The variational Poisson cohomology

THE V ARIA TIONAL POISSON COHOMO LOGY ALBER TO DE SOLE 1 AND VI CTOR G. KAC 2 T o the m emory of Boris Kup ershmidt (11/27/1 946 – 12/12 /2010 ) Abstra ct. It is well kno wn that the va lidit y of the so called Lenard- Magri scheme of integrabilit y of a bi-Hamiltonian PDE can be estab- lished if one has some precise information on the corresponding 1st v ari- ational P oisson cohomology for on e of th e tw o Hamiltonian op erators. In the ﬁrst part of th e pap er w e ex plain how to introduce v arious coho- mology complexes, including L ie sup eralgebra and P oisson cohomology complexes, and basic and red u ced Lie conformal algebra and Poisson vertex a lgebra cohomolog y complexes, b y making use o f the correspond- ing u niversal Lie sup eralgebra or Lie conformal sup eralgebra. The most relev ant are certain subcomplexes of the b asic and reduced Poisson ver- tex algebra cohomology complexes, whic h we iden tify (non-canonically) with the generalized de Rham complex and the generalized v ariati onal complex. In the second part of th e pap er we compute th e cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequ ence, w e compu te the cohomology of the generalized vari a- tional complex for an y quasiconstant coeﬃcient Hamiltonian op erator with inve rtible leading co eﬃcient. F or the latter we use some diﬀerential linear algebra dev eloped in the App end ix. 1. Dipartimento di Matematica , Universit` a di Roma “La Sapienza”, 00185 Roma, Italy desole@mat.uniroma1.it . 2. Department of Mathematics, M.I.T., Cam bridge, MA 02139, USA k ac@math .mit.edu . A.D.S. w as partially supp orted by PRIN and AST grants. V .K. was partially supported by an NSF grant, and an ER C advanced grant. Parts of this wo rk were done while A.D.S. w as visiting the Department of Mathematics of M.I.T., while V .K. was v isiting the newly created Center for Mathematics and Theoretical Physics in R ome, and A.D.S and V.K. w ere visiting the MSC and the Department of Mathematics of Tsinghua U niversit y in Beijing. 1 2 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Contents 1. In tro duction 4 2. The univ ersal L ie sup eralgebra W ( V ) for a ve ctor sup er s pace V , and Lie su p eralgebra cohomolo gy 12 2.1. The u niv ersal L ie sup eralgebra W ( V ) 12 2.2. The sp ace W ( V , U ) as a redu ction of W ( V ⊕ U ) 14 2.3. Prolongations 15 2.4. Lie su p eralgebra structures 15 2.5. Lie su p eralgebra mo dules and cohomology complexes 16 3. The un iversal (o dd) P oisson sup eralge bra f or a commutati v e asso ciativ e su p eralgebra A and Poisson sup eralgebra cohomology 17 3.1. The u niv ersal o dd Po isson sup eralgebra Π W as (Π A ) 18 3.2. P oisson sup eralge bra structures and P oisson s up eralgebra cohomology complexes 21 3.3. The u niv ersal Poisson su p eralgebra W as ( A ) an d o dd P oisson sup er algebra structures on A 22 4. The Lie sup eralge bra W ∂ ( V ) for an F [ ∂ ]-modu le V , and L ie conformal sup eralgebra cohomolo gy 23 4.1. The Lie sup eralgebra W ∂ ( V ) 24 4.2. The sp ace W ∂ ( V , U ) as a reduction of W ∂ ( V ⊕ U ) 27 4.3. Lie conformal sup eralgebra structur es 28 4.4. Lie conformal sup eralgebra m o dules and cohomology complexes 28 5. The Lie sup eralg ebra W ∂ , as ( V ) for a comm u tativ e asso cia tiv e diﬀeren tial sup eralgebra V and PV A cohomology 29 5.1. P oisson v er tex sup eralgebra structur es 30 5.2. Odd P oisson vertex sup eralgebra structures 31 6. The unive rsal Lie conformal sup eralgebra f W ∂ ( V ) for an F [ ∂ ]- mo dule V , and th e basic Lie conformal sup eralge bra cohomology complexes 32 6.1. The u niv ersal L ie conformal su p eralgebra f W ∂ ( V ) 32 6.2. The b asic Lie conformal sup eralgebra cohomology complex 43 6.3. Extension to inﬁ nitely generated F [ ∂ ]-mo du les 43 7. The universal (o dd ) P oisson v er tex sup eralge bra for a diﬀeren tial sup er algebra V and basic PV A cohomology 45 7.1. The u niv ersal o dd PV A f W ∂ , as (Π V ) 45 7.2. The u niv ersal P V A f W ∂ , as ( V ) 51 8. Algebras of diﬀerentia l functions and the v ariational complex 51 8.1. Algebras of diﬀerential fu nctions 51 8.2. de Rham complex e Ω • ( V ) and v ariational complex Ω • ( V ) 53 8.3. Exactness of the v ariational complex 55 9. The Lie su p eralgebra of v ariational p olyvec tor ﬁelds and PV A cohomology 55 9.1. The Lie sup eralgebra of v ariational p olyv ector ﬁelds W v ar (Π V ) 55 THE V ARIA TIONAL POISSON COHOMOLOGY 3 9.2. PV A structures on an algebra of diﬀerentia l functions and cohomology complexes 59 9.3. Identi ﬁcation of W v ar (Π V ) w ith Ω • ( V ) 60 9.4. The v ariational and Poisson cohomology complexes in terms of lo cal p olydiﬀerenti al op erators 61 9.5. Generalized v ariational complexes 65 10. The univ ersal od d PV A f W v ar (Π V ) f or an algebra of d iﬀeren tial functions, and b asic P V A cohomolo gy 65 10.1. The Lie conform al alge bra CV ect( V ) of conformal v ector ﬁelds 65 10.2. The univ ersal o dd PV A f W v ar (Π V ) 67 10.3. PV A structures on an algebra of diﬀerentia l fun ctions and basic cohomolo gy complexes 71 10.4. Iden tiﬁcation of f W v ar (Π V ) with e Ω • ( V ) 72 10.5. Generalized de Rh am complexes 72 11. Computation of the v ariational P oisson cohomolo gy 74 11.1. F ormalit y of the generaliz ed de Rham complex 74 11.2. Cohomology of the generalized v ariational complex 80 11.3. Explicit description of H 0 (Ω • ( V ) , δ K ) and H 1 (Ω • ( V ) , δ K ). 93 App end ix A. Systems of linear d iﬀeren tial equations and (p oly)diﬀeren tial op erators 95 A.1. Lemmas on diﬀerential op erators 95 A.2. Linear algebra ov er a diﬀerenti al ﬁeld 97 A.3. Linearly closed d iﬀerential ﬁ elds 103 A.4. Main r esults 106 A.5. Generalizat ion to p olydiﬀerential op erators 111 References 131 4 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 1. Introduction The theory of Poi sson v ertex algebras is a v ery conv enient framew ork for the theory of Hamiltonian partial d iﬀeren tial equations [ BDSK ]. First, let us introd uce some ke y notions. Let V be a unital comm utativ e asso ciativ e algebra with a d eriv ation ∂ . The space g − 1 = V /∂ V is called the space of Hamiltonian functions , th e image of h ∈ V in g − 1 b eing denoted b y R h . The Lie algebra g 0 of all deriv ations of V , comm uting w ith ∂ , is called the Lie algebra of e volutionary ve c tor ﬁelds . Its action on V descends to g − 1 . A λ - br acket on V is a linear map V ⊗ V → F [ λ ] ⊗ V , a ⊗ b 7→ { a λ b } , satisfying the follo wing three prop erties ( a, b ∈ V ): (sesquilinearit y) { ∂ a λ b } = − λ { a λ b } , { a λ ∂ b } = ( ∂ + λ ) { a λ b } , (sk ew commutati vit y) { b λ a } = −{ a − ∂ − λ b } , where ∂ is mo ved to the left, (Leibniz rule) { a λ bc } = { a λ b } c + b { a λ c } . Denote by g 1 the space of all λ -brac k ets on V . One of the basic constructions of the presen t pap er is the Z -graded Lie sup er algebra (1.1) W ∂ , as (Π V ) = (Π g − 1 ) ⊕ g 0 ⊕ (Π g 1 ) ⊕ . . . , where g − 1 , g 0 and g 1 are as ab o v e and Π g i stands for the space g i with re- v ersed parity . F or R f , R g ∈ Π g − 1 , X , Y ∈ g 0 and H ∈ Π g 1 the commutato rs are deﬁ n ed as follo ws: R f , R g  = 0 , (1.2)  X, R f  = R X ( f ) , (1.3) [ X, Y ] = X Y − Y X , (1.4)  { . λ . } H , R f  ( g ) = { f λ g } H   λ =0 , (1.5) { f λ g } [ X,H ] = X ( { f λ g } H ) − { X ( f ) λ g } H − { f λ , X ( g ) } H . (1.6) In Section 5 w e construct explicitly the whole Lie sup eralgebra W ∂ , as (Π V ), but f or applicatio ns to H amiltonian PDE one needs only t he condition [ H , K ] = 0 for H , K ∈ Π g 1 , wh ic h is as follo ws ( f , g , h ∈ V ): (1.7) {{ f λ g } K λ + µ h } H − { f λ { g µ h } K } H + { g µ { f λ h } K } H + ( H ↔ K ) = 0 . A λ -brac ket { . λ . } = { . λ . } H is called a Poisson λ -brac k et if [ H, H ] = 0, i.e., one has (Jacobi iden tit y) { f λ { g µ h }} − { g µ { f λ h }} = {{ f λ g } λ + µ h } . THE V ARIA TIONAL POISSON COHOMOLOGY 5 The diﬀeren tial alg ebra V , endow ed with a P oisson λ -brac ket , is called a Poisson vertex algebr a (PV A) [ DSK1 ]. Tw o P oisson λ -brac ket s { . λ . } H and { . λ . } K on V are called c omp atible if ( 1.7 ) holds, whic h means that their sum is a P oisson λ -b r ac ket as well. One of the key prop erties of a PV A V is th at the v ecto r space V /∂ V carries a we ll-deﬁned Lie algebra str ucture, given by (1.8) R f , R g  = R { f λ g }   λ =0 , f , g ∈ V . Moreo ver, V is a left mo dule ov er the Lie algebra V /∂ V with the we ll-deﬁned action (1.9) { R f , g } = { f λ g }   λ =0 , f , g ∈ V , b y deriv atio ns, commuting with ∂ , of the asso ciativ e p ro duct in V and of the λ -brac k et. In particular, all the d eriv ations X f = { R f , . } of V are ev olutionary; they are called Hamiltonian ve ctor ﬁelds . Tw o Hamilto nian fu n ctions R f and R g are said to b e i n involution if (1.10)  R f , R g  = 0 . Giv en a Hamiltonian function R h ∈ V /∂ V and a P oisson λ -brac k et on V , the corresp ondin g Hamiltonian equation is d eﬁned by the Hamiltonian vecto r ﬁeld X h : (1.11) du dt =  R h, u  , u ∈ V . The equation ( 1.11 ) is calle d in tegrable if R h is con tained in an inﬁnite- dimensional ab elian sub algebra of the Lie algebra V /∂ V with brac k et ( 1.8 ). Pic king a b asis R h 0 = R h , R h 1 R h 2 , . . . of this ab elian subalgebra, we obtain a hierarch y of inte grable Hamiltonian equations (1.12) du dt n = R h n , u  , n ∈ Z + , whic h are compatible since the corresp ond ing Hamiltonian v ector ﬁelds X h n comm ute. The basic d evice for pro ving integ rabilit y of a Hamiltonian equation is th e so-calle d Lenard -Magri sc h eme, w hic h is th e follo wing simp le observ ation, ﬁrst men tioned in [ GGKM ] and [ Lax ]; a su rv ey of related results up to th e early 90’s can b e found in [ Dor ]. Supp ose that the diﬀerentia l algebra V is endo w ed with t w o λ -brac kets { . λ . } H and { . λ . } K and assume that: (1.13) R h n , u  H = R h n +1 , u  K , n ∈ Z + , u ∈ V , for some Hamiltonian functions R h n ∈ V /∂ V . Th en all these Hamilton- ian fu nctions are in inv olution with resp ect to b oth brack ets { . λ . } H and { . λ . } K on V /∂ V . Note that w e do not need to assume that the λ -brack ets are Poi sson nor that th ey are compatible. These assu mptions ente r wh en w e try to pro v e the existence of the sequence, R h n , satisfying ( 1.13 ), as we explain b elow. 6 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Indeed, let H, K ∈ Π g 1 b e t w o compatible Poisson λ -b rac k ets o n V . Since [ K, K ] = 0, it follo ws that (ad K ) 2 = 0, h ence w e ma y consider the variational c ohomolo gy c omplex ( W ∂ , as (Π V ) = L ∞ j = − 1 W j , ad K ), w here W − 1 = Π g − 1 , W 0 = g 0 , W 1 = Π g 1 , . . . . By d eﬁ nition, X ∈ W 0 is close d if, in view of ( 1.6 ), it is a deriv ation of the λ -brac ket { . λ . } K , and it is exact if, in view of ( 1.5 ), X = { h λ ·} K   λ =0 for some h ∈ W − 1 . No w we can ﬁnd a solution to ( 1.13 ), by indu ction on n as follo ws. By Jacobi iden- tit y in W ∂ , as (Π V ) w e hav e [ K , [ H , h n ]] = − [ H , [ K, h n ]], whic h, by ind uctiv e assumption, equals − [ H , [ H , h n − 1 ]] = 0, since [ H, H ] = 0 and H ∈ W 1 is o dd. Th us, the elemen t [ H , h n − 1 ] ∈ W 0 is closed. If this closed elemen t is exact, i.e., it equals [ K , h n ] for some h n ∈ W − 1 , we complete the n th step of induction. In general we hav e (1.14) [ H , h n − 1 ] = [ K, h n ] + a n , where a n is a representati v e of the corresp onding cohomology class. Lo oking at ( 1.14 ) more carefully , on e often can p ro v e that one can tak e a n = 0, so the Lenard -Magri scheme still works. The cohomological approac h to the L enard-Magri sc heme was prop osed long ago in [ Kra ] and [ Ol1 ]. Ho wev er, no mac hiner y has b een deve lop ed in order to compute this cohomology . In the present pap er w e d evelo p suc h a mac hinery by in trod u cing a “co v ering” complex ( f W ∂ , as , ad K ) of the complex ( W ∂ , as , ad K ), whose cohomology is muc h easier to compute, and then study in detail the corresp onding long exact sequence. What does this ha v e to do with the classical Hamilto nian PDE, like the KdV equation? In order to explain this, consider the algebra of d iﬀeren tial p olynomials R ℓ = F [ u ( n ) i   i = 1 , . . . , ℓ ; n ∈ Z + ] w