as a subalgebra of the Lie algebra g l ( Z ) = g l N ℓ . Recal l, from Theorem 3.5 that H 0 K ≃ A 0 K consists of elemen ts of the form Q =  P j P ∗ ij ( ∂ ) u j  i ∈ I ∈ V ℓ , w here P ( ∂ ) =  P ij ( ∂ )  i ∈ I is an ℓ × ℓ matrix differen tial op erator of order at most N − 1 solving equation (3.23). Moreo v er, b y (3.13), the br ack et of an elemen t Q ∈ H 0 K as ab o v e and an elemen t C ∈ Z − 1 K = H − 1 K ⊂ V /∂ V , is giv en by [ Q, C ] = X i,j ∈ I Z  P ∗ ij ( ∂ ) u j  δ C δ u i = X i,j ∈ I Z u i P ij ( ∂ ) δ C δ u j . Hence, b y th e id en tificatio n (4.3) , the corresp ond ing action of Q ∈ H 0 K on Z ⊂ F ℓ is simply giv en by the standard action of the ℓ × ℓ matrix differen tial op erator P ( ∂ ) on F ℓ . By Lemmas 5.2 and 5.3 and b y Prop osition 5.5, h ·|· i 0 K ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 25 is a nondegenerate symmetric b ilinear f orm on Z , and by Lemma 5.4 it is in v arian t with r esp ect to this action of Q ∈ H 0 K on Z . Hence, the image of H 0 K via the ab o ve em b edding H 0 K → gl ( Z ), is a subalgebra of s o ( Z , h ·|· i 0 K ). Due to transitivit y of th e Z -graded Lie su p eralgebra H • K ( V ), it emb eds in the full p rolongation of the pair  Z , so ( Z , h ·|· i 0 K )  , whic h, b y Pr op osition 2.2, is isomorphic to e H ( N ℓ, S ), w h ere S is the N ℓ × N ℓ matrix of the b ilinear form h ·|· i 0 K , in some b asis. By Theorem 3.5, d im C H k K =  N ℓ k +2  , wh ic h is equal to dim C e H k ( N ℓ, S ). W e thus conclude that the Z -graded Lie sup eralgebras H • K ( V ) and e H ( N ℓ, S ) are isomorphic. R emark 5.6 . Th e same arguments as ab o v e s h o w that, without an y as- sumption on the algebra of differential fun ctions V and on the differen- tial field F (with subfi eld of constan ts C ), and for ev ery Hamiltonian op- erator K (n ot n ecessarily quasiconstan t n or with in v ertible leading co effi- cien t), we h a ve an injective homomorphism of Z -graded Lie su p eralgebras H • K ( V ) / E H • K ( V ) → W ( n ), wh er e n = d im C ( H − 1 K ). 6. Transla tion inv ariant v aria tional Poiss on co homology In the p revious sections we stud ied the v ariational P oisson cohomology e H • K ( V ) in the sim p lest case when the d ifferential field of quasiconstan ts F ⊂ V is linearly closed. In this section we consider the other extreme case, often stud ied in literature – the translation inv arian t case, when F = C . 