ith the deriv atio n ∂ , deﬁn ed on generators b y ∂ ( u ( n ) i ) = u ( n +1) i . Here one should th ink of the u i as fu nc- tions, dep end ing on a parameter t (time), in one indep enden t v ariable x , whic h is a co ordinate on a 1-dimensional manifold M , and of ∂ as the deriv- ativ e by x , so that u ( n ) i is the n th deriv ativ e of u i . F urthermore, one should think of R h ∈ R ℓ /∂ R ℓ as R M h dx since R ℓ /∂ R ℓ pro vides the u niv ersal space in whic h integ ration b y p arts h olds. It is straigh tforw ard to c hec k that equation ( 1.11 ) can b e w r itten in the follo win g equiv alen t, but more familiar, form: (1.15) du dt = H ( ∂ ) δ h δ u , where δh δu is the vec tor of v ariational deriv ativ es (1.16) δ h δ u i = X n ∈ Z + ( − ∂ ) n ∂ h ∂ u ( n ) i , and H ( ∂ ) = ( H ij ( ∂ )) ℓ i,j =1 is the ℓ × ℓ matrix diﬀerential op erator with en tries H ij ( ∂ ) = { u j ∂ u i } → . Here the arro w means that ∂ should b e mo ved to the righ t. It is n ot d iﬃcult to sh o w that the sk ew comm utativit y of the λ -brac k et THE V ARIA TIONAL POISSON COHOMOLOGY 7 is equiv alen t to skew adjointness of the diﬀerential op erator H ( ∂ ), and, in addition, the v alidity of the Jacobi iden tit y of the λ -brac k et is, by deﬁn ition, equiv alen t to H ( ∂ ) b eing a Hamiltonian op erator. F u rthermore, the brac k et ( 1.8 ) on R ℓ /∂ R ℓ tak es the f amiliar form (1.17) { R f , R g } = Z δ g δ u ·  H ( ∂ ) δ f δ u  , and one can sho w that this is a Lie algebra br ac ket if an d only if H ( ∂ ) is a Hamiltonian op erator [ BDSK ]. Giv en λ -brac k ets { u i λ u j } = −{ u j − ∂ − λ u i } ∈ R ℓ [ λ ] of an y p air of generators u i , u j , one can extend them uniquely to a λ -brack et on R ℓ , wh ic h is giv en b y the follo w ing explicit formula [ DSK1 ] (1.18) { f λ g } = X 1 ≤ i,j ≤ ℓ m,n ∈ Z + ∂ g ∂ u ( n ) j ( ∂ + λ ) n { u i ∂ + λ u j } → ( − ∂ − λ ) m ∂ f ∂ u ( m ) i . This λ -brac k et deﬁnes a PV A structure on R ℓ if and only if the Jacobi iden tit y holds for an y tr iple of generators u i , u j , u k [ BDSK ]. The simplest examp le of a Hamiltonian op erator is the Gardner-F addeev- Zakharo v (GFZ) op erator K ( ∂ ) = ∂ . It is the observ ation in [ Gar ] that the KdV equation (1.19) du dt = 3 uu ′ + cu ′′′ , c ∈ F , can b e written in a Hamiltonian form (1.20) du dt = D δ h 1 δ u , where h 1 = 1 2 ( u 3 + cuu ′′ ) , and it is the subsequ en t pro of in [ FZ ] that KdV is a completely in tegrable Hamiltonian equation, that triggered the theory of Hamiltonian PDE. T h e corresp ondin g λ -brac k et on R 1 is, of course, giv en by the form ula { u λ u } = λ , extended to R 1 ⊗ R 1 → F [ λ ] ⊗ R 1 b y ( 1.18 ). In a subsequent pap er [ Mag ], Magri sh o w ed that the op erator H ( ∂ ) = u ′ + 2 u∂ + c∂ 3 is Hamiltonia n for all c ∈ F , that it is compatible with the GFZ op erato r, and that the KdV equation can b e written in a diﬀerent Hamiltonian form (1.21) du dt = ( u ′ + 2 u∂ + c∂ 3 ) δ h 1 δ u , where h 1 = 1 2 u 2 . Moreo ver, he explained ho w to use this to p ro v e the v alidit y of the Lenard- Magri scheme 1 , whic h ga v e a new pro of of in tegrabilit y of KdV and some other equations. 1 Since in the literature t he names Lenard and Magri scheme are alternatively used, w e decided to call it the Lenard- Magri scheme. The history of Lenard’s con tribution is colorfully describ ed in [ PS ], where one can also ﬁn d an extensive list of sub seq u ent publications on the sub ject. 8 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Of course, the λ -brac k et corresp onding to the Magri op erator is give n b y (1.22) { u λ u } = ( ∂ + 2 λ ) u + cλ 3 , whic h d eﬁnes (via ( 1.18 )) a PV A structure on R 1 for all v alues of c ∈ F . The reader can ﬁnd a d etailed exp osition of the app lications of PV A to Hamiltonian PDE in th e p ap er [ BDSK ], where, in p articular, some suﬃcient conditions for the v alidit y of the Lenard-Magri scheme and its generaliza- tions are found and applied to the pr o of of in tegrabilit y of man y imp ortan t equations. How eve r man y Hamilt onian equations remain out of reac h of the metho ds of [ BDSK ], but w e think that the cohomologica l approac h is more p o w erful (though less elementa ry) and we are planning to d emonstrate this in a subsequ ent pap er . In ord er to mak e our id eas clearer (or, p erhaps, m ore confu sing) we b egin the pap er with a long digression, whic h go es from Section 2 thr ough Section 10 , to a general approac h to v arious cohomology theories (in fact, the reader, in terested only in ap p lications to the theory of int egrable Hamiltonian PDE, can, without muc h diﬃculty , jump to Section 11 ). In Section 2 , give n a vec tor sup ersp ace V , w e consider the universal Z - graded Lie sup eralgebra W ( V ) = ⊕ j ≥− 1 W j ( V ) with W − 1 ( V ) = V . Uni- v ersalit y here is understo o d in the s en se that, giv en an y other Z -graded Lie sup er algebra g = ⊕ j ≥− 1 g j with g − 1 = V , there exists a unique, gradin g preserving homomorphism g → W ( V ), identica l on V . It is easy to sh o w that W j ( V ) = Hom( S j +1 ( V ) , V ) for all j ≥ − 1, and one can write down explicitly the Lie su p eralgebra brac k et. In p articular, W 0 ( V ) = End V and W 1 ( V ) = Hom( S 2 V , V ), so that an y ev en elemen t of the vec tor sup erspace W 1 ( V ) deﬁnes a commutati v e sup eralgebra structure on V (and this corre- sp ond en ce is b ijectiv e). On th e other han d , as observe d in [ CK ], an y o d d element X of the v ector sup er s pace W 1 (Π V ) deﬁ nes an ske w comm u tativ e sup eralgebra structure on V b y the formula (1.23) [ a, b ] = ( − 1) p ( a ) X ( a ⊗ b ) , a, b ∈ V , where p is the parit y on V . Moreo v er, this is a Lie sup eralgebra structure if and only if [ X , X ] = 0 in W (Π V ). Thus, giv en a Lie sup eralgebra structure on V , considering the corresp onding element X ∈ W 1 (Π V ), we obtain a cohomology complex ( C • = ⊕ j ∈ Z C j , ad X ), where C j = W j +1 (Π V ), and it turns out that C • = C • ( V , V ) coincides with the cohomolog y complex of the L ie su p eralgebra V with coeﬃcients in the adjoin t represen tation. More generally , giv en a mo d ule M ov er the Lie s u p eralgebra V , one considers, instead of V , the L ie sup eralgebra V ⋉ M with M an ab elian ideal, and by a simple redu ction pro cedure constructs the cohomolo gy of the Lie sup eral- gebra V with co eﬃcien ts in M . This constru ction for V p urely ev en go es bac k to the pap er [ NR ] on deformation theory . In Section 3 , assuming th at V ca rries a stru cture of a commutativ e as- so ciativ e sup eralge bra, we let W as − 1 (Π V ) = Π V , W as 0 (Π V ) = Der V , the THE V ARIA TIONAL POISSON COHOMOLOGY 9 subalgebra of all deriv ations of the sup eralgebra V in the sup eralge bra End Π V = End V (the su p erscript “as” stands for “asso ciativ e”). Let W as (Π V ) = ⊕ j ≥− 1 W as j (Π V ) b e the full prolongation in the Lie sup eral- gebra W (Π V ), deﬁned inductiv ely for j ≥ 1 by W as j (Π V ) = n a ∈ W j (Π V )    [ a, W − 1 (Π V )] ⊂ W as j − 1 (Π V ) o . Then o dd elemen ts X in W as 1 (Π V ), suc h that [ X, X ] = 0, b ijectiv ely corre- sp ond to P oisson alge bra stru ctures on V (with the giv en comm utativ e asso- ciativ e sup eralgebra stru cture). In this case the complex ( W as (Π V ) , ad X ) is the Po isson cohomolo gy complex of the P oisson sup eralge bra V (in tro d uced in [ Lic ]). Inciden tally , on e can introd uce a comm utativ e associativ e p ro duct on W as (Π V ), making it (along with the Lie sup eralg ebra br ac ket) an o dd Pois- son (= Gerstenhab er) sup eralgebra. Here w e observe a remark able du alit y when p assing from Π V t o V : W as ( V ) is an (ev en) P oisson sup eralgebra, whereas the o dd elemen ts of W as 1 ( V ) corresp ond to o dd Poisson sup eralge- bra stru ctures on V . Next, in Section 4 w e consider the case when V carries a stru cture of an F [ ∂ ] -mo dule. He re and th r oughout th e pap er F [ ∂ ], as usual, denotes the algebra of p olynomials in an (eve n) indeterminate ∂ . Motiv ated by the construction of the univ ersal Lie sup eralgebra W (Π V ), we construct a Z - graded Lie sup eralgebra W ∂ (Π V ) = ⊕ ∞ k = − 1 W ∂ k (Π V ), which, to s ome exten t, pla ys the same role in the theory of Lie conformal alge bra as W (Π V ) pla ys in the theory of Lie algebras (explained ab o v e). Recall that a Lie conformal algebra is an F [ ∂ ]-mo dule, endo w ed w ith the λ -brac k et, satisfying sesquilinearity , sk ew commutativi t y and Jacobi id en tit y (in tro d uced ab o v e). In other words, a L ie conformal algebra is an analogue of a Lie algebra in the same w a y as a P oisson vertex algebra is an analogue of a Po isson algebra. W e let W ∂ − 1 (Π V ) = Π( V /∂ V ) and W ∂ 0 (Π V ) = En d F [ ∂ ] V , and constru ct W ∂ (Π V ) as a prolongation in W (Π( V /∂ V )) (not necessarily full), so that o dd elemen ts X ∈ W ∂ 1 (Π V ) parameterize sesquilinear ske w comm u tativ e λ - brac k ets on V , and the λ -brac k et satisﬁes the Jacobi identi t y (i.e., d eﬁnes on V a Lie conformal algebra str u cture) if and only if [ X, X ] = 0. In the same wa y as in the Lie algebra case, w e obtain a cohomology complex ( W ∂ (Π V ) , ad X ), pro vided that [ X, X ] = 0 f or an od d element X ∈ W ∂ 1 (Π V ), and this complex (after the shift by 1), is the Lie conform al algebra cohomology complex with co eﬃcients in the adjoint repr esen tatio n. In the same wa y , by a r eduction, w e reco v er the Lie conformal algebra cohomolog y complex with co eﬃcient s in an y repr esen tatio n, studied in [ BKV ], [ BD AK ], [ DSK2 ]. Next, in S ection 5 w e consider the case w hen V carries b oth, a structure of an F [ ∂ ]-mod ule, an d a compatible with it comm utativ e alge bra structure, in other words, V is a diﬀerent ial algebra. Then in the same wa y as ab ov e, 10 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 w e construct the Lie sup eralgebra W ∂ , as (Π V ) (cf. ( 1.1 )) as a Z -graded subalgebra of W ∂ (Π V ), for whic h W ∂ , as − 1 (Π V ) = W ∂ − 1 (Π V ) = Π( V /∂ V ), W ∂ , as 0 (Π V ) = Der F [ ∂ ] V ⊂ W ∂ 0 (Π V ) = End F [ ∂ ] V , and W ∂ , as 1 (Π V ) is su c h that its odd elemen ts X parameterize all λ -brac kets on the diﬀeren tial alge- bra V , so that those satisfying [ X , X ] = 0 corresp ond to PV A s tr uctures on V . This explains the strange notation ( 1.1 ) of this Lie sup eralgebra. In Section 6 we constru ct the universal Lie conformal sup eralge bra f W ∂ ( V ) for a ﬁn itely generated F [ ∂ ]-sup er m o dule V , and in Section 7 w e construct the univ ersal o dd Poisson v ertex algebra f W ∂ , as (Π V ) for a ﬁnitely generated diﬀeren tial sup eralgebra V . These constructions are v ery similar in spirit to the constru ctions of the univ ersal Lie su p eralgebras W ∂ ( V ) and W ∂ , as ( V ), from Sectio ns 4 and 5 resp ectiv ely . Th e ﬁn itely generated assumption is needed in order for the corresp onding λ -br ac kets to b e p olynomial in λ . Note that in th e deﬁn ition of the Lie algebra brac k et on V /∂ V , and its represent ation on V , as w ell in the discussion of the Lenard -Magri sc h eme, w e needed only that V is a Lie conformal algebra. Ho wev er, for p ractical applications one usually uses PV A’s, and, in fact some sp ecia l kin d of PV A’s, whic h are diﬀeren tial algebra extensions of R ℓ with the λ -brac k et giv en b y form ula ( 1.18 ). F or such a PV A V w e construct, in Sections 9 a sub algebra of the Lie algebra W ∂ , as (Π V ) W v ar (Π V ) = ⊕ j ≥− 1 W v ar j , where W v ar − 1 = W ∂ , as − 1 , but W v ar j for j ≥ 0 ma y b e smaller. F or example W v ar 0 consists of d eriv ations of the form P 1 ≤ j ≤ ℓ n ∈ Z + P j,n ∂ ∂ u ( n ) j , commuting with ∂ , and it is these deriv ations that are called in v ariational calculus ev olutionary v ecto r ﬁelds. Next, W v ar 1 consists of all λ -brac k ets of the form ( 1.18 ), etc. W e call elemen ts of W v ar k the variationa l k -ve ctor ﬁelds . There has b een an extensive discussion of v ariational p oly-v ector ﬁelds in the literature. The earliest r eferen ce w e kno w of is [ Kup ], see also the b o ok [ Ol2 ]. One of the later r eferences is [ IVV ]; the idea to u se Cartan’s prolongation comes from this pap er. In order to solv e the Lenard-Magri sc h eme ( 1.13 ) o v er a d iﬀeren tial func- tion extension V of R ℓ with the λ -brac k ets { . , . } H and { . , . } K of the form as in ( 1.1 8 ), one has to compute the cohomology of the complex ( W v ar (Π V ) , ad K ), where K ∈ W v ar 1 is su ch that [ K, K ] = 0, as we ex- plained ab o v e. In order to compute this variational Poisson c ohom olo gy , w e construct, in Section 10 a Z + -graded Lie conformal sup eralg ebra (whic h is actually a subalgebra of the o dd PV A f W ∂ , as (Π V )) f W v ar (Π V ) = ⊕ j ≥− 1 f W v ar j with f W v ar − 1 = V , for wh ich the asso ciated Lie su p eralgebra is W v ar (Π V ). Since the Lie sup eralgebra W v ar (Π V ) acts on the Lie c onformal sup eralgebra THE V ARIA TIONAL POISSON COHOMOLOGY 11 f W v ar (Π V ), in p articular, K acts, pro viding it with a diﬀeren tial d K , com- m uting with the action of ∂ . W e th us ha v e an exact sequ ence of complexes: (1.24) 0 → ( ∂ f W v ar ( V ) , d K ) → ( f W v ar ( V ) , d K ) → ( W v ar ( V ) , ad K ) → 0 , so that we can study th e corresp onding cohomology long exact sequence. T o actually p erform calculat ions, w e identi fy (non-canonicall y) the space f W v ar (Π V ) with the space e Ω • ( V ) of the de Rham complex, and the sp ace W v ar (Π V ) with the space of the r educed de Rh am complex = v ariatio nal complex Ω • ( V ). W e th us get the “generalized” de Rham complex ( e Ω • ( V ) , d K ) and the “generalize d” v ariati onal complex (Ω • ( V ) , ad K ). The ordinary de Rham and v ariational complexes are not, strictly sp eaking, sp ecial cases, since they corresp ond to K = I , whic h is not a sk ewadjoin t op erator. How ev er, in the case wh en the diﬀeren tial op erator K is quasiconstan t, i.e., ∂ ∂ u ( n ) i ( K ) = 0 for all i, n , the construction of these complexes is still v alid. In Section 11 we completely solv e the problem of computation of coho- mology of the generalized de Rham complex ( e Ω • ( V ) , d K ) in the case when V is a normal algebra of diﬀeren tial functions and K is a qu asiconstan t ma- trix diﬀerentia l op erator with in v ertible leading co eﬃcien t. F or that w e use “local” homotop y op erators, similar to those introdu ced in [ BDSK ] for the de Rh am complex. After that, as in [ BDSK ], we study the cohomolog y long exact sequence corresp ondin g to the short exact s equ ence ( 1.24 ). As a result w e get a com- plete description of the cohomolo gy of th e generalize d v ariational complex for an arbitrary quasiconstan t ℓ × ℓ matrix diﬀerent ial op erator K of order N with inv ertible leading coeﬃcien t. In fact, we ﬁnd simple explicit formulas for r epresen tativ es of cohomology classes, and we pr ov e that dim H k (Ω • ( V ) , ad K ) =  N ℓ k + 1  , pro vided that qu asiconstants form a linearly closed diﬀerentia l ﬁeld. These results lead to fu rther progress in the app licatio n of the Lenard -Magri sc heme (w ork in progress). In the sp ecial case when K is a constan t co eﬃcien t order 1 sk ew adjoin t matrix diﬀeren tial op erator, it is prov ed in [ Get ] that the v ariational Po isson cohomology complex is formal. Our explicit description of the long exact s equ ence in terms of p olydif- feren tial op erators leads to some p r oblems on systems of linear d iﬀeren tial equations of arb itrary order in the same n um b er of unkno wns. In the Ap- p endix we devel op some diﬀerentia l linear algebra in order to s olve these problems. All v ector spaces are consid er ed o v er a ﬁeld F of c haracteristic zero. T en- sor pr o ducts, d irect su ms, and Hom’s are considered ov er F , unless otherwise sp eciﬁed. 12 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 W e wish to thank A. Kiselev for dra wing our atten tion to the cohomo- logica l app roac h to the Lenard-Magri sc h eme, I. Krasilshc hik for corresp on- dence, and A. Maﬀei for u seful discus sions. 2. The universal Lie superal gebra W ( V ) for a ve ctor supers p ace V , and Lie superalge bra coh omology Recall that a ve ctor sup ers pace is a Z / 2 Z -graded vecto r space U = U ¯ 0 ⊕ U ¯ 1 . If a ∈ U α , w here α ∈ Z / 2 Z = { ¯ 0 , ¯ 1 } , one sa ys that a has parit y p ( a ) = α . In this case we sa y that the su p ersp ace U has parity p . By a sup er algebra stru cture on U we alw a ys mean a p arity pr eserving pr o duct U ⊗ U → U, a ⊗ b 7→ ab , w hic h is called commutativ e (resp. skew commuta tiv e) if ba = ( − 1) p ( a ) p ( b ) ab (resp . ba = − ( − 1) p ( a ) p ( b ) ab ). An endomorphism of U is called ev en (r esp . o dd) if it preserve s (resp. rev erses) the parity . The sup erspace E n d( U ) of all endomorphisms of U is en do w ed with a Lie sup eralge bra structure b y th e formula: [ A, B ] = A ◦ B − ( − 1) p ( A ) p ( B ) B ◦ A . One d enotes by Π U the su p ersp ace obtained from U b y reversing the parit y , n amely Π U = U as a v ector sp ace, with parit y ¯ p ( a ) = p ( a ) + ¯ 1. One deﬁnes a structure of a ve ctor sup ers p ace on the tens or algebra T ( U ) o v er U by additivit y . The symmetric, (resp ectiv ely exterior) su p eralgebra S ( U ) (resp. V ( U )) is deﬁned as the quotien t of the tens or su p eralgebra T ( U ) b y the relations u ⊗ v − ( − 1) p ( u ) p ( v ) v ⊗ u (resp . u ⊗ v + ( − 1) p ( u ) p ( v ) v ⊗ u ). Note that S (Π U ) is the same as V U as a ve ctor space, but n ot as a ve ctor sup er s pace. 2.1. The univ ersal Lie sup eralgebra W ( V ) . Let V b e a ve ctor s u p er- space with parit y ¯ p (the reason for this notation will b e clear later). W e r ecall the construction of the universal Lie sup eralge bra W ( V ) asso ciated to V . Let W k ( V ) = Hom ( S k +1 ( V ) , V ), the sup erspace of ( k + 1)-linear s u p ersym - metric functions on V with v alues in V , and let W ( V ) = L ∞ k = − 1 W k ( V ). Again, w e denote its parit y by ¯ p . W e endo w this vect or sup ersp ace with a structure of a Z -graded Lie sup eralgebra as follo ws. If X ∈ W h ( V ), Y ∈ W k − h ( V ), w ith h ≥ − 1 , k ≥ h − 1, we deﬁne X  Y to b e the fol- lo w ing elemen t in W k ( V ): (2.1) X  Y ( v 0 , . . . , v k ) = X i 0 < ··· n . T ypical examples of algebras of diﬀerent ial functions are: the ring of translation in v arian t diﬀeren tial p olynomials, R ℓ = F [ u ( n ) i | i ∈ I , n ∈ Z + ], where ∂ ( u ( n ) i ) = u ( n +1) i , and the ring of diﬀerentia l p olynomials, R ℓ [ x ] = F [ x, u ( n ) i | i ∈ I , n ∈ Z + ], where ∂ x = 1 and ∂ u ( n ) i = u ( n +1) i . Other examples can b e constru cted starting from R ℓ or R ℓ [ x ] b y taking a lo calization by some multiplicat iv e su bset S , or an algebraic extension obtained b y adding solutions of some p olynomial equations, or a diﬀerentia l extension obtained b y adding solutions of some d iﬀeren tial equations. I n all these examples, and more generally in an y algebra of diﬀeren tial functions extension of R ℓ , the action of ∂ : V → V is given b y ∂ = ∂ ∂ x + X i ∈ I ,n ∈ Z + u ( n +1) i ∂ ∂ u ( n ) i , wh ic h implies th at (8.4) F ∩ ∂ V = ∂ F . Indeed, if f ∈ V has, in s ome v ariable u i , d iﬀerential order n ≥ 0, then ∂ f has d iﬀeren tial order n + 1, hence it do es not lie in F . The variation al derivative δ δu : V → V ⊕ ℓ is d eﬁned by (8.5) δ f δ u i := X n ∈ Z + ( − ∂ ) n ∂ f ∂ u ( n ) i . It follo ws immediately from ( 8.2 ) that (8.6) δ δ u i ( ∂ f ) = 0 , for every i ∈ I and f ∈ V , namely , ∂ V ⊂ Ker δ δu . A ve ctor ﬁeld is, b y d eﬁ nition, a deriv ation of V of the form (8.7) X = X i ∈ I ,n ∈ Z + P i,n ∂ ∂ u ( n ) i , P i,n ∈ V . W e d enote by V ect( V ) the space of all v ector ﬁelds, w h ic h is clearly a sub- algebra of th e Lie algebra Der( V ) of all deriv ations of V . A v ecto r ﬁeld X is THE V ARIA TIONAL POISSON COHOMOLOGY 53 called evolutionary if [ ∂ , X ] = 0, and w e denote b y V ect ∂ ( V ) the Lie sub alge- bra of all evolutio nary v ector ﬁelds. Namely , V ect ∂ ( V ) = V ect( V ) ∩ Der ∂ ( V ). By ( 8.1 ), a vec tor ﬁeld X is evol utionary if and only if it has the form (8.8) 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 . As in [ BDSK ], w e denote b y V ℓ the space of ℓ × 1 column vect ors with entrie s in V , and b y V ⊕ ℓ the subs p ace of ℓ × 1 column vecto rs with only ﬁnitely many non-zero en tries. 8.2. de Rham complex e Ω • ( V ) and v ariational complex Ω • ( V ) . He re w e describ e th e explicit construction of th e complex of v ariational calculus follo win g [ DSK2 ]. Recall that the de R ham c omplex e Ω • ( V ) is d eﬁned as the free comm uta- tiv e sup eralgebra o ver V with o dd generators δ u ( n ) i , i ∈ I , n ∈ Z + and the diﬀeren tial δ d eﬁ ned further. The algebra e Ω • ( V ) consists of ﬁn ite sums of the form (8.9) e ω = X i 1 ,...,i k ∈ I m 1 ,...,m k ∈ Z + P m 1 ...m k i 1 ...i k δ u ( m 1 ) i 1 ∧ · · · ∧ δ u ( m k ) i k , P m 1 ...m k i 1 ...i k ∈ V . W e h a ve a natural Z + -grading e Ω • ( V ) = L k ∈ Z + e Ω k ( V ) d eﬁned b y letting elemen ts in V h av e d egree 0, while the generators δ u ( n ) i ha v e degree 1. Th e space e Ω k ( V ) is a free mo du le o v er V with a basis consisting of the element s δ u ( m 1 ) i 1 ∧ · · · ∧ δ u ( m k ) i k , with ( m 1 , i 1 ) > · · · > ( m k , i k ) (with r esp ect to the lex- icographic ord er). In p articular e Ω 0 ( V ) = V and e Ω 1 ( V ) = L i ∈ I ,n ∈ Z + V δ u ( n ) i . W e let δ be an o dd deriv ation of degree 1 of e Ω • ( V ), suc 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 immed iate to chec k that δ 2 = 0 and that, for e ω ∈ e Ω k as in ( 8.9 ), w e hav e (8.10) δ ( e ω ) = X j ∈ I ,n ∈ Z + X i 1 ,...,i k ∈ I m 1 ,...,m k ∈ Z + ∂ P m 1 ...m k i 1 ...i k ∂ u ( n ) j δ u ( n ) j ∧ δ u ( m 1 ) i 1 ∧ · · · ∧ δ u ( m k ) i k . The sup ersp ace e Ω • ( V ) has a structure of an F [ ∂ ]-mo dule, wh ere ∂ acts as an ev en d eriv ation of th e w edge pr o duct, whic h extends the action on V = e Ω 0 ( V ), and comm utes with δ . Since ∂ comm u tes w ith δ , we may consider the corresp ondin g reduced complex Ω • ( V ) = e Ω • ( V ) /∂ e Ω • ( V ) = L k ∈ Z + Ω k ( V ), kno wn as the variation al c omplex . By an abuse of notation, we denote by δ the corresp onding d iﬀeren tial on Ω • ( V ). 54 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 W e id en tify the space e Ω k ( V ) w ith the space of skewsymmetric arr ays , i.e. arra ys of p olynomial s (8.11) P =  P i 1 ,...,i k ( λ 1 , . . . , λ k )  i 1 ,...,i k ∈ I , where P i 1 ,...,i k ( λ 1 , . . . , λ k ) ∈ F [ λ 1 , . . . , λ k ] ⊗ V are zero for all but ﬁnitely man y c hoices of indexes, and are ske wsymmetric with resp ect to sim ulta- neous p ermutati ons of the v ariables λ 1 , . . . , λ k and the ind exes i 1 , . . . , i k . The identi ﬁcation is obtained b y asso ciating P in ( 8.11 ) to e ω in ( 8.9 ) , where P m 1 ,...,m k i 1 ,...,i k is the co eﬃcien t of λ m 1 1 . . . λ m k k in P i 1 ,...,i k ( λ 1 , . . . , λ k ). The form ula for th e diﬀerential δ : e Ω k ( V ) → e Ω k +1 ( V ) gets translated as f ollo ws: (8.12) ( δ P ) i 0 ,...,i k ( λ 0 , . . . , λ k ) = k X α =0 ( − 1) α X n ∈ Z + ∂ P i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . ., λ k ) ∂ u ( n ) i α λ n α . In this language the F [ ∂ ]-mod u le structure of e Ω • ( V ) is giv en b y (8.13) ( ∂ P ) i 1 ,...,i k ( λ 1 , . . . , λ k ) = ( ∂ + λ 1 + · · · + λ k ) P i 1 ,...,i k ( λ 1 , . . . , λ k ) , so that the reduced space Ω k ( V ) = e Ω k ( V ) /∂ e Ω k ( V ) gets naturally identiﬁed with the sp ace of arrays ( 8.11 ), where P i 1 ,...,i k ( λ 1 , . . . , λ k ) are considered as elemen ts of F − [ λ 1 , . . . , λ k ] ⊗ F [ ∂ ] V . Th e diﬀeren tial δ on Ω • ( V ) is giv en by the same formula ( 8.12 ). F or example, Ω 0 ( V ) = V /∂ V , and Ω 1 ( V ) is naturally identiﬁed with V ⊕ ℓ , thanks to the canonical isomorp hism F [ λ ] ⊗ F [ ∂ ] V ≃ V . Under these iden tiﬁ- cations, the map δ : V /∂ V → V ⊕ ℓ coincides with the v ariational deriv ativ e ( 8.5 ): δ ( R f ) = δ f δ u , where, as in the previous sections, we denote b y f 7→ R f the canonical quo- tien t map V → V /∂ V . F urthermore, Ω 2 ( V ) is naturally id en tiﬁed with the space of sk ew adjoint ℓ × ℓ -matrix diﬀerentia l op erators S ( ∂ ) =  S ij ( ∂ )  i,j ∈ I . This iden tiﬁcatio n is obtained by mappin g P =  P m,n ∈ Z + P m,n i,j λ m µ n  i,j ∈ I to th e op erator S ( ∂ ) giv en b y S ij ( ∂ ) = P m,n ∈ Z + ( − ∂ ) n ◦ P m,n i,j ∂ m . With these iden tiﬁcatio ns, formula ( 8.12 ) for th e diﬀeren tial of F ∈ V ⊕ ℓ = Ω 1 ( V ) b ecomes δ F = − D F ( ∂ ) + D ∗ F ( ∂ ) , where (8.14) D F ( ∂ ) =  X n ∈ Z + ∂ F i ∂ u j ∂ n  i,j ∈ I is the F r e chet derivative of F , and D ∗ F ( ∂ ) is the adjoin t diﬀeren tial op erator. THE V ARIA TIONAL POISSON COHOMOLOGY 55 8.3. E xa ctness of the v ariational complex. Recall f r om [ BDSK ] that an algebra of diﬀeren tial functions V is calle d normal if w e ha ve ∂ ∂ u ( m ) i  V m,i  = V m,i for all i ∈ I , m ∈ Z + , wh ere we let (8.15) V m,i := n f ∈ V    ∂ f ∂ u ( n ) j = 0 if ( n, j ) > ( m, i ) in lexicographic order o . W e also d enote V m, 0 = V m − 1 ,ℓ , and V 0 , 0 = F . The algebras R ℓ and R ℓ [ x ] are ob viously normal. Moreo v er , any their extension V can b e furth er extended to a normal algebra. Conv ersely , in [ DSK2 ] it is pro v ed that an y norm al algebra of diﬀeren tial functions V is automatica lly a diﬀerentia l algebra extension of R ℓ . In [ BDSK ] w e p ro v ed the follo w ing result (see also ( 8.4 )): Theorem 8.2. If V i s a normal algebr a of diﬀer ential functions, then (a) H k ( e Ω • ( V ) , δ ) = 0 for k ≥ 1 , and H 0 ( e Ω • ( V ) , δ ) = F , (b) H k (Ω • ( V ) , δ ) = 0 for k ≥ 1 , and H 0 (Ω • ( V ) , δ ) = F /∂ F . In p articular, δf δu = 0 if and only if f ∈ ∂ V + F , and F ∈ V ⊕ ℓ is in the image of δ δu if and only if its F r e chet derivative D F ( ∂ ) is selfadjoint . 