6.1. Upp er b ound of the dimensio n of the translation inv ariant v ariational P oisson cohomology . Let V b e a normal algebra of differen- tial functions, and assume that it is tr anslation invariant , i.e. the d ifferen tial field F of quasiconstan ts coincides with the fi eld C of constant s. Let K ( ∂ ) b e an ℓ × ℓ m atrix d ifferen tial op erator of order N , with coefficien ts in Mat ℓ × ℓ ( C ), and with in v ertible leading co efficien t K N . F or k ≥ − 1, denote by e H k the space of arra ys  P i 0 ,...,i k ( λ 0 , . . . , λ k )  i 0 ,...,i k ∈ I with en tries P i 0 ,...,i k ( λ 0 , . . . , λ k ) ∈ C [ λ 0 , . . . , λ k ], of degree at most N − 1 in eac h v ariable, whic h are sk ewsymmetric with resp ect to sim ultaneous p erm u - tations of the indices i 0 , . . . , i k and the v ariables λ 0 , . . . , λ k (in the n otation of [DSK 11], e H k = e Ω k − 1 0 , 0 ). In particular, e H − 1 = C . No te that, for k ≥ − 1, w e ha v e (6.1) dim C e H k =  N ℓ k + 1  . The long exact s equence [DSK 11, eq.(11.4)] b eco mes (in the notation of the pr esen t pap er): (6.2) 0 → C β − 1 − → H − 1 K γ − 1 − → e H 0 α 0 − → e H 0 β 0 − → . . . · · · γ k − 1 − → e H k α k − → e H k β k − → H k K γ k − → e H k +1 α k +1 − → e H k +1 β k +1 − → . . . 26 ALBER TO DE SOLE, VICTOR G. KAC F or eve ry k ≥ − 1, w e hav e dim C ( H k K ) = dim C (Ker γ k ) + dim C (Im γ k ). By exactness of the sequence (6.2), w e ha v e th at dim C (Im γ k ) = dim C (Ker α k +1 ), and dim C (Ker γ k ) = dim C (Im β k ). Moreo ver, d im C (Im β − 1 ) = 1 and, for k ≥ 0, w e ha v e, again b y exactness of (6.2), that d im C (Im β k ) = dim C e H k − dim C (Ker β k ) = dim C e H k − dim C (Im α k ) = dim C (Ker α k ). Hence, using (6.1) w e conclude that (6.3) dim C ( H − 1 K ) = 1 + dim C (Ker α 0 ) ≤ N ℓ + 1 , and, f or k ≥ 0 (by th e T artaglia-P ascal triangle), (6.4) dim C ( H k K ) = dim C (Ker α k ) + d im C (Ker α k +1 ) ≤  N ℓ + 1 k + 2  . Recalling equation (4.1), we h a ve H − 1 K = Z − 1 K = R f ∈ V /∂ V   K ( ∂ ) δf δu = 0  . By Lemma 4.4(b) we ha v e a su rjectiv e map δ δu : H − 1 K → Ker  K ( ∂ )   C ℓ  . Recall that, if V is a normal alge bra of d ifferen tial functions, w e ha ve Ker  δ δu : V → V ℓ  = C + ∂ V [BDSK09]. It follo ws that Ker  δ δu   H − 1 K  = Ker  δ δu   V /∂ V  ≃ C . Therefore, H − 1 K = R C ⊕  R uA   A ∈ Ker( K 0 ) ⊂ C ℓ  , where, u = ( u 1 , . . . , u ℓ ), and K 0 = K (0) is th e constan t co efficien t of the differen tial op erator K ( ∂ ). Hence, (6.5) dim C ( H − 1 K ) = 1 + dim C (Ker K 0 ) = 1 + ℓ − rk( K 0 ) . In conclusion, the inequalit y in (6.3) is a strict inequalit y unless K ( ∂ ) has or- der 1 with K 0 = 0, i.e. K ( ∂ ) = S ∂ , where S ∈ Mat ℓ × ℓ ( C ) is a nondegenerate matrix. R emark 6.1 . Th e map α k : e H k → e H k can b e constructed as follo ws [DSK11]. Let P =  P i 0 ,...,i k ( λ 0 , . . . , λ k )  i 0 ,...,i k ∈ I b e in e H k , i.e. P i 0 ,...,i k ( λ 0 , . . . , λ k ) are p olynomials of degree at most N − 1 in eac h v ariable λ i with co