9. The Lie superalgeb ra of v aria tional p ol yvector f ields and PV A coh omology Let V b e an algebra of diﬀeren tial fun ctions extension of the algebra of diﬀeren tial p olynomials R ℓ = F [ u ( n ) i | i ∈ I , n ∈ Z + ]. Recal l fr om S ection 5.1 the Z -graded Lie su p eralgebra W ∂ , as (Π V ), obtained as a prolongatio n of the Lie algebra Der ∂ ( V ) of d eriv ations of V , comm uting with ∂ , in the u niv ersal Lie su p eralgebra W ∂ (Π V ). In Section 9.1 we in trod uce a smaller Z -graded subalgebra of W ∂ (Π V ), which we call the Lie sup eralgebra of v ariational p olyv ector ﬁelds, denoted by W v ar (Π V ) = L ∞ k = − 1 W v ar k . It is obtained as a prolongation of the Lie su balgebra of evolutio nary v ector ﬁelds V ect ∂ ( V ) ⊂ Der ∂ ( V ), in tro duced in Section 8.1 . W e then identify in Section 9.3 the space W v ar (Π V ) with the space Ω • ( V ) in tr o duced in Section 8.2 , and we relate the corresp ondin g cohomology complexes. 9.1. The Lie sup eralgebra of v ariational p olyv ector ﬁelds W v ar (Π V ) . Recall that the sup ersp ace W ∂ k (Π V ), of p arit y k mo d 2, consists of maps X : V ⊗ ( k + 1) → F − [ λ 0 , . . . , λ k ] ⊗ F [ ∂ ] V , satisfying sesquilinearit y: (9.1) X λ 0 ,...,λ k ( f 0 , . . . ∂ f i . . . , f k ) = − λ i X λ 0 ,...,λ k ( f 0 , . . . , f k ) , i = 0 , . . . , k , and skewsymmetry: (9.2) X λ σ (0) ,...,λ σ ( k ) ( f σ (0) , . . . , f σ ( k ) ) = sign( σ ) X λ 0 ,...,λ k ( f 0 , . . . , f k ) , σ ∈ S k +1 . In th e sp ecial case wh en V = R ℓ = F [ u ( n ) i | i ∈ I , n ∈ Z + ], the Leibn iz ru le ( 5.3 ) implies the f ollo wing master e quation for an elemen t X ∈ W ∂ , as k (Π V ), 56 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 whic h expresses the action of X on V ⊗ ( k + 1) in terms of its action on ( k + 1)- tuples of generators: (9.3) X λ 0 ,...,λ k ( f 0 , . . . , f k ) = X i 0 ,...,i k ∈ I m 0 ,...,m k ∈ Z +  e ∂ ∂ λ 0 ∂ f 0 ∂ u ( m 0 ) i 0  . . . . . .  e ∂ ∂ λ k ∂ f k ∂ u ( m k ) i k  ( − λ 0 ) m 0 . . . ( − λ k ) m k X λ 0 ,...,λ k ( u i 0 , . . . , u i k ) . Here w e are using the follo wing form ula: (9.4)  e ∂ ∂ λ f  P ( λ ) := X n ∈ Z + 1 n ! ( ∂ n f ) ∂ n P ( λ ) ∂ λ n = P ( λ + ∂ ) → f . In general, for an arbitrary algebra of diﬀeren tial functions V con taining R ℓ , w e deﬁne the space W v ar k of variational k - ve ctor ﬁelds as the su bspace of W ∂ k (Π V ) consisting of element s X satisfying the master equation ( 9.3 ). By ( 9.4 ), another form of equation ( 9.3 ) is the f ollo wing (for eac h s = 0 , . . . , k ): (9.5) X λ 0 ,...,λ k ( f 0 , . . . , f k ) = X i ∈ I , m ∈ Z + X λ 0 ,...,λ s + ∂ , . ..,λ k ( f 0 , . . . , s ˇ u i , . . . , f k ) → ( − λ s − ∂ ) m ∂ f s ∂ u ( m ) i , where s ˇ u i means th at u i is p ut in place of f s . Note that the master equation implies that X satisﬁes the Leibniz rule ( 5.3 ). Th us, W v ar k is a sub space of W ∂ , as k (Π V ). Prop osition 9.1. The sup ersp ac e W v ar (Π V ) is a Z -gr ade d sub algebr a of the Lie sup er algebr a W ∂ , as (Π V ) . Pr o of. The pr o of of this statemen t is similar to that of Prop osition 5.1 . W e need to pro ve t hat, if X ∈ W v ar h and Y ∈ W v ar k − h , then [ X, Y ] = X  Y − ( − 1) h ( k − h ) Y  X lies in W v ar k , namely , it satisﬁes the master equa- tion ( 9.3 ), or, equiv alen tly , equation ( 9. 5 ) for s = 0 , . . . , k . W e observe that, since [ X, Y ] λ 0 ,...,λ k ( f 0 , . . . , f k ) is sk ewsymmetric with resp ect to simultane- ous p erm utations of the v ariables λ i and the elemen ts f i , it suﬃces to prov e that [ X, Y ] satisﬁes equation ( 9.5 ) for s = 0. W e ha ve, by a straigh tforward computation,  X  Y  λ 0 ,...,λ k ( f 0 , f 1 , . . . , f k ) = X j ∈ I ,n ∈ Z +  X  Y  λ 0 + ∂ , λ 1 ,...,λ k ( u j , f 1 , . . . , f k ) → ( − λ 0 − ∂ ) n ∂ f 0 ∂ u ( n ) j + X i,j ∈ I , m,n ∈ Z + X i 1 < ··· ( m, i ) , ∂ n + N λ α P i 1 ,...,i k ( λ 1 , . . . , λ k ) = 0 , if ( n, i α ) > ( m, i ) , for all i 1 , . . . , i k ∈ I , where th e inequ alities are understo o d in the lexi- cographic order. In other wo rds, the co eﬃcien ts of all the p olynomials P i 1 ,...,i k ( λ 1 , . . . , λ k ) lie in V m,i , and, moreo v er, P i 1 ,...,i k ( λ 1 , . . . , λ k ) has de- gree at most m + N (resp. m − 1 + N ) in eac h v ariable λ α with i α ≤ i (resp. i α > i ). W e also let e Ω k n, 0 ( V ) = e Ω k n − 1 ,ℓ ( V ) for n ≥ 1. Finally , w e let e Ω • 0 , 0 = L k ∈ Z + e Ω k 0 , 0 , where e Ω k 0 , 0 is a subspace of e Ω k ( V ) consisting of arr a ys P =  P i 1 ,...,i k ( λ 1 , . . . , λ k )  i 1 ,...,i k ∈ I (sk ewsymmetric with THE V ARIA TIONAL POISSON COHOMOLOGY 75 resp ect to sim ultaneous p ermutati ons of indexes and v ariables), whose en- tries P i 1 ,...,i k ( λ 1 , . . . , λ k ) are p olynomials of degree at most N − 1 in eac h v ariable λ 1 , . . . , λ k , with quasiconstan t co eﬃcient s. In particular, e Ω 0 0 , 0 = F . By the skewsymmetry condition, (11.2) e Ω k 0 , 0 = 0 if k > N ℓ . Note that, if K ( ∂ ) is an ℓ × ℓ matrix diﬀerent ial op erator with quasi- constan t coeﬃcient s, then the d iﬀeren tial δ K deﬁned in ( 10.11 ) is zero on e Ω • 0 , 0 , so th at ( e Ω • 0 , 0 , 0) is a sub complex of the generalized de Rham complex ( e Ω • ( V ) , δ K ) Note also that e Ω • ( V ) is n atur ally a v ector space o v er F , and the diﬀeren tial δ K is F -linear. Also, all the e Ω k m,i ( V ) are F -linear sub spaces. R emark 11.1 . Due to formula ( 9.3 ) (cf. Prop osition 10.4 ) the restriction of the λ -br ac ket f rom f W v ar (Π V ) = e Ω • ( V ) to the subspace e Ω • 0 , 0 is zero. Hence, e Ω • 0 , 0 is a sub algebra of the Lie conformal sup eralgebra e Ω • ( V ). In this section we pr ov e the follo wing generalization of Theorem 8.2 (a): Theorem 11.2. L et V b e a normal algebr a of diﬀer ential functions and as- sume that the sub algebr a of quasic onstants F ⊂ V is a ﬁeld. L et K ( ∂ ) =  K ij ( ∂ )  i,j ∈ I b e an ℓ × ℓ matrix diﬀer ential op er ator of or der N with qua- sic onstant c o eﬃci e nts, and invertible le ading c o eﬃcient K N ∈ Mat ℓ × ℓ ( F ) . Then: (a) The inclusion ( e Ω • 0 , 0 , 0) ⊂ ( e Ω • ( V ) , δ K ) , is a quasiisomorphism of c om- plexes, i.e. it induc es a c anonic al Lie c onformal sup er algebr a isomor- phism of c ohomolo gy: H k ( e Ω • ( V ) , δ K ) ≃ e Ω k 0 , 0 . (b) F or k ≥ 0 , H k ( e Ω • ( V ) , δ K ) is a ve ctor sp ac e over F of dimension  N ℓ k  . The p ro of of Theorem 11.2 consists of seve ral steps. First, we pro v e three lemmas wh ic h will b e used in its pro of. In analog y with ( 9.19 ), for S =  S ij  i,j ∈ I ∈ Mat ℓ × ℓ ( F ), we deﬁne th e map Φ S : e Ω • ( V ) → e Ω • ( V ) , P 7→ Φ S ( P ), giv en b y the follo wing equation (11.3) (Φ S P ) i 1 ,...,i k ( λ 1 , . . . , λ k ) = X j 1 ,...,j k ∈ I P j 1 ,...,j k ( λ 1 + ∂ 1 , . . . , λ k + ∂ k ) S j 1 i 1 . . . S j k i k , where, as usual, ∂ α denotes ∂ acting on S j α i α . Lemma 11.3. (a) F or every S ∈ Mat ℓ × ℓ ( F ) , we have Φ S ( e Ω k ( V )) ⊂ e Ω k ( V ) and Φ S ( e Ω k 0 , 0 ) ⊂ e Ω k 0 , 0 . (b) If K ( ∂ ) is an ℓ × ℓ matrix diﬀer ential op er ator with quasic onstant c o e f- ﬁcients, then Φ S ( δ K P ) = δ K ◦ S Φ S ( P ) , 76 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 wher e ( K ◦ S )( ∂ ) =  P r ∈ I K ir ( ∂ ) ◦ S r j  i,j ∈ I . In other wor ds, Φ S is a homomo rphism of c omplexes: ( e Ω • ( V ) , δ K ) → ( e Ω • ( V ) , δ K ◦ S ) . (c) F or S, T ∈ Mat ℓ × ℓ ( F ) , we have Φ S ◦ Φ T = Φ T S . (d) If S ∈ Mat ℓ × ℓ ( F ) is an invertible matr ix, then Φ S : ( e Ω • ( V ) , δ K ) ∼ − → ( e Ω • ( V ) , δ K ◦ S ) , is an i somorph ism of c omplexes, which r estricts to an automorp hism of the sub c omplex ( e Ω • 0 , 0 , 0) . Pr o of. P art (a) is clear. Part (b) is pro v ed by a straigh tforw ard computation using the deﬁnitions ( 10.11 ) and ( 11.3 ) of the diﬀeren tial δ K and the map Φ S . P art (c) is again straight forw ard, using the deﬁ n ition ( 11.3 ) of Φ S . Finally , part (d) immed iately follo ws from (a), (b) and (c).  Lemma 11.4. L et K ( ∂ ) =  K ij ( ∂ )  i,j ∈ I b e an ℓ × ℓ matrix diﬀer ential op er ator of or der N with quasic onstant c o eﬃcients, and assume that its le ading c o eﬃcient K N ∈ Mat ℓ × ℓ ( F ) is diagonal. Then, δ K  e Ω k m,i ( V )  ⊂ e Ω k +1 m,i ( V ) , for every k ∈ Z + , m ∈ Z + , i ∈ I . Pr o of. Let P =  P i 1 ,...,i k ( λ 1 , . . . , λ k )  i 1 ,...,i k ∈ I ∈ e Ω k m,i ( V ). By the d eﬁnition ( 10.11 ) of δ K , w e hav e, using the assumption that K ( ∂ ) has quasiconstan t co eﬃcien ts, ∂ ∂ u ( n ) j ( δ K P ) i 0 ,...,i k ( λ 0 , . . . , λ k ) = k X α =0 ( − 1) α X r ∈ I ,p ∈ Z + ∂ 2 P i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . . , λ k ) ∂ u ( p ) r ∂ u ( n ) j ( λ α + ∂ ) p K r,i α ( λ α ) , whic h is zero if ( n, j ) > ( m, i ) by the assumption on P . Next, w e ha ve (11.4) ∂ n + N λ α ( δ K P ) i 0 ,...,i k ( λ 0 , . . . , λ k ) = X β 6 = α ( − 1) β X r ∈ I ,p ∈ Z + ∂ ∂ u ( p ) r  ∂ n + N λ α P i 0 , β ˇ ...,i k ( λ 0 , β ˇ . . ., λ k )  ( λ β + ∂ ) p K r,i α ( λ α ) +( − 1) α X r ∈ I m X p =0 ∂ P i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . . , λ k ) ∂ u ( p ) r ∂ n + N λ α ( λ α + ∂ ) p K r,i α ( λ α ) . The ﬁ rst sum in the RHS of ( 11.4 ) is zero for ( n, j ) > ( m, i ) b y th e assu m p- tion on P . Moreo ve r, the second sum in the RHS of ( 11.4 ) is zero for n > m since, by assumption, K ( λ ) h as degree N . Let then n = m in this su m. W e ha v e ∂ m + N λ α ( λ α + ∂ ) p K r,i α ( λ α ) = ( m + N )! δ p,m ( K N ) r,i α . THE V ARIA TIONAL POISSON COHOMOLOGY 77 Since, by assump tion, K N is a diagonal matrix, and since ∂ P i 1 ,...,i k ( λ 1 ,...,λ k ) ∂ u ( m ) r is zero for r > i , w e conclude that the second sum in the RHS of ( 11.4 ) is zero f or i α > i , as requir ed.  No w w e are going to use the assumption that that V is normal and F ⊂ V is a ﬁeld. Giv en i ∈ I , m ∈ Z + , choose an F -subsp ace U m,i ⊂ V m,i complemen tary to the k er n el of the map ∂ ∂ u ( m ) i : V m,i → V m,i . By normalit y of V , ∂ ∂ u ( m ) i restricts to an F -linear isomorp hism ∂ ∂ u ( m ) i : U m,i ∼ − → V m,i , and w e denote b y R du ( m ) i · : V m,i ∼ − → U m,i ⊂ V m,i the in v erse F -linear map, so that ∂ ∂ u ( m ) i R du ( m ) i f = f for ev ery f ∈ V m,i . Clearly , if we c hange the c hoice of the complement ary subs pace U m,i , the in tegral R du ( m ) i f ∈ V m,i of f ∈ V m,i c hanges b y addin g an elemen t of V m,i − 1 . W e extend the an tideriv ativ e R du ( m ) i · to the space of p olynomials in λ 1 , . . . , λ k with co eﬃcients in V m,i b y app lying it to co eﬃcien ts. Clearly , the op erators ∂ λ α and R du ( m ) i · , acting on F [ λ 1 , . . . , λ k ] ⊗ V m,i , comm u te. W e deﬁne the lo c al homotopy op er ators h m,i : e Ω k +1 m,i ( V ) → e Ω k m,i ( V ) , k ≥ 0, b y the follo w ing formula (11.5) ( h m,i P ) i 1 ,...,i k ( λ 1 , . . . , λ k ) = R du ( m ) i ∂ m + N µ ( m + N )! P i,i 1 ,...,i k ( µ, λ 1 , . . . , λ k ) . Lemma 11.5. L et P =  P i 0 ,...,i k ( λ 0 , . . . , λ k )  i 0 ,...,i k ∈ I ∈ e Ω k +1 m,i ( V ) . Then: (a) h m,i P ∈ e Ω k m,i ( V ) . (b) If P ∈ e Ω k +1 m,i − 1  ⊂ e Ω k +1 m,i ( V )  , then h m,i P = 0 . (c) If K ( ∂ ) is an ℓ × ℓ matrix diﬀer ential op er ator of or der N with quasi- c onstant c o eﬃcients and le ading c o eﬃci ent 1 I , the op er ator h m,i satisﬁes the fol lowing hom otopy c ondition: (11.6) h m,i ( δ K P ) + δ K ( h m,i P ) − P ∈ e Ω k m,i − 1 ( V ) . Pr o of. Clearly , ( h m,i P ) i 1 ,...,i k ( λ 1 , . . . , λ k ) is sk ewsymmetric w ith resp ect to sim ultaneous p erm u tations of th e v ariables λ 1 , . . . , λ k and of the ind exes i 1 , . . . , i k . Moreo v er, b y assumption on P and b y the deﬁnition of R du ( m ) i · , the co eﬃcien ts of all the p olynomials ( h m,i P ) i 1 ,...,i k ( λ 1 , . . . , λ k ) lie in V m,i . F urther m ore, if ( n, i α ) > ( m, i ) with α ∈ { 1 , . . . , k } , we ha v e, ∂ n + N λ α ( n + N )! ( h m,i P ) i 1 ,...,i k ( λ 1 , . . . , λ k ) = R du ( m ) i ∂ m + N µ ( m + N )! ∂ n + N λ α ( n + N )! P i,i 1 ,...,i k ( µ, λ 1 , . . . , λ k ) = 0 , b y the assumption on P . Hence, h m,i P ∈ e Ω k m,i , pro ving (a). Part (b) is clear since, by deﬁnition, P ∈ e Ω k +1 m,i − 1 are suc h that P i,i 1 ,...,i k ( µ, λ 1 , . . . , λ k ) 78 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 is a p olynomial of degree at most m + N − 1 in the v ariable µ . W e are left to p ro v e part (c). By ( 10.11 ) and ( 11.5 ), we ha v e, f or P ∈ e Ω k +1 m,i ( V ), (11.7) ( δ K h m,i P ) i 0 ,...,i k ( λ 0 , . . . , λ k ) = k X β =0 X j ∈ I ,n ∈ Z + ∂ ∂ u ( n ) j R du ( m ) i  ( λ β + ∂ ) n K j i β ( λ β )  ∂ m + N λ β ( m + N )! P i 0 ,..., β ˇ i ,...,i k ( λ 0 , . . . , λ k ) (here w e used the fact that R du ( m ) i is F -linear and K ij ( λ ) h as co eﬃcien ts in F ), and (11.8) ( h m,i δ K P ) i 0 ,...,i k ( λ 0 , . . . , λ k ) = − k X β =0 X j ∈ I ,n ∈ Z + R du ( m ) i ∂ ∂ u ( n ) j  ( λ β + ∂ ) n K j i β ( λ β )  ∂ m + N λ β ( m + N )! P i 0 ,..., β ˇ i ,...,i k ( λ 0 , . . . , λ k ) + R du ( m ) i ∂ ∂ u ( m ) i P i 0 ,...,i k ( λ 0 , . . . , λ k ) . By Lemma 11.4 and p arts (a) and (b), we kno w that δ K ( h m,i P ) , h m,i ( δ K P ) and P all lie in e Ω k +1 m,i . Hence, in order to prov e equation ( 11.6 ) w e only need to p ro v e the follo win g t w o iden titie s: (11.9) ∂ ∂ u ( m ) i  ( h m,i δ K P ) i 0 ,...,i k ( λ 0 , . . . , λ k ) + ( δ K h m,i P ) i 0 ,...,i k ( λ 0 , . . . , λ k )  = ∂ ∂ u ( m ) i P i 0 ,...,i k ( λ 0 , . . . , λ k ) ; ∂ m + N λ α ( m + N )!  ( h m,i δ K P ) i 0 ,...,i k ( λ 0 , . . . , λ k ) + ( δ K h m,i P ) i 0 ,...,i k ( λ 0 , . . . , λ k )  = ∂ m + N λ α ( m + N )! P i 0 ,...,i k ( λ 0 , . . . , λ k ) , if i α = i . The ﬁrs t identit y of ( 11.9 ) follo ws immediately from equ ations ( 11.7 ) and ( 11.8 ), using that ∂ ∂ u ( m ) i ◦ R du ( m ) i f = f for every f ∈ V m,i . The second iden- tit y in ( 11.9 ) follo ws by a straigh tforward computation using the follo win g t w o facts. Since, by assumption, the leading coeﬃcient of K ( ∂ ) is 1 I, we ha v e, for ( n, j ) ≤ ( m, i ), ∂ m + N λ α ( m + N )!  ( λ α + ∂ ) n K j i ( λ α )  = δ n,m δ j,i . Moreo ver, b y the skewsymmetry condition on P , we ha v e, if i α = i β = i f or β 6 = α , ∂ m + N λ α ∂ m + N λ β P i 0 ,...,,i k ( λ 0 , . . . , λ k ) = 0 .  THE V ARIA TIONAL POISSON COHOMOLOGY 79 Pr o of of The or em 11.2 . By Lemma 11.3 (d), w e hav e isomorphism of com- plexes Φ K − 1 N : ( e Ω • , δ K ) → ( e Ω • , δ K ◦ K − 1 N ), whic h induces an automorphism of the sub complex ( e Ω • 0 , 0 , 0). Hence, replacing K ( ∂ ) b y ( K ◦ K − 1 N )( ∂ ), it s uﬃces to p ro v e (a) for K ( ∂ ) with leading co eﬃcien t 1 I. Let P ∈ e Ω k ( V ) b e su ch that δ K P = 0. F or some i ∈ I , m ∈ Z + w e hav e P ∈ e Ω k m,i ( V ) and, b y Lemma 11.5 (c), we ha v e P = δ K ( h m,i P ) + P 1 , for some P 1 ∈ e Ω k m,i − 1 ( V ) s uc h that δ K P 1 = 0. Rep eating the same argument ﬁnitely man y times, w e get that P = δ K Q + R , for some Q ∈ e Ω k − 1 m,i ( V ) and R ∈ e Ω k 0 , 0 . Hence, Ker  δ K : e Ω k ( V ) → e Ω k +1 ( V )  = δ k  e Ω k − 1 ( V )  + e Ω k 0 , 0 . T o pro v e part (a) it remains to sho w that δ k  e Ω k − 1 ( V )  ∩ e Ω k 0 , 0 = 0 . Let P = δ K Q ∈ e Ω k 0 , 0 , for some Q ∈ e Ω k − 1 m,i . By Lemma 11.5 (c), w e hav e Q = δ K ( h m,i Q ) + h m,i ( δ K Q ) + Q 1 , for some Q 1 ∈ e Ω k − 1 m,i − 1 , and, b y Lemma 11.5 (b), w e ha v e h m,i ( δ K Q ) = h m,i P = 0. Hence, P = δ K Q 1 ∈ δ K e Ω k − 1 m,i − 1 ( V ). Re- p eating the s ame argument ﬁ nitely man y times, w e then get P ∈ δ K e Ω k − 1 0 , 0 = 0. Next, we prov e part (b). An element P ∈ e Ω k 0 , 0 is u niquely determined by the collection of p olynomials P 1 , .. , 1 | {z } n 1 ,..., ℓ, .. , ℓ | {z } n ℓ ( λ 1 , . . . , λ k ) ∈ F [ λ 1 , . . . , λ k ] ⊗ F , where n 1 , . . . , n ℓ ≥ 0 are suc h that n 1 + · · · + n ℓ = k , whic h ha v e degree at most N − 1 in eac h v ariable λ α , α = 1 , . . . , k , and, for eve ry i = 1 , . . . , ℓ , are sk ewsymmetric in the v ariables λ n 1 + ··· + n i − 1 +1 , . . . , λ n 1 + ··· + n i . Hence, e Ω k 0 , 0 is a vec tor space o ver F of dimension R k , give n by (11.10 ) R k = X n 1 ,...,n ℓ ∈ Z + n 1 + ··· + n ℓ = k ℓ Y i =1 C ( N , n i ) , where C ( N , n ) is the dimension of the space of skewsymmetric p olynomials in n v ariables of degree at most N − 1 in eac h v ariable, i.e. C ( N , n ) =  N n  . T aking the generating s eries of b oth sides of equation ( 11.10 ), w e then get ∞ X k =0 R k z k =  ∞ X n =0  N n  z n  ℓ = (1 + z ) N ℓ , whic h imp lies R k =  N ℓ k  , as r equired.  80 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 11.2. Cohomology of t he generalized v ariational complex. Recall from Prop osition 10.8 (b) that we hav e a short exact sequence of complexes 0 → ( ∂ e Ω • ( V ) , δ K ) α → ( e Ω • ( V ) , δ K ) β → (Ω • ( V ) , δ K ) → 0 , where α is the inclusion map, and β is the canonical quotien t map e Ω • ( V ) → e Ω • ( V ) /∂ e Ω • ( V ) = Ω • ( V ). It indu ces a long exact sequence in cohomology: (11.11 ) 0 → H 0 ( ∂ e Ω • , δ K ) α 0 → H 0 ( e Ω • , δ K ) β 0 → H 0 (Ω • , δ K ) γ 0 → H 1 ( ∂ e Ω • , δ K ) α 1 → . . . . . . γ k − 1 → H k ( ∂ e Ω • , δ K ) α k → H k ( e Ω • , δ K ) β k → H k (Ω • , δ K ) γ k → H k +1 ( ∂ e Ω • , δ K ) α k +1 → . . . Recall that, by Th eorem 11.2 , for ev ery k ∈ Z + , we ha v e a canonical ident i- ﬁcation H k ( e Ω • ( V ) , δ K ) = e Ω k 0 , 0 . Next, w e wan t to d escrib e H k ( ∂ e Ω • ( V ) , δ K ) and the map α k : H k ( ∂ e Ω • ( V ) , δ K ) → H k ( e Ω • ( V ) , δ K ). This is given by the follo win g Lemma 11.6. (a) The inclusion ( ∂ e Ω • 0 , 0 , 0) ⊂ ( ∂ e Ω • ( V ) , δ K ) , is a quasiiso- morphism of c omplexes, i.e. it induc es c anonic al isomorphisms: H k ( ∂ e Ω • ( V ) , δ K ) ≃ ∂ e Ω k 0 , 0 ≃  F / C for k = 0 e Ω k 0 , 0 for k ≥ 1 . (b) Under the identiﬁc ations H 0 ( e Ω • ( V ) , δ K ) = F in The or em 11.2 and H 0 ( ∂ e Ω • ( V ) , δ K ) = F / C in p art (a), the map α 0 induc es the map α 0 : F / C → F g iven by a + C 7→ ∂ a . (c) F or k ≥ 1 , identifying H k ( e Ω • ( V ) , δ K ) = e Ω k 0 , 0 = H k ( ∂ e Ω • ( V ) , δ K ) as in The or em 11.2 and in p art (a), the map α k induc es the endomorphism α k ∈ End  e Ω k 0 , 0  deﬁne d as fol lows. F or P ∈ e Ω k 0 , 0 , ther e exist Q ∈ e Ω k ( V ) and (a unique) R ∈ e Ω k 0 , 0 such that ∂ P = δ K Q + R . Then, (11.12 ) α k ( P ) = R . (d) Assuming that the le ading c o eﬃc i ent of K ( ∂ ) is 1 I , we c an write α k explicitly using the lo c al homotopy op er ators ( 11.5 ) : (11.13 ) α k ( P ) = (1 − δ K ◦ h 0 , 1 )(1 − δ K ◦ h 0 , 2 ) . . . (1 − δ K ◦ h 0 ,ℓ ) ∂ P . Pr o of. The map ∂ : Ω k ( V ) → Ω k ( V ) is injectiv e for k ≥ 1, w hile, for k = 0, w e ha v e e Ω 0 ( V ) = V and Ker( ∂ | V ) = C , the algebra of constants. Since ∂ and δ K comm ute, it follo w s that K er  δ K   ∂ e Ω k ( V )  = ∂ Ker  δ K   e Ω k ( V )  ⊂ e Ω k ( V ) for all k ≥ 0, and δ K  ∂ e Ω k − 1 ( V )  = ∂ δ K  e Ω k − 1 ( V )  ⊂ e Ω k ( V ) for k ≥ 1. Hence, we get H 0 ( ∂ e Ω • ( V ) , δ K ) = Ker  δ K   ∂ e Ω 0 ( V )  = ∂ K er  δ K   e Ω 0 ( V )  = ∂ F ≃ F / C . THE V ARIA TIONAL POISSON COHOMOLOGY 81 In the second last equalit y we used the fact that Ker  δ K   e Ω 0 ( V )  = F , b y the deﬁnition ( 10.11 ) of δ K and that, by assump tion, K has inv ertible lead- ing co eﬃcient. Moreo ver, the last isomorphism ab o ve is ind u ced by the surjectiv e m ap ∂ : F → ∂ F . Similarly , for k ≥ 1, w e ha ve H k ( ∂ e Ω • ( V ) , δ K ) = Ker  δ K   ∂ e Ω k ( V )  δ K  ∂ e Ω k − 1 ( V )  = ∂ Ker  δ K   e Ω k ( V )  ∂ δ K  e Ω k − 1 ( V )  ≃ Ker  δ K   e Ω k ( V )  δ K  e Ω k − 1 ( V )  = H k ( e Ω • ( V ) , δ K ) ≃ e Ω k 0 , 0 . In the third iden tit y w e us ed the injectiv eness of ∂ . This pro v es p art (a). The map α 0 : H 0 ( ∂ e Ω( V ) , δ K ) → H 0 ( e Ω( V ) , δ K ) is in duced by the in - clusion map ∂ e Ω 0 ( V ) = ∂ V ⊂ V = e Ω 0 ( V ). Sin ce H 0 ( e Ω( V ) , δ K ) = F and H 0 ( ∂ e Ω( V ) , δ K ) = ∂ F , the map α 0 coincides with the inclusion map ∂ F ⊂ F . P art (b) follo ws from the identiﬁcati on F / C ≃ ∂ F via the map a + C 7→ ∂ a . F or p art (c) we use a similar argumen t. The isomorph ism H k ( e Ω( V ) , δ K ) ≃ e Ω k 0 , 0 giv en b y Theorem 11.2 (a) maps P + δ K ( e Ω k − 1 ( V )) ∈ H k ( e Ω( V ) , δ K ) to the un ique element R ∈ e Ω k 0 , 0 suc h that P − R ∈ δ K ( e Ω k − 1 ( V )), and the in v erse map sends P ∈ e Ω k 0 , 0 to P + δ K ( e Ω k − 1 ( V )) ∈ H k ( e Ω( V ) , δ K ). S im- ilarly , we ha v e the canonical isomorphism, H k ( ∂ e Ω( V ) , δ K ) ≃ e Ω k 0 , 0 , which maps ∂ P + δ K ( ∂ e Ω k − 1 ( V )) ∈ H k ( ∂ e Ω( V ) , δ K ) to the uniqu e elemen t R ∈ e Ω k 0 , 0 suc h that P − R ∈ δ K ( e Ω k − 1 ( V )), and the inv erse map sends P ∈ e Ω k 0 , 0 to ∂ P + δ K ( ∂ e Ω k − 1 ( V )) ∈ H k ( ∂ e Ω( V ) , δ K ). Equation ( 11.12 ) follo ws from the fact that the map α k : H k ( ∂ e Ω( V ) , δ K ) → H k ( e Ω( V ) , δ K ) is indu ced b y the inclusion map ∂ e Ω k ( V ) ⊂ e Ω k ( V ), i.e. it send s ∂ P + δ K ( ∂ e Ω k − 1 ( V )) ∈ H k ( ∂ e Ω( V ) , δ K ) to ∂ P + δ K ( e Ω k − 1 ( V )) ∈ H k ( e Ω( V ) , δ K ). W e are left to prov e part (d). Giv en P =  P i 1 ,...,i k ( λ 1 , . . . , λ k )  i 1 ,...,i k ∈ I ∈ e Ω k 0 , 0 , the en tries of the arra y ∂ P ∈ e Ω k ( V ) are the p olynomials ( ∂ + λ 1 + · · · + λ k ) P i 1 ,...,i k ( λ 1 , . . . , λ k ), whic h hav e quasiconstan t co eﬃcien ts and h a v e degree at most N in eac h λ i . Hence, ∂ P ∈ e Ω k 0 ,ℓ ( F ) ⊂ e Ω k 0 ,ℓ ( V ). It follo ws by Lemma 11.5 (c) that (1 I − δ K ◦ h 0 , 1 )(1 I − δ K ◦ h 0 , 2 ) . . . (1 I − δ K ◦ h 0 ,ℓ ) ∂ P lies in e Ω k 0 , 0 . Since, ob viously , this element d iﬀers from ∂ P by an exact elemen t, w e conclude, by part (c), that it coincides with α k ( P ).  Using Theorem 11.2 and Lemma 11.6 , the long exact sequence ( 11.11 ) b ecomes (11.14 ) 0 → F / C ∂ → F β 0 → H 0 (Ω • ( V ) , δ K ) γ 0 → e Ω 1 0 , 0 α 1 → e Ω 1 0 , 0 β 1 → . . . · · · γ k − 1 → e Ω k 0 , 0 α k → e Ω k 0 , 0 β k → H k (Ω • ( V ) , δ K ) γ k → e Ω k +1 0 , 0 α k +1 → e Ω k +1 0 , 0 β k +1 → . . . 