efficients in C , skewsymmetric with resp ect to sim ultaneous p ermutatio ns in th e in- dices i 0 , . . . , i k and th e v ariables λ 0 , . . . , λ k . Then, there exist a unique elemen t α k ( P ) := R =  R i 0 ,...,i k ( λ 0 , . . . , λ k )  i 0 ,...,i k ∈ I ∈ e H k and a (unique) arra y Q =  Q j,i 1 ,...,i k ( λ 1 , . . . , λ k )  j,i 1 ,...,i k ∈ I , wh ere Q j,i 1 ,...,i k ( λ 1 , . . . , λ k ) are p olynomials of d egree at most N − 1 in eac h v ariable, with co efficien ts in C , sk ewsymmetric with resp ect of simultaneous p erm u tations of the ind ices i 1 , . . . , i k and the v ariables λ 1 , . . . , λ k , s u c h that the follo wing iden tit y h olds in C [ λ 0 , . . . , λ k ]: (6.6) ( λ 0 + · · · + λ k ) P i 0 ,...,i k ( λ 0 , . . . , λ k ) = R i 0 ,...,i k ( λ 0 , . . . , λ k ) + k X α =0 ( − 1) α X j ∈ I Q j,i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . ., λ k ) K j i α ( λ α ) . ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 27 Hence, Ker( α k ) is in b ijection with the space Σ k of arr a ys Q as ab o ve , satisfying the condition: k X α =0 ( − 1) α X j ∈ I Q j,i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . ., λ k ) K j i α ( λ α ) ∈ ( λ 0 + · · · + λ k ) C [ λ 0 , . . . , λ k ] . F or example, Σ 0 =  Q ∈ C ℓ   K T 0 Q = 0  , hence its dimens ion equ als dim C (Ker α 0 ) = dim(Ker K 0 ) = ℓ − r k( K 0 ) (in accordance w ith (6.5)). F ur- thermore, Σ 1 consists of p ol ynomials Q ( λ ) with coefficients in Mat ℓ × ℓ ( C ), of degree at most N − 1, suc h that K T ( − λ ) Q ( λ ) = Q T ( − λ ) K ( λ ) . R emark 6.2 . It is clear from Remark 6.1 that, wh ile in the linearly closed case, the Lie su p eralgebra H • K ( V ) d ep ends only on ℓ and the ord er N of K ( ∂ ), in th e translation inv ariant case F = C the dimen s ion of H • K ( V ) dep end s essentia lly on the op erat or K ( ∂ ). Hence, in this sense, the c hoice of an algebra V ov er a linearly closed differentia l field F seems to b e a more natural one. This is the k ey message of the pap er. In the next section w e stud y in more detail the v ariati onal P oisson co- homology H k K , and its Z -graded Lie sup eralgebra structure, for a “h ydr- dynamic t yp e” Hamilt onian operator, i.e . for K ( ∂ ) = S ∂ , where S ∈ Mat ℓ × ℓ ( C ) is nondegenerate and symm etric. 6.2. T ranslation inv ariant v ariational P oisson cohomology for K = S ∂ . As in the previous section, let V b e a translation inv arian t n ormal algebra of different ial f unctions, with field of constan ts C (whic h coincides with th e fi eld of quasiconstan ts). Let S ∈ Mat ℓ × ℓ ( C ) b e nondegenerate and symmetric, and consider the Hamiltonian op erator K ( ∂ ) = S ∂ . F or k ≥ − 1, w e den ote by Λ k +1 the space of sk ewsymmetric ( k + 1)- linear forms on C ℓ , i.e . th e sp ace of arra ys B =  b i 0 ,...,i k  i 0 ,...,i k ∈ I , to - tally sk ewsymmetric with r esp ect to p ermutatio ns of the indices i 0 , . . . , i k . F or k ≥ 0, w e also denote b y Λ k +1 S the space of arr a ys of the form A =  a j,i 1 ,...,i k  j,i 1 ,...,i k ∈ I , w hic h are sk ewsymmetric with resp ect to p ermutations of the indices i 1 , . . . , i k , and whic h satisfy the equation X j ∈ I s i 0 ,j a j,i 1 ,i 2 ...,i k = − X j ∈ I a j,i 0 ,i 2 ,...,i k s j,i 1 . Clearly , dim C (Λ k +1 S ) = dim C (Λ k +1 ) =  ℓ k +1  for ev ery k ≥ − 1. F or example, Λ 0 = C , Λ 1 S = Λ 1 = C ℓ , Λ 2 is the space of sk ewsymmetric ℓ × ℓ matrices o v er C , and Λ 2 S =  A ∈ Mat ℓ × ℓ ( C )   A T S + S A = 0  = so ( ℓ, S ) . Giv en A =  a j,i 0 ,...,i k  j,i 0 ,...,i k ∈ I ∈ Λ k +2 S , w e denote uA =  X j ∈ I u j a j,i 0 ,...,i k  i 0 ,...,i k ∈ I ∈ W var k . 28 ALBER TO DE SOLE, VICTOR G. KAC Let A • = L ∞ k = − 1 A k , wh ere A k = Λ k +1 ⊕  uA   A ∈ Λ k +2 S  ⊂ W var k , k ≥ − 1 . Theorem 6.3. L et V b e trnslation invariant normal algebr a of differ ential functions, and let K ( ∂ ) = S ∂ , wher e S is a symmetric nonde gener ate ℓ × ℓ matrix over C . Then: (a) A • is a sub algebr a of the Z -gr ade d Lie su p er algebr a Z • K ( V ) , c omplemen- tary to the ide al B • K ( V ) . In p articular, we have the fol lowing de c omp o- sition of Z k K in a dir e ct sum of ve ctor sp ac es over C : Z k K = A k ⊕ B k K . (b) W e have an isomo rphism of Z -gr ade d Lie sup er algebr as (cf. Se c ction 2.2): H • K ( V ) = A • ≃ e H ( ℓ + 1 , e S ) , wher e e S is the ( ℓ + 1) × ( ℓ + 1) matrix obtaine d fr om S by adding a zer o r ow and c olumn. In p articular, dim C ( H k K ) =  l +1 k +2  . Pr o of. F or B ∈ Λ k +1 , w e obviously hav e δ K B = 0. Moreo v er, it is immediate to chec k, using the formula (3.19) for δ K , that, if A ∈ Λ k +2 S , then δ K ( uA ) = 0. Hence, A k ⊂ Z k K for eve ry k ≥ − 1. Next, we compute the b ox pro duct (3.10) b et ween tw o elemen ts of A • . Let B ⊕ uA ∈ Λ h +1 ⊕ u Λ h +2 S = A h , and D ⊕ uC ∈ Λ k − h +1 ⊕ u Λ k − h +2 S = A k − h . W e ha v e B  D = 0, uA  D = 0, moreo ver, B  uC ∈ Λ k +1 ⊂ A k and uA  uC ∈ u Λ k +2 S ⊂ A are giv en by (6.7) ( B  uC ) i 0 ,...,i k = X σ ∈S h,k sign( σ ) X j ∈ I b j,i σ ( k − h +1) ,...,i σ ( k ) c j,i σ (0) ,...,i σ ( k − h ) , ( uA  uC ) i 0 ,...,i k = X σ ∈S h,k sign( σ ) X i,j ∈ I u i a i,j,i σ ( k − h +1) ,...,i σ ( k ) c j,i σ (0) ,...,i σ ( k − h ) . W e thus conclud e that A • = L k ≥− 1 A k is a subalgebra of the Z -graded Lie sup eralgebra Z • ( V ) ⊂ W v ar (Π V ). Since S − 1 ,k +1 = ∅ , w e hav e that A − 1  A • = 0. Moreo v er, S − 1 ,k +1 = { 