82 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Next, we stud y the maps β k , k ∈ Z + . First, it is clear that β 0 : e Ω 0 0 , 0 = F → H 0 (Ω • ( V ) , δ K ) ⊂ V /∂ V , is giv en by f 7→ R f . In particular, β 0 = 0 if and only if ∂ F = F . F or k ∈ Z + , let us consider th e map β k +1 : e Ω k +1 0 , 0 → H k +1 (Ω • ( V ) , δ K ). Let P ∈ e Ω k +1 0 , 0 , i.e. P =  P i 0 ,...,i k ( λ 0 , . . . , λ k )  i 0 ,...,i k ∈ I is a sk ewsymmetric arra y with resp ect to sim ultaneous p erm utations of the indices i 0 , . . . , i k and the v ariables λ 0 , . . . , λ k and, for eac h k -tuple ( i 0 , . . . , i k ), P i 0 ,...,i k ( λ 0 , . . . , λ k ) ∈ F [ λ 0 , . . . , λ k ] ⊗ F is a p olynomial of degree at most N − 1 in eac h v ariable λ i . Then, by deﬁn ition, β k +1 ( P ) ∈ H k (Ω • ( V ) , δ K ) is (11.15 ) β k +1 ( P ) =  P i 0 ,...,i k ( λ 0 , . . . , λ k )  i 0 ,...,i k ∈ I + δ K  Ω k ( V )  , where P i 0 ,...,i k ( λ 0 , . . . , λ k ) should no w b e view ed as an element of the sp ace F − [ λ 0 , . . . , λ k ] ⊗ F [ ∂ ] F . Note that the space of exact element s δ K  Ω k ( V )  con tains all arrays (11.16 )  k X α =0 ( − 1) α X j ∈ I K ∗ i α ,j ( λ 0 + α ˇ . . . + λ k + ∂ ) Q j i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . . , λ k )  i 0 ,...,i k ∈ I , where Q j i 1 ,...,i k ( λ 1 , . . . , λ k ) ∈ F [ λ 1 , . . . , λ k ] ⊗F are p olynomials with quasicon- stan t co eﬃcien ts, and they are sk ewsymmetric with resp ect to simultaneo us p ermutatio ns of i 1 , . . . , i k and λ 1 , . . . , λ k . Ind eed, recalling the deﬁnition ( 9.29 ) of δ K , we hav e that δ K applied to the arra y (11.17 )  X j ∈ I Q j i 1 ,...,i k ( λ 1 , . . . , λ k ) u j  i 1 ,...,i k ∈ I ∈ Ω k ( V ) , giv es ( 11.16 ). F or example, for k = 0, giv en P =  P i ( λ )  i ∈ I ∈ e Ω 1 0 , 0 , we ha v e β 1 ( P ) =  p i  i ∈ I + δ K  Ω 0 ( V )  , wh ere p i = P ∗ i (0) ∈ F . On the other hand, for Q j = f j ∈ F , the arra y ( 11.16 ) b ecome s  P j ∈ I K ∗ i,j ( ∂ ) f j  i ∈ I . Hence, β 1 = 0 pro vided that the map K ∗ ( ∂ ) : F ℓ → F ℓ is su rjectiv e. In general, for k ≥ 0, let P =  P i 0 ,...,i k ( λ 0 , . . . , λ k )  i 0 ,...,i k ∈ I ∈ e Ω k +1 0 , 0 , i.e. P is a sk ewsymmetric array with resp ect to sim ultaneous p erm u tations of the ind ices i 0 , . . . , i k and the v ariables λ 0 , . . . , λ k and suc h that, for eac h k - tuple ( i 0 , . . . , i k ), P i 0 ,...,i k ( λ 0 , . . . , λ k ) ∈ F [ λ 0 , . . . , λ k ] ⊗ F is a p olynomial of degree at most N − 1 in eac h v ariable λ i . By equ ation ( 11.15 ) and form ulas ( 11.16 ) and ( 11.17 ), w e ha v e th at β k +1 ( P ) = 0 pro vided that there exists a collect ion of ℓ sk ews ymmetric arra ys in k v ariables  Q j i 1 ,...,i k ( λ 1 , . . . , λ k )  i 1 ,...,i k ∈ I , THE V ARIA TIONAL POISSON COHOMOLOGY 83 indexed by j = 1 , . . . , ℓ , where Q j i 1 ,...,i k ( λ 1 , . . . , λ k ) are p olynomials with quasiconstan t co eﬃcient s, su c h that (11.18 ) k X α =0 ( − 1) α X j ∈ I K ∗ i α ,j ( λ 0 + α ˇ . . . + λ k + ∂ ) Q j i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . ., λ k ) − P i 0 ,i 1 ,...,i k ( λ 0 , λ 1 , . . . , λ k ) ∈ ( λ 0 + · · · + λ k + ∂ ) F [ λ 0 , . . . , λ k ] ⊗ F . Recall Deﬁnition A.3.1 from App endix A.3 of a linearly closed diﬀerential ﬁeld. Theorem 11.7. L et V b e a normal algebr a of diﬀe r ential functions, and as- sume that the algebr a of quasic onsta nts F ⊂ V is a line arly close d diﬀer ential ﬁeld. L e t K ( ∂ ) b e an ℓ × ℓ matrix diﬀer ential op er ator with quasic onsta nt c o eﬃcients and invertible le ading c o eﬃcient. Then β k = 0 for every k ≥ 0 and every ℓ ≥ 1 . Pr o of. The facts that β 0 = 0 and β 1 = 0 were p oin ted out ab o v e. Using notation ( A.5.7 ) and ( A.5.8 ) in App end ix A.5 , w e hav e that the arra y ( 11. 16 ) is equal to h K ∗ ◦ Q i − . Hence, condition ( 11.18 ), after replacing λ 0 b y − λ 1 − · · · − λ k − ∂ , b ecomes P = h K ∗ ◦ Q i − . Hence, the assertion that β k +1 ( P ) = 0 follo ws f r om T h eorem A.5.11 in the App endix.  Example 11.8 . I n the case k = 1 and ℓ = 1, the pr ob lem of solving equation ( 11.18 ) b ecomes: for P ( λ ) ∈ F [ λ ] skew adjoint, i.e. P ∗ ( λ ) := P ( − λ − ∂ ) = − P ( λ ), we w an t to ﬁnd Q ( λ ) ∈ F [ λ ], suc h that P ( λ ) = Q ∗ ( λ + ∂ ) K ( λ ) − K ∗ ( λ + ∂ ) Q ( λ ). Solutions for certain choice s of K ( λ ) are the follo wing: (1) K ( λ ) = 1: tak e Q ( λ ) = 1 2 P ( λ ), (2) K ( λ ) = λ : tak e Q ( λ ) such that ∂ Q ( λ ) = P ( λ ), (3) K ( λ ) = λ 2 : tak e Q ( λ ) = Q ∗ ( λ ) suc h that ( ∂ + 2 λ ) ∂ Q ( λ ) = P ( λ ), (4) K ( λ ) = λ 3 : tak e Q ( λ ) = ( λ − ∂ ) α + R ( λ ), with R ( λ ) = R ∗ ( λ ), s uc h that ( ∂ + 2 λ )  − ∂ 3 α + 2( λ 2 + λ∂ + ∂ 2 ) R ( λ )  = P ( λ ). By the exact sequence ( 11.14 ), β k = 0 implies that γ k : H k (Ω • ( V ) , δ K ) → e Ω k +1 0 , 0 is an em b edding, and its image coincides with the k ernel of the en- domorphism α k +1 of the C -v ector sp ace e Ω k +1 0 , 0 . Hence, in order to compute the v ariational Po isson cohomolog y , we need to stu d y the maps α k +1 and γ k . In p articular, we will use the r esults of App endix A.5 to compute the dimension o ver C of Ker( α k +1 ), whic h b y the ab o v e observ ations coincides with the dimension of H k (Ω • ( V ) , δ K ), and , for eac h elemen t C ∈ Ker( α k +1 ), w e will ﬁ n d a representati v e of γ − 1 ( C ) ∈ H k (Ω • ( V ) , δ K ) in e Ω k ( V ). T o start with, we n eed the follo w ing: Lemma 11.9. Supp ose that the algebr a of diﬀer ential functions V is an ex- tension of the algebr a of diﬀer ential p olynomials F  u ( n ) i   i ∈ I , n ∈ Z +  , for 84 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 a diﬀer ential ﬁeld F . Then, ther e exists a dir e ct sum (over F ) de c omp osition (11.19 ) V = F ⊕  M i ∈ I ,n ∈ Z + F u ( n ) i  ⊕ U , wher e U ⊂ V is an F -line ar subsp ac e of V suc h that (11.20 ) ∂ ∂ u ( m ) j U ⊂  M i ∈ I ,n ∈ Z + F u ( n ) i  ⊕ U , for al l j ∈ I , m ∈ Z + . Pr o of. Consider the map: F  u ( n ) i   i ∈ I , n ∈ Z +  ։ F given b y ev aluati ng at u ( n ) i = 0 , ∀ i ∈ I , n ∈ Z + , and extend it to a linear o v er F map ε : V ։ F . Let then U := n g ∈ V    ε ( g ) = 0 , ε  ∂ g ∂ u ( n ) i  = 0 ∀ i ∈ I , n ∈ Z + o . Clearly , for ev ery f ∈ V w e ha ve f − ε ( f ) − X i ∈ I ,n ∈ Z + ε  ∂ f ∂ u ( n ) i  u ( n ) i ∈ U , so that V = F +  L i ∈ I ,n ∈ Z + F u ( n ) i  + U . Moreo v er, if f = α + X i ∈ I ,n ∈ Z + β i,n u ( n ) i + g = 0 , with α, β i,n ∈ F and g ∈ U , then α = ε ( f ) = 0, and β i,n = ε  ∂ f ∂ u ( n ) i  = 0, so that V admits the direct sum d ecomp osition ( 11.19 ). Let then f ∈ V an d consider its decomposition giv en b y ( 11.19 ): f = α + P i ∈ I ,n ∈ Z + β i,n u ( n ) i + g , w here α, β i,n ∈ F and g ∈ U . W e ha ve ε ( f ) = α , pro ving that (11.21 ) Ker( ε ) =  M i ∈ I ,n ∈ Z + F u ( n ) i  ⊕ U ⊂ V . T o conclude, we note that, by the deﬁ n ition of U , if g ∈ U then ∂ g ∂ u ( n ) i ∈ Ker( ε ) for every i ∈ I , n ∈ Z + , whic h, together with ( 11.21 ), giv es ( 11.20 ) .  Recall from Ap p endix A.5.1 that a k -diﬀeren tial op erator on F ℓ is an arra y P =  P i 0 ,i 1 ,...,i k ( λ 1 , . . . , λ k )  i 0 ,i 1 ,...,i k ∈ I , whose en tries are p olynomials in λ 1 , . . . , λ k with co eﬃcient s in F , and it is said to b e sk ewsymmetric if the en tries P i 0 ,i 1 ,...,i k ( λ 1 , . . . , λ k ) are skewsymmetric with resp ect to sim ultane- ous p erm utations of th e in d ices i 1 , . . . , i k and th e v ariables λ 1 , . . . , λ k . Giv en an ℓ × ℓ matrix diﬀerential op er ator K ( ∂ ), we denote b y Σ k ( K ) the space of sk ewsymmetric k -diﬀeren tial op erators on F ℓ whose en tries are p olynomials of degree at most N − 1 in eac h v ariable λ 1 , . . . , λ k , solving equation ( A.5.52 ). F or example Σ 0 ( K ) consists of elemen ts P ∈ F ℓ solving K ( ∂ ) P = 0. By THE V ARIA TIONAL POISSON COHOMOLOGY 85 Theorem A.5.12 , if F is a linearly closed diﬀeren tial ﬁeld, then Σ k ( K ) is a v ecto r sp ace o v er C of dimension  N ℓ k +1  . Theorem 11.10. L et k ∈ Z + . L et V b e a normal algebr a of diﬀer ential functions, and assume that the algebr a of quasic onstants F ⊂ V is a line arly close d diﬀer ential ﬁeld. L et K ( ∂ ) b e an ℓ × ℓ matrix diﬀer ential op e r ator of or der N with quasic onstant c o eﬃcients and invertible le ading c o eﬃci e nt K N ∈ Mat ℓ × ℓ ( F ) . (a) Ther e is a c ano nic al isomorphism of C -ve ctor sp ac es φ k : Σ k ( K ∗ ) → Ker( α k +1 ) , deﬁne d as fol lows: given P ∈ Σ k ( K ∗ ) , we let φ k ( P ) = C ∈ Ker( α k +1 ) , wher e (11.22 ) k X α =0 ( − 1) α X j ∈ I P j,i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . . , λ k ) K j,i α ( λ α ) = ( λ 0 + λ 1 + · · · + λ k + ∂ ) C i 0 ,i 1 ,...,i k ( λ 0 , λ 1 , . . . , λ k ) , for al l indic es i 0 , . . . , i k ∈ I (e quality in F [ λ 0 , . . . , λ k ] ⊗ F ). (b) Ther e is a c anonic al isomorphism χ k : Σ k ( K ∗ ) ≃ H k (Ω • ( V ) , δ K ) deﬁne d as fol low s: given P ∈ Σ k ( K ∗ ) , we let χ k ( P ) ∈ H k (Ω • ( V ) , δ K ) b e the c ohomolo g y class with r epr esentative (11.23 )  X j ∈ I P j,i 1 ,...,i k ( λ 1 , . . . , λ k ) u j  i 1 ,...,i k ∈ I ∈ e Ω k ( V ) . In p articular, (11.24 ) dim C ( H k (Ω • ( V ) , δ K )) =  N ℓ k + 1  . (c) The maps γ k : H k (Ω • ( V ) , δ K ) → Ker( α k +1 ) in the exact se quenc e ( 11.14 ) and φ k : Σ k ( K ∗ ) → Ker( α k +1 ) ar e c omp atible in the sense that (11.25 ) φ k = γ k ◦ χ k . F or C ∈ Ker( α k +1 ) , let φ − 1 k ( C ) =  P i 0 ,i 1 ,...,i k ( λ 1 , . . . , λ k )  i 0 ,i 1 ,...,i k ∈ I ∈ Σ k ( K ∗ ) . Then, the arr ay ( 11.23 ) in e Ω k ( V ) is a r e pr esentative of the c ohomolo g y class γ − 1 k ( C ) ∈ H k (Ω • ( V ) , δ K ) . Pr o of. First, w e pro v e that the map φ k giv en b y ( 11.22 ) is w ell d eﬁ ned. Let P ∈ Σ k ( K ∗ ), so that h K ∗ ◦ P i − = 0. By equation ( A.5.10 ) from the App end ix, we can r ewrite this condition by sa ying th at (11.26 ) k X α =0 ( − 1) α X j ∈ I P j,i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . . , λ k ) K j,i α ( λ α ) b ecomes zero if w e replace λ 0 b y − λ 1 − · · · − λ k − ∂ , with ∂ acting from the left. I n other w ords, ( 11.26 ), as an element of F [ λ 0 , λ 1 , . . . , λ k ] ⊗ F , is equal to (11.27 ) ( λ 0 + λ 1 + · · · + λ k + ∂ ) C i 0 ,i 1 ,...,i k ( λ 0 , λ 1 , . . . , λ k ) , 86 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 where C =  C i 0 ,i 1 ,...,i k ( λ 0 , λ 1 , . . . , λ k )  i 0 ,i 1 ,...,i k ∈ I is a sk ewsymmetric arr a y whose en tries are p olynomials with quasiconstan t co eﬃcien ts of degree less than or equal to N − 1, i.e. C ∈ e Ω k +1 0 , 0 . F ur thermore, we claim that C lies in K er( α k +1 ). Indeed, ta king Q ∈ e Ω k ( V ) b e the arra y ( 11.23 ), w e ha v e, by ( 10.11 ), that ( δ K Q ) i 0 ,...,i k ( λ 0 , . . . , λ k ) is equal to ( 11.26 ). Hence, the equalit y of ( 11.26 ) and ( 11.27 ) implies that δ K Q = ∂ C . Therefore, b y Lemma 11.6 (c), we conclude that α k +1 ( C ) = 0, proving that φ k is w ell deﬁned. W e next prov e that the map φ k : Σ k ( K ∗ ) → Ker( α k +1 ) is injectiv e. Since, b y assu mption, P i 0 ,...,i k ( λ 1 , . . . , λ k ) h as d egree less than or equal to N − 1 in eac h v ariable, th e co eﬃcien t of λ N 0 in ( 11.26 ) is (11.28 ) X j ∈ I P j,i 1 ,...,i k ( λ 1 , . . . , λ k )( K N ) j,i 0 . T o sa y that φ k ( P ) = 0 is equiv alen t to say that ( 11.26 ), view ed as an elemen t of F [ λ 0 , . . . , λ k ] ⊗ F , is identica lly zero for all ind ices i 0 , . . . , i k ∈ I . In particular ( 11 .28 ) is zero. Since, b y assumption, K N is an in v ertible matrix, it follo w s that P i 0 ,i 1 ,...,i k ( λ 1 , . . . , λ k ) = 0, for all ind ices i 0 , . . . , i k . Hence, φ k is in j ectiv e. T o complete the pro of of part (a) w e are left with sho wing that the map φ k is surjectiv e. Let C =  C i 0 ,...,i k ( λ 0 , . . . , λ k )  i 0 ,...,i k ∈ I b e an elemen t of Ker( α k +1 ). By Lemma 11.6 (c), there exists an elemen t Q ∈ e Ω k ( V ) su c h that (11.29 ) ∂ C = δ K Q . By Lemma 11.9 , w e can assume that the co eﬃcien ts of Q are linear in the v ariables u ( n ) j , i.e. (11.30 ) Q i 1 ,...,i k ( λ 1 , . . . , λ k ) = M X n =0 X j ∈ I P n j,i 1 ,...,i k ( λ 1 , . . . , λ k ) u ( n ) j with M ∈ Z + and P n j,i 1 ,...,i k ( λ 1 , . . . , λ k ) ∈ F [ λ 1 , . . . , λ k ] ⊗ F . In deed, the quasiconstan t p art of Q is killed b y the diﬀerent ial δ K , while δ K applied to the U -p art of Q has zero quasiconstan t p art. Note that, if Q is as in ( 11.30 ), then equation ( 11.29 ) b ecomes (11.31 ) k X α =0 ( − 1) α X j ∈ I M X n =0 P n j,i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . . , λ k )( λ α + ∂ ) n K j,i α ( λ α ) = ( λ 0 + · · · + λ k + ∂ ) C i 0 ,...,i k ( λ 0 , . . . , λ k ) ∈ F [ λ 0 , . . . , λ k ] ⊗ F , for all c hoices of indices i 0 , . . . , i k ∈ I . In order to prov e sur jectivit y of φ k , we w ill sho w that w e can c ho ose Q as in ( 11.30 ) with M = 0 and P 0 j,i 1 ,...,i k ( λ 1 , . . . , λ k ) of d egree at m ost N − 1 in eac h v ariable λ 1 , . . . , λ k , suc h that equation ( 11.31 ) holds w ith the giv en THE V ARIA TIONAL POISSON COHOMOLOGY 87 C ∈ Ker( α k +1 ). In this case, by the deﬁnition ( 11.22 ) of the map φ k , P =  P 0 i 0 ,i 1 ,...,i k ( λ 1 , . . . , λ k )  i 0 ,i 1 ,...i k ∈ I is an elemen t of Σ k ( K ∗ ) suc h that φ k ( P ) = C . W e w ill achiev e the desired form of Q in three steps: ﬁrst w e reduce to the case when all p olynomials P n j,i 1 ,...,i k ( λ 1 , . . . , λ k ) hav e degree less than or equal to M + N in eac h v ariable λ α ; th en w e reduce to the case when M = 0; ﬁnally we reduce to the case w hen th e p olynomials P 0 j,i 1 ,...,i k ( λ 1 , . . . , λ k ) h a v e degree at most N − 1 in eac h v ariable. Note that in the case k = 0 we only need to do the second step. Let k ≥ 1 a nd le t d b e the maximal degree in one o f the v ariables λ 1 , . . . , λ k of all the p olynomials P n j,i 1 ,...,i k ( λ 1 , . . . , λ k ) for n = 0 , . . . , M and j, i 1 , . . . , i k ∈ I , and assume that d > M + N . By taking separately all terms in whic h some of the v ariables λ α are raised to the p o w er d , we can write (11.32 ) P n j,i 1 ,...,i k ( λ 1 , . . . , λ k ) = R n j,i 1 ,...,i k λ d 1 . . . λ d k + X 1 ≤ β ≤ k R n,β j,i 1 ,...,i k ( λ β ) λ d 1 β ˇ . . . λ d k + X 1 ≤ β <γ ≤ k R n,β , γ j,i 1 ,...,i k ( λ β , λ γ ) λ d 1 β ˇ . . . γ ˇ . . . λ d k + · · · + R n j,i 1 ,...,i k ( λ 1 , . . . , λ k ) = k X q =0 X 1 ≤ β 1 < ··· <β q ≤ k R n,β 1 ,...,β q j,i 1 ,...,i k ( λ β 1 , . . . , λ β q ) λ d 1 β 1 ,...,β q ˇ . . . . . . λ d k , where R n,β 1 ,...,β q j,i 1 ,...,i k ( λ β 1 , . . . , λ β q ) are p olynomials with qu asiconstan t co eﬃ- cien ts of degree strictly less than d in eac h v ariable. Then equation ( 11.31 ) b ecomes k X α =0 ( − 1) α X j ∈ I M X n =0 k X q =0 X 0 ≤ β 1 < ··· <β q ≤ k ( β h <α<β h +1 ) λ d 0 α,β 1 ,...,β q ˇ . . . . . . λ d k × R n,β 1 +1 ,...,β h +1 ,β h +1 ,...