1 } . Hence, for d ⊕ uC ∈ C ⊕ u C ℓ = A − 1 and B ⊕ uA ∈ Λ k +1 ⊕ u Λ k +2 S = A k , w e ha v e [ B ⊕ uA, d ⊕ uC ] = B  ( uC ) ⊕ ( uA  uC ) ∈ Λ k ⊕ u Λ k +1 S = A k − 1 , with entries (6.8) [ B , uC ] i 1 ,...,i k = ( B  uC ) i 1 ,...,i k = X j ∈ I b j,i 1 ,...,i k c j , [ uA, uC ] i 1 ,...,i k = ( uA  uC ) i 1 ,...,i k = X i,j ∈ I u i a i,j,i 1 ,...,i k c j . It is clear, from form ula (6.8) , that [ B ⊕ uA, uC ] = 0 for ev ery C ∈ C ℓ if and only if A = 0 and B = 0. Hence A • is a transitiv e Z -graded L ie sup eralgebra. ESSENTIAL V ARIA TIONAL POISSON COHO M OLOGY 29 Since [ B k K , Z − 1 K ] = 0, it follo ws, in particular, th at A k ∩ B k K = 0 for ev ery k ≥ − 1. Hence A k coincides with its image in H k K ( V ), and A • can b e view ed as a su balgebra of the Z -graded Lie sup eralge bra H • K ( V ). Th erefore (by the T artaglia-P ascal triangle) dim C H k K ≥ dim C A k =  ℓ +1 k +2  . Sin ce, b y (6.4), dim C H k K ≤  ℓ +1 k +2  , we conclude that all th ese inequalities are equalities, and that H • ( V ) ≃ A • are isomorphic Z -graded Lie su p eralgebras. T o conclude, in view of Pr op osition 2.2, w e need to pro v e th at A • is the full p rolongation of the p air ( C ℓ +1 , so ( ℓ + 1 , e S ), where e S is the ( ℓ + 1) × ( ℓ + 1) matrix obtained adding a zero r o w and column to S . W e hav e C ℓ +1 = C ⊕ C ℓ , and so ( ℓ + 1 , e S ) = n  0 B T 0 A     B ∈ C ℓ , A ∈ s o ( ℓ, S ) o ≃ C ℓ ⊕ so ( ℓ, S ) , with th e Lie brac k et of B ⊕ A ∈ C ℓ ⊕ so ( ℓ, S ) and d ⊕ C ∈ C ⊕ C ℓ giv en b y (6.9) [ B + A, d + C ] = B · C ⊕ AC ∈ C ⊕ C ℓ . By definition, we ha v e A 0 = Λ 1 ⊕ u Λ 2 S = C ℓ ⊕ u · so ( ℓ, S ), and the action of B ⊕ uA ∈ C ℓ ⊕ u · so ( ℓ, S ) on d ⊕ uC ∈ C ⊕ u C ℓ = A − 1 , giv en by (6.8), is [ B ⊕ uA, d ⊕ uC ] i = B · C ⊕ uAC . Namely , in view of (6.9), it is induced b y the n atural act ion of so ( ℓ + 1 , e S ) ≃ C ℓ ⊕ so ( ℓ, S ) on C ⊕ C ℓ . Hence, A − 1 ⊕ A 0 ≃ ( C ⊕ C ℓ ) ⊕ ( C ℓ ⊕ so ( n, S )). Sin ce A • is a transitive Z -graded Lie sup eralgebra, it is a subalgebra of the full p rolongation of ( C ℓ +1 , so ( ℓ + 1 , e S ). On the other hand, by Prop ositio n 2.2 the fu ll prolongation of ( C ℓ +1 , so ( ℓ + 1 , e S ) is isomorph ic to e H ( ℓ + 1 , e S ), and dim C e H ( ℓ + 1 , e S ) = 2 ℓ +1 − 1 = P k ≥− 1 dim C A k . Hence, A • m ust b e isomorphic to e H ( ℓ + 1 , e S ), as we wan ted.  Corollary 6.4. Under the assumptions of The or em 6.3, the essential vari- ational c ohomolo gy E H • K ( V ) is zer o. Pr o of. It im m ediately f ollo ws fr om the transitivit y of the Z -graded Lie su- p eralgebra H • K ( V ).  