β q j,i 0 , α ˇ ...,i k ( λ β 1 , . . . , λ β q )( λ α + ∂ ) n K j,i α ( λ α ) = ( λ 0 + · · · + λ k + ∂ ) C i 0 ,...,i k ( λ 0 , . . . , λ k ) . W e can rewrite the ab o ve equation in the f ollo wing equiv alen t form (11.33 ) k X q =0 X 0 ≤ β 0 < ··· <β q ≤ k q X r =0 ( − 1) β r X j ∈ I M X n =0 λ d 0 β 0 ,...,β q ˇ . . . . . . λ d k × R n,β 0 +1 ,...,β r − 1 +1 ,β r +1 ,...,β q j,i 0 , β r ˇ ... ,i k ( λ β 0 , r ˇ . . . , λ β q ) × ( λ β r + ∂ ) n K j,i β r ( λ β r ) = ( λ 0 + · · · + λ k + ∂ ) C i 0 ,...,i k ( λ 0 , . . . , λ k ) . Note that the RHS ab o v e has degree at most N in eac h v ariable λ 1 , . . . , λ k . Hence, by lo oking at the co eﬃcient of λ d 1 . . . λ d k in b oth sides of equation 88 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 ( 11.33 ), w e get X j ∈ I M X n =0 R n j,i 1 ,...,i k ( λ 0 + ∂ ) n K j,i 0 ( λ 0 ) = 0 . Since, b y assumption, the leading coeﬃcien t K N of the diﬀeren tial operator K ( ∂ ) is an inv ertible matrix, one easily gets that R n j,i 1 ,...,i k = 0 for all n = 0 , . . . , M and j, i 1 , . . . , i k ∈ I . Hence, in the LHS of ( 11.33 ) the term with q = 0 v anishes. Next, for 0 ≤ β 0 < β 1 ≤ k , by looking at the co eﬃcient of λ d 0 β 0 ,β 1 ˇ . . . . . . λ d k in b oth s id es of equation ( 11.33 ), w e get T β 0 ,β 1 ( λ β 0 , λ β 1 ) := ( − 1) β 0 X j ∈ I M X n =0 R n,β 1 j,i 0 , β 0 ˇ ... ,i k ( λ β 1 )( λ β 0 + ∂ ) n K j,i β 0 ( λ β 0 ) +( − 1) β 1 X j ∈ I M X n =0 R n,β 0 +1 j,i 0 , β 1 ˇ ... , i k ( λ β 0 )( λ β 1 + ∂ ) n K j,i β 1 ( λ β 1 ) = 0 . On the other hand, the term with q = 1 in the LHS of ( 11.33 ) is exactly X 0 ≤ β 0 <β 1 ≤ k T β 0 ,β 1 ( λ β 0 , λ β 1 ) λ d 0 β 0 ,β 1 ˇ . . . . . . λ d k , hence, it v anishes, and the sum o ver q in the LHS of ( 11.33 ) starts with q = 2. Rep eating the same argument several times, w e conclude that all the terms with q ≤ k − 1 in th e LHS of ( 11.33 ) v anish, hence the equation b ecomes (11.34 ) k X α =0 ( − 1) α X j ∈ I M X n =0 R n, 1 ,...,k j,i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . . , λ k )( λ α + ∂ ) n K j,i α ( λ α ) = ( λ 0 + · · · + λ k + ∂ ) C i 0 ,...,i k ( λ 0 , . . . , λ k ) . Comparing equations ( 11.31 ) and ( 11.34 ), we can r eplace the p olynomials P n j,i 1 ,...,i k ( λ 1 , . . . , λ k ) by th e p olynomials R n, 1 ,...,k j,i 1 ,...,i k ( λ 1 , . . . , λ k ), w h ic h h a v e degree strictly less than d . Hence, r ep eating this argumen t sev eral times, w e ma y assume that the degree of all p olynomials P n j,i 1 ,...,i k ( λ 1 , . . . , λ k ) is less than or equal to M + N , concluding the ﬁrst step. In the second step w e w an t to reduce to the case wh en M = 0. F or this, assuming M ≥ 1, w e will reduce to the case when M is replaced by M − 1. W e ﬁ nd an expansion of P similar to the one discussed in equation ( 11.32 ). Using the fact that K ( ∂ ) has order N and its leading coeﬃcient K N is an in v ertible matrix, we can write (11.35 ) P n j,i 1 ,...,i k ( λ 1 , . . . , λ k ) = k X q =0 X j 1 ,...,j k ∈ I X 1 ≤ β 1 < ··· <β q ≤ k Q n,β 1 ,...,β q j,j 1 ,...,j k ( λ β 1 , . . . , λ β q ) × δ j β 1 ,i β 1 . . . δ j β q ,i β q  ( λ 1 + ∂ ) M K j 1 ,i 1 ( λ 1 )  β 1 ...β q ˇ . . . . . .  ( λ k + ∂ ) M K j k ,i k ( λ k )  , THE V ARIA TIONAL POISSON COHOMOLOGY 89 where Q n,β 1 ,...,β q j,j 1 ,...,j k ( λ β 1 , . . . , λ β q ) are p olynomials with quasiconstan t coeﬃ- cien ts of d egree strictly less than M + N in eac h v ariable λ β 1 , . . . , λ β q . Th en, equation ( 11.31 ) b ecomes k X α =0 X j ∈ I M X n =0 k X q =0 X j 0 , α ˇ ...,j k ∈ I X 0 ≤ β 1 < ··· <β q ≤ k ( β h <α<β h +1 ) ( − 1) α Q n,β 1 +1 ,...,β h +1 ,β h +1 ,...,β q j,j 0 , α ˇ ...,j k ( λ β 1 , . . . , λ β q ) × δ j β 1 ,i β 1 . . . δ j β q ,i β q  ( λ 0 + ∂ ) M K j 0 ,i 0 ( λ 0 )  α,β 1 ...β q ˇ . . . . . .  ( λ k + ∂ ) M K j k ,i k ( λ k )  ×  ( λ α + ∂ ) n K j,i α ( λ α )  = ( λ 0 + · · · + λ k + ∂ ) C i 0 ,...,i k ( λ 0 , . . . , λ k ) , or, rearranging terms app ropriately , w e can rewrite it in the follo wing equiv- alen t form (11.36 ) M X n =0 k X q =0 q X r =0 X 0 ≤ β 0 < ··· <β q ≤ k ( − 1) β r X j 0 ,...,j k ∈ I Q n,β 0 +1 ,...,β r − 1 +1 ,β r +1 ,...,β q j β r ,j 0 , β r ˇ ... ,j k ( λ β 0 , r ˇ . . . , λ β q ) × δ j β 0 ,i β 0 r ˇ . . . δ j β q ,i β q  ( λ 0 + ∂ ) M K j 0 ,i 0 ( λ 0 )  β 0 ...β q ˇ . . . . . .  ( λ k + ∂ ) M K j k ,i k ( λ k )  ×  ( λ β r + ∂ ) n K j β r ,i β r ( λ β r )  = ( λ 0 + · · · + λ k + ∂ ) C i 0 ,...,i k ( λ 0 , . . . , λ k ) . Note that the RHS has d egree at most N in eac h v ariable λ 1 , . . . , λ k . By lo oking at the co eﬃcien t of λ M + N 0 . . . λ M + N k in b oth sides of equation ( 11.36 ), w e get, since M ≥ 1, k X α =0 ( − 1) α X j 0 ,...,j k ∈ I Q M j α ,j 0 , α ˇ ...,j k ( K N ) j 0 ,i 0 . . . ( K N ) j k ,i k = 0 . Since K N is inv ertible, we d educe that T i 0 ,...,i k := k X α =0 ( − 1) α Q M i α ,i 0 , α ˇ ...,i k = 0 . On the other h and, the term w ith q = 0 and n = M in the L HS of ( 11.36 ) is equal to X j 0 ,...,j k ∈ I T j 0 ,...,j k  ( λ 0 + ∂ ) M K j 0 ,i 0 ( λ 0 )  . . .  ( λ k + ∂ ) M K j k ,i k ( λ k )  , hence it v anish es. Next, for k ≥ 1 ﬁx α ∈ { 1 , . . . , k } an d consider the co eﬃcien t of λ M + N 0 α ˇ . . . λ M + N k in b oth sides of equation ( 11.36 ). In the RHS we get 0 since M ≥ 1, while in the LHS there are only t w o contributions, one coming from q = 0 90 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 and n ≤ M − 1, and the other coming fr om q = 1 and n = M . W e thus get M − 1 X n =0 ( − 1) α X j 0 ,...,j k ∈ I Q n j α ,j 0 , α ˇ ...,j k ( K N ) j 0 ,i 0 α ˇ . . . ( K N ) j 0 ,i 0  ( λ α + ∂ ) n K j α ,i α ( λ α )  + k X β =0 ( β <α ) ( − 1) β X j 0 ,...,j k ∈ I Q M ,α j β ,j 0 , β ˇ ...,j k ( λ α ) δ j α ,i α ( K N ) j 0 ,i 0 α ˇ . . . ( K N ) j k ,i k + k X β =0 ( β >α ) ( − 1) β X j 0 ,...,j k ∈ I Q M ,α +1 j β ,j 0 , β ˇ ...,j k ( λ α ) δ j α ,i α ( K N ) j 0 ,i 0 α ˇ . . . ( K N ) j k ,i k = 0 . Again, sin ce K N is in v er tib le, we d educe th at T α i 0 ,...,i k ( λ α ) := k X β =0 ( β <α ) ( − 1) β Q M ,α i β ,i 0 , β ˇ ...,i k ( λ α ) + k X β =0 ( β >α ) ( − 1) β Q M ,α +1 i β ,i 0 , β ˇ ...,i k ( λ α ) + M − 1 X n =0 X j ∈ I ( − 1) α Q n j,i 0 , α ˇ ...,i k  ( λ α + ∂ ) n K j,i α ( λ α )  = 0 . On the other hand, the sum of the term with q = 1 and n = M together with the terms with q = 0 and n ≤ M − 1 in the LHS of ( 11.36 ) is equal to X j 0 ,...,j k ∈ I k X α =0 T α j 0 ,...,j k ( λ α ) δ j α ,i α ×  ( λ 0 + ∂ ) M K j 0 ,i 0 ( λ 0 )  α ˇ . . .  ( λ k + ∂ ) M K j k ,i k ( λ k )  , hence it v anish es. Rep eating the same argument sev eral times, at eac h step w e prov e that the sum of the term with q = q 0 + 1 and n = M together with the terms with q = q 0 and n ≤ M − 1 v anishes. As a resu lt, in the LHS of equation ( 11.36 ) only the terms with q = k and n < M su rviv e. Hence, equation ( 11.36 ) b ecomes M − 1 X n =0 k X α =0 ( − 1) α X j ∈ I Q n, 1 ,...,k j,i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . . , λ k )( λ α + ∂ ) n K j,i α ( λ α ) = ( λ 0 + · · · + λ k + ∂ ) C i 0 ,...,i k ( λ 0 , . . . , λ k ) . This is the same as ( 11.31 ) with the p olynomials P n j,i 1 ,...,i k ( λ 1 , . . . , λ k ) re- placed b y 0 for n = M , and b y th e p olynomials Q n, 1 ,...,k j,i 1 ,...,i k ( λ 1 , . . . , λ k ) for n < M . T his completes the second step. So f ar, we show ed that w e can c ho ose Q in ( 11.30 ) of th e form (11.37 ) Q i 1 ,...,i k ( λ 1 , . . . , λ k ) = X j ∈ I P j,i 1 ,...,i k ( λ 1 , . . . , λ k ) u j , THE V ARIA TIONAL POISSON COHOMOLOGY 91 where P j,i 1 ,...,i k ( λ 1 , . . . , λ k ) are p olynomials with qu asiconstan t co eﬃcients of d egree at most N in eac h v ariable. In this case equation ( 11.31 ) reads (11.38 ) k X α =0 ( − 1) α X j ∈ I P j,i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . . , λ k ) K j,i α ( λ α ) = ( λ 0 + · · · + λ k + ∂ ) C i 0 ,...,i k ( λ 0 , . . . , λ k ) ∈ F [ λ 0 , . . . , λ k ] ⊗ F . T o complete the pro of of part (a), we are left with sho wing that w e can c hoose the p olynomials P j,i 1 ,...,i k ( λ 1 , . . . , λ k ) to b e of degree at most N − 1 in eac h v ariable λ α , suc h that equation ( 11.38 ) still holds. As b efore, we expand the p olynomials P j,i 1 ,...,i k ( λ 1 , . . . , λ k ) as in ( 11.35 ) (11.39 ) P j,i 1 ,...,i k ( λ 1 , . . . , λ k ) = k X q =0 X j 1 ,...,j k ∈ I X 1 ≤ β 1 < ··· <β q ≤ k Q β 1 ,...,β q j,j 1 ,...,j k ( λ β 1 , . . . , λ β q ) × δ j β 1 ,i β 1 . . . δ j β q ,i β q K j 1 ,i 1 ( λ 1 ) β 1 ...β q ˇ . . . . . . K j k ,i k ( λ k ) , where the p olynomials Q β 1 ,...,β q j,j 1 ,...,j k ( λ β 1 , . . . , λ β q ) ha v e d egree strictly less than N . Then, equation ( 11.38 ) reads (11.40 ) k X q =0 X j 0 ,...,j k ∈ I k X α =0 X 0 ≤ β 1 < ··· <β q ≤ k ( β h <α<β h +1 ) ( − 1) α Q β 1 +1 ,...,β h +1 ,β h +1 ,...,β q j α ,j 0 , α ˇ ...,j k ( λ β 1 , . . . , λ β q ) × δ j β 1 ,i β 1 . . . δ j β q ,i β q K j 0 ,i 0 ( λ 0 ) β 1 ...β q ˇ . . . . . . K j k ,i k ( λ k ) = ( λ 0 + · · · + λ k + ∂ ) C i 0 ,...,i k ( λ 0 , . . . , λ k ) . W e then pro ceed as in step tw o. Note that, since by assumption the p olyno- mials C i 0 ,...,i k ( λ 0 , . . . , λ k ) ha v e d egree at most N − 1 in eac h v ariable, in the righ t hand side of ( 11.40 ) in eac h monomial at most one v ariable λ α app ears in degree N . Therefore, for k ≥ 1, comparing the co eﬃcient of λ N 0 . . . λ N k in b oth sid es of ( 11.40 ), and u sing the fact that K N is inv ertible, we get T i 0 ,...,i k := k X α =0 ( − 1) α Q i α ,i 0 , α ˇ ...,i k = 0 , for every choice of indices i 0 , . . . , i k ∈ I . But the term with q = 0 in the LHS of ( 11.40 ) is X j 0 ,...,j k ∈ I T j 0 ,...,j k K j 0 ,i 0 ( λ 0 ) . . . K j k ,i k ( λ k ) , hence it v anishes. Similarly , f or k ≥ 2, giv en β ∈ { 0 , . . . , k } and comparing the coeﬃcient of λ N 0 β ˇ . . . λ N k in b oth sides of ( 11.40 ), we get, again using the 92 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 fact th at K N is inv ertible, T β i 0 ,...,i k ( λ β ) := X α<β ( − 1) α Q β i α ,i 0 , α ˇ ...,i k ( λ β ) + X α>β ( − 1) α Q β +1 i α ,i 0 , α ˇ ...,i k ( λ β ) = 0 , for ev ery choi ces of indices i 0 , . . . , i k ∈ I . and the term with q = 1 in the LHS of ( 11.40 ) is exact ly k X β =0 X j 0 ,...,j k ∈ I T β j 0 ,...,j k δ j β ,i β K j 0 ,i 0 ( λ 0 ) β ˇ . . . K j k ,i k ( λ k ) , hence it v anishes. Rep eating the same argument several times, w e p ro v e that all th e terms in the LHS of ( 11.40 ) with q ≤ k − 1 v anish. Note that the same argument alwa ys works for q ≤ k − 1 since the monomial λ N 0 β 1 ...