R emark 6.5 . If S is a nond egenerate, but not n ecessarily symmetric, ℓ × ℓ matrix, we still h av e an isomorph ism of vec tor sp aces H k K ≃ A k , but H • K ( V ) is not, in general, a Lie sup eralgebra. R emark 6.6 . The description of H • K ( V ), as a v ecto r sp ace, for K = S ∂ with S symmetric nondegenerate matrix ov er C , agrees with the results of S .-Q. Liu and Y. Z hang [LZ 11, LZ11pr]. Referen ces [BDSK09] A . Barak at, A. De Sole, and V.G. Kac, Poisson vertex algebr as i n the the ory of Hami ltonian e quat ions , Japan. J. Math. 4 , (2009) 141–252. [DMS05] L. Degiov anni, F. Magri, and V. Sciacca, On deformation of Poisson manifolds of hydr o dynamic typ e Comm. Math. Phys. 253 , (2005) no. 1, 1–24. 30 ALBER TO DE SOLE, VICTOR G. KAC [DSK06] A. De Sole, and V.G. Kac, Finite vs. affine W -algebr as , Japan. J. Math. 1 , (2006) 137–261 . [DSK09] A. De Sole, and V.G. Kac, Lie c onformal algebr a c ohomolo gy and the variational c omplex , Commun. Math. Ph ys. 292 , (2009) 667–71 9. [DSK11] A. De Sole, an d V.G. Kac, The variat ional Poisson c ohomolo gy , (2011) arXiv:1106.00 82 . [DSKW10] A. D e S ole, V.G. Kac, and M. W akimoto, On cl assific ation of Poisson vertex algebr as , T ransform. Groups 15 (2010), n o. 4, 883–907. [Dor93] I.Y a. Dorfman, Dir ac structu r es and i nte gr abil ity of nonli ne ar evolution e quations , Nonlinear Sci. Theory Appl. (John Wiley & Sons, 1993) 176 pp. [Get02] E.Getzler, A Darb oux the or em for Hamiltonian op er ators in the formal c alculus of variations , Du ke Math. J. 111 (2002), no. 3, 535–56 0. [GS64] V.W. Guillemin, S . Sternberg, An algebr aic mo del of tr ansitive differ ential ge om- etry Bull. Amer. Math. So c. 70 (1964) 16–47. [Kac77] V.G. Kac, Li e sup er algebr as Adva nces in Math. 26 (1977), no. 1, 8–96. [Kra88] I.S. Krasilshc hik, Schouten br ackets and c anonic al algebr as , Lecture Notes in Math. 1334 , (Springer V erlag, New Y ork 1988). [Kup80] B.A. Kup ershmidt, Ge ometry of jet bund les and the struc tur e of L agr angian and Hamiltonian f ormalisms , in Ge ometric metho ds in Mathematic al Physics , Lecture Notes in Math. 775 , (Springer V erlag, N ew Y ork 1980) 162–218. [Lic77] A . Lichnerow icz, Le s varietes de Poisson et leur algebr es de Li e asso cie es , J. Diff. Geom. 12 , (1977) 253–300 . [LZ11] S.-Q . Liu, and Y. Zh ang, Jac obi struct ur es of evolutionary p artial differ ential e qua- tions , Adv. Math. 227 (2011), 73–130. [LZ11pr] S .-Q. Liu, and Y. Zhang, priv ate comm unication, Beijing, June 2011. [Mag78] F. Magri, A simple mo del of the inte gr able Ham iltonian e quation , J. Math. Phys. 19 , (1978) 129–13 4. [Olv87] P .J. Olver, BiHamiltonian systems , in Or dinary and p artial differ ential e quations , Pitman Research Notes in Mathematics Series 157 , (L on gman Scientific an d T ec hnical, New Y ork 1987) 176–1 93.