β q ˇ . . . . . . λ N k con tains at least t w o v ariables raised to the p o wer N . In conclusion, equation ( 11.40 ) is equiv alen t to the same equation where in the LHS we only k eep the term with q = k : X j ∈ I k X α =0 ( − 1) α Q 1 ,...,k j,i 0 , α ˇ ...,i k ( λ 0 , α ˇ . . . , λ k ) K j,i α ( λ α ) = ( λ 0 + · · · + λ k + ∂ ) C i 0 ,...,i k ( λ 0 , . . . , λ k ) . In other w ord s, w e can r eplace the p olynomials P j,i 1 ,...,i k ( λ 1 , . . . , λ k ) deﬁned in ( 11.37 ) b y the p olynomials Q 1 ,...,k j,i 1 ,...,i k ( λ 1 , . . . , λ k ), which hav e degree at most N − 1 in eac h v ariable, without changing the RHS of equation ( 11.38 ). This completes the pro of of part (a). The dimension form u la ( 11.2 4 ) follo ws from the ﬁrst assertion in part (b) and Th eorem A.5.12 . Note that, if P ∈ Σ k ( K ∗ ), then δ K applied to the arra y ( 11.23 ) is in the image of ∂ , hen ce the arra y ( 11 .23 ) deﬁnes a cohomology class in Ω k ( V ). Therefore, χ k is a wel l-deﬁned map: Σ k ( K ∗ ) → H k (Ω • ( V ) , δ K ). In order to complete the pro of of b oth parts (b ) and (c), we only need to chec k that the map χ k : Σ k ( K ∗ ) → H k (Ω • ( V ) , δ K ) giv en b y ( 11.23 ) satisﬁes equation ( 11.25 ). Ind eed, the map φ k : Σ k ( K ∗ ) → Ker( α k +1 ) is an isomorphism by p art (a), and th e map γ k : H k (Ω • ( V ) , δ K ) → Ker( α k +1 ) is an isomorph ism by T heorem 11.7 and the long exact s equence ( 11.14 ). Hence, equation ( 11.25 ) implies that the map χ k m ust b e an isomorphism as we ll. Also, the last assertion in part (c) is clear since, b y ( 11.25 ), w e ha v e γ − 1 k = χ k ◦ φ − 1 k . Before pro ving equation ( 11.25 ), let us recall the usual homologic al alge- bra deﬁnition of the b ou n dary map γ k in the long exact sequence ( 11.11 ). F or [ ω ] ∈ H k (Ω • ( V ) , δ K ), w e hav e γ k ([ ω ]) =  α − 1 δ K β − 1 ( ω )  ∈ H k +1 ( ∂ e Ω • ( V ) , δ K ) . THE V ARIA TIONAL POISSON COHOMOLOGY 93 In other words, let [ ω ] ∈ H k (Ω • ( V ) , δ K ) b e the class of a clo sed elemen t ω ∈ Ω k ( V ). Since β : e Ω k ( V ) → Ω k ( V ) is surjectiv e, th ere exists η ∈ e Ω k ( V ) suc h that β ( η ) = ω . Since, by assu mption, δ K ω = 0, w e ha v e that δ K ( η ) ∈ Ker( β ) = Im( α ). Hence, there exists ζ ∈ ∂ e Ω k +1 ( V ) s uc h that δ K ( η ) = α ( ζ ). Since α is injectiv e, δ K ( ζ ) = 0, and w e let γ k ([ ω ]) = [ ζ ] ∈ H k +1 ( ∂ e Ω • ( V ) , δ K ). Using the iden tiﬁcation of the exact sequence ( 11.11 ) with ( 11.14 ), the con- struction of the map γ k can b e describ ed as follo ws. Consider a sk ewsymmet- ric arr a y P =  P i 1 ,...,i k ( λ 1 , . . . , λ k )  i 1 ,...,i k ∈ I ∈ e Ω k ( V ) such that, when view ed as an elemen t in Ω k ( V ) (i.e. wh en w e view its entries in F − [ λ 1 , . . . , λ k ] ⊗ F [ ∂ ] V ), it is closed: δ K P = 0 in Ω k +1 ( V ). By Th eorem 11.2 (a) there exists a unique s kewsymmetric arra y C =  C i 0 ,...,i k ( λ 0 , . . . , λ k )  i 0 ,...,i k ∈ I ∈ e Ω k +1 0 , 0 , where C i 0 ,...,i k ( λ 0 , . . . , λ k ) are p olynomials with quasiconstan t co eﬃcients of degree at most N − 1 in eac h v ariable λ i , su c h that δ K P = ∂ ( C + δ K Q ) for some Q ∈ e Ω k ( V ). Th en γ k ([ P ]) = C . Giv en P ∈ Σ k ( K ∗ ), the arr a y φ k ( P ) = C ∈ Ker( α k +1 ) is deﬁn ed by equation ( 11.22 ). On the other h and, a repr esen tativ e of the cohomology class χ k ( P ) ∈ H k (Ω • ( V ) , δ K ) is th e arra y Q ∈ e Ω k ( V ) giv en by ( 11.23 ), and, by the ab o v e observ ations, the array γ k ( χ k ( P )) = C 1 ∈ Ker( α k +1 ) is deﬁned by the equation δ K Q = ∂ C 1 . Since this equation for C 1 coincides with equation ( 11.22 ) for C , w e conclude that C 1 = C , p ro ving form ula ( 11.25 ).  11.3. Explicit description of H 0 (Ω • ( V ) , δ K ) and H 1 (Ω • ( V ) , δ K ) . As- sume th at V is a normal algebra of d iﬀeren tial fu nctions and F ⊂ V is a linearly clo sed diﬀerential ﬁ eld, so that T heorem 11.10 holds. It is easy to see from the deﬁnition ( 9.29 ) of the action of δ K on Ω 0 = V /∂ V that H 0 (Ω • ( V ) , δ K ) = n R f ∈ V /∂ V    K ∗ ( ∂ ) δ f δ u = 0 o . The space Σ 0 ( K ∗ ) describ ed b efore Th eorem 11.10 is Σ 0 ( K ∗ ) = n P ∈ F ℓ    K ∗ ( ∂ ) P = 0 o , and the isomorph ism χ 0 : Σ 0 ( K ∗ ) → H 0 (Ω • ( V ) , δ K ), deﬁn ed in T h eorem 11.10 (b), is giv en b y χ 0 ( P ) = Z X j ∈ I P j u j . It is immediate to c heck that the v ariational deriv ativ e of R P j P j u j is P , hence, if P lies in Σ 0 ( K ∗ ), then χ 0 ( P ) lies in H 0 (Ω • ( V ) , δ K ). Recall fr om Section 8.2 that Ω 1 ( V ) is naturally id en tiﬁed with V ⊕ ℓ . Under this id en tiﬁcation, th e space of exact elemen ts in Ω 1 ( V ) is B 1 (Ω • ( V ) , δ K ) = n K ∗ ( ∂ ) δ f δ u    R f ∈ V /∂ V o . 94 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Moreo ver, it is not hard to chec k using the deﬁnition ( 9.29 ) of δ K , that the space of closed elemen ts in Ω 1 ( V ) is Z 1 (Ω • ( V ) , δ K ) = n F ∈ V ℓ    D F ( ∂ ) ◦ K ( ∂ ) = K ∗ ( ∂ ) ◦ D ∗ F ( ∂ ) o , where D F ( ∂ ) is the F rechet deriv ative ( 8.14 ) and D ∗ F ( ∂ ) its adj oin t matrix diﬀeren tial op erator. O n the other hand , it is easy to see that the space Σ 1 ( K ∗ ) consists of matrix diﬀeren tial op erators P =  P ij ( ∂ )  i,j ∈ I with qua- siconstan t co eﬃcient s and of order at most N − 1, solving the equation (11.41 ) K ∗ ( ∂ ) ◦ P ( ∂ ) = P ∗ ( ∂ ) ◦ K ( ∂ ) . The isomorphism χ 1 : Σ 1 ( K ∗ ) → H 1 (Ω • ( V ) , δ K ) deﬁn ed in Theorem 11.10 (b) is give n by χ 1 ( P ) = F + δ K (Ω 0 ( V )), wh ere (11.42 ) F =  X j ∈ I P ∗ ij ( ∂ ) u j  i ∈ I ∈ V ℓ . It is not h ard to c hec k that, if F is as in ( 11.42 ), then its F rec het deriv ativ e is D F ( ∂ ) = P ∗ ( ∂ ) , hence, if P satisﬁes equation ( 11.41 ), then F lies in Z 1 (Ω 1 ( V ) , δ K ). R emark 11.11 . Recall that, if H and K are compatible Hamiltonian op era- tors, the Lenard sc heme is the follo wing recurrent relation: H ( ∂ ) δ h n δ u = K ( ∂ ) δ h n +1 δ u , or, equiv alen tly , [ H , R h n ] = [ K , R h n +1 ] , in the Lie sup eralgebra W v ar (Π V ) ≃ Ω • ( V ). The Hamilto nian functions R h n are constructed b y indu ction on n ∈ Z + . In fact, as explained in the in tro- duction (see equation ( 1.14 )), assum ing that we h av e constructed R h j , j = 0 , . . . , n − 1 satisfying the Lenard r ecurrence form ula, then [ H , R h n − 1 ] is a closed elemen t of (Ω 1 ( V ) , δ K ). Hence, b y equation ( 11.42 ), there exist a Hamiltonian fu nction R h n ∈ V /∂ V and a un ique P ∈ Σ 1 ( K ∗ ), i.e. a m atrix diﬀeren tial op erator P =  P ij ( ∂ )  i,j ∈ I of order at most N − 1 with quasi- constan t co eﬃcient s solving ( 11.41 ), suc h that the follo win g equation holds in V ℓ : (11.43 ) [ H , R h n − 1 ] = [ K , R h n ] +  X j ∈ I P ∗ ij ( ∂ ) u j  i ∈ I . In order to complete the n -th step of the Lenard scheme, we ha v e to sho w that P = 0. F or this, the follo wing observ ations may b e used. First note that, since [ H , K ] = 0, ad H indu ces a w ell deﬁ n ed linear map H k (Ω • ( V ) , δ K ) → H k +1 (Ω • ( V ) , δ K ), hence, thanks to the isomorphism χ k : Σ k ( K ∗ ) → H k (Ω • ( V ) , δ K ) deﬁn ed in Th eorem 11.10 , we get an indu ced linear map α H k : Σ k ( K ∗ ) → Σ k +1 ( K ∗ ). On the other hand , applying ad H to b oth sides of equation ( 11.43 ), w e get that (ad H )  P j ∈ I P ∗ ij ( ∂ ) u j  i ∈ I is THE V ARIA TIONAL POISSON COHOMOLOGY 95 an exact elemen t of (Ω 2 ( V ) , δ K ), or, equiv alen tly , α H 1 ( P ) = 0. Th us, in order to apply the Lenard sc heme at the n -th step, it s uﬃces to s h o w that Ker( α H 1 ) = 0. Recalling form ula ( 9.15 ) for the action of ad H on V ℓ = Ω 1 ( V ), it is not hard to sho w that th e condition that α H 1 is inj ectiv e translates to th e condition that, if X P ∗ ( ∂ ) u ( H ) − H ( ∂ ) ◦ P ( ∂ ) − P ∗ ( ∂ ) ◦ H ( ∂ ) = − K ( ∂ ) ◦ D ∗ F ( ∂ ) − D F ( ∂ ) ◦ K ( ∂ ) , for s ome P ∈ Σ 1 ( K ∗ ) and F ∈ V ℓ , then P = 0. Appendix A. Syst ems of l inear differential equa tions and (pol y)diff erential ope ra tors In this App endix w e p ro v e some facts ab ou t matrix diﬀeren tial and p oly- diﬀeren tial op erators needed in the computation of the v ariational P oisson cohomology (cf. Section 11.2 ). In order to establish these facts, we use the theory of systems of linear diﬀeren tial equ ations in sev eral u nkno wns. This theory has b een deve lop ed b y a num b er of authors , see [ Ler ], [ V ol ], [ Huf ], [ SK ], [ Miy ]. Our exp osition (whic h w e dev eloped b efore b ecoming aw are of the ab o v e references) is giv en in the spir it of d iﬀeren tial algebra, as the rest of the pap er. A.1. L emmas on diﬀerential op erators. Let M b e a unital asso ciativ e (not necessarily comm utativ e) algebra, with a deriv ation ∂ . Consid er the algebra of diﬀeren tial op erators M [ ∂ ]. Its elemen ts are expressions of the form (A.1.1) P ( ∂ ) = N X n =0 a n ∂ n , a n ∈ M , whic h are m ultiplied according to the ru le ∂ ◦ a = a∂ + a ′ . If a N 6 = 0, then w e say that P ( ∂ ) has or der ord( P ) = N and we call a N ∈ M its le ading c o eﬃcient . Lemma A.1.1. If the diﬀer ential op er ator (A.1.2) M X m =0 ∂ m +1 ◦ a m ∂ m + N X n =0 ∂ n ◦ b n ∂ n ∈ M [ ∂ ] is zer o, then al l the elements a m and b n ar e zer o. Henc e, in a diﬀer ential op er ator of the form ( A.1.2 ) , the elements a m and b n ar e uniqu ely deter- mine d. Pr o of. In the con tr ary case, t wo things can happ en: either a M 6 = 0 and 2 M + 1 > 2 N , or b N 6 = 0 and 2 N > 2 M + 1. In the ﬁrst case, the op erator ( A.1.2 ) has order 2 M + 1, and the leading coeﬃcient is a M , a cont radiction. Similarly in the second case.  Lemma A.1.2. L et p, q ∈ Z + and a ∈ M . Then 96 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 (a) for p > q ∈ Z + ∂ p ◦ a∂ q = [( p + q − 1) / 2] X m = q  p − m − 1 m − q  ∂ m +1 ◦ a ( p + q − 2 m − 1) ∂ m + [( p + q ) / 2] X m = q +1  p − m − 1 m − q − 1  ∂ m ◦ a ( p + q − 2 m ) ∂ m ; (b) for p < q ∈ Z + ∂ p ◦ a∂ q = [( p + q − 1) / 2] X m = p γ p,q m ∂ m +1 ◦ a ( p + q − 2 m − 1) ∂ m + [( p + q ) / 2] X m = p δ p,q m ∂ m ◦ a ( p + q − 2 m ) ∂ m , wher e γ p,q m and δ p,q m ar e inte gers. Pr o of. (a). By in d uction on p − q . F or p − q = 1, the statemen t is immediate to c hec k. F or p − q = 2, w e ha v e: ∂ q +2 ◦ a∂ q = ∂ q +1 ◦ a ′ ∂ q + ∂ q +1 ◦ a∂ q +1 , whic h agrees with our claim. F or p − q ≥ 3, we ha v e, b y ind uction, ∂ p ◦ a∂ q = ∂ p − 1 ◦ a ′ ∂ q + ∂ p − 1 ◦ a∂ q +1 = [( p + q − 2) / 2] X m = q  p − m − 2 m − q  ∂ m +1 ◦ a ( p + q − 2 m − 1) ∂ m + [( p + q − 1) / 2] X m = q +1  p − m − 2 m − q − 1  ∂ m ◦ a ( p + q − 2 m ) ∂ m + [( p + q − 1) / 2] X m = q +1  p − m − 2 m − q − 1  ∂ m +1 ◦ a ( p + q − 2 m − 1) ∂ m + [( p + q ) / 2] X m = q +2  p − m − 2 m − q − 2  ∂ m ◦ a ( p + q − 2 m ) ∂ m = [( p + q − 1) / 2] X m = q  p − m − 1 m − q  ∂ m +1 ◦ a ( p + q − 2 m − 1) ∂ m + [( p + q − 1) / 2] X m = q +1  p − m − 1 m − q − 1  ∂ m ◦ a ( p + q − 2 m ) ∂ m . In the last iden tit y w e us ed the T artaglia-P ascal triangle. (b). It follo ws fr om (a), since, by the b inomial form ula, ∂ p ◦ a∂ q = q − p X h =0  q − p h  ( − 1) h ∂ q − h ◦ a ( h ) ∂ p .  THE V ARIA TIONAL POISSON COHOMOLOGY 97 Corollary A.1.3. Any diﬀer ential op er ator P ( ∂ ) ∈ M [ ∂ ] of or der less than or e qu al to N c an b e written, in a unique way, i n any of these thr e e forms: P ( ∂ ) = N X n =0 a n ∂ n = N X n =0 ∂ n ◦ b n = [( N − 1) / 2] X m =0 ∂ m +1 ◦ c n ∂ m + [ N/ 2] X n =0 ∂ n ◦ d n ∂ n . Pr o of. Existence is clear. Uniqueness of the ﬁrst tw o f orms is clea r, and the third one is Lemma A.1.1 .  Supp ose next that M has an anti- in v olution a 7→ a ∗ , comm u ting with ∂ . Example A.1.4 . If M = M at ℓ × ℓ ( F ), wh ere F is a comm utativ e diﬀerenti al algebra, we let a ∗ = a T , the transp ose matrix. W e extend ∗ to an an ti-in volutio n of M [ ∂ ] b y letting  N X n =0 a n ∂ n  ∗ = N X n =0 ( − ∂ ) n ◦ a ∗ n . W e sa y that P ( ∂ ) is selfadjoin t (resp ectiv ely skew adjoint) if P ∗ ( ∂ ) = P ( ∂ ) (resp. P ∗ ( ∂ ) = − P ( ∂ )). Lemma A.1.5. (a) If S ( ∂ ) is a skewadjoint op er ator of or der less than or e qual to N , then it c an b e written, in a uniqu e way, in the form (A.1.3) S ( ∂ ) = [( N − 1) / 2] X m =0 ∂ m ◦  ∂ ◦ a m + a m ∂  ∂ m + [ N/ 2] X m =0 ∂ m ◦ b m ∂ m wher e a m = a ∗ m and b m = − b ∗ m . (b) If S ( ∂ ) is a selfadjoint op er ator of or der less than or e qual to N , then it c an b e written, in a unique way, in the form S ( ∂ ) = [( N − 1) / 2] X m =0 ∂ m ◦  ∂ ◦ a m + a m ∂  ∂ m + [ N/ 2] X m =0 ∂ m ◦ b m ∂ m wher e a m = − a ∗ m and b m = b ∗ m . Pr o of. Use the third form of Corollary A.1.3 and compute S − S ∗ (resp. S + S ∗ ).  A.2. L inear algebra ov er a diﬀeren tial ﬁeld. Let F b e a diﬀeren tial ﬁeld, i.e. a ﬁeld with a deriv ation ∂ , and let C = { c ∈ F | ∂ c = 0 } ⊂ F b e the su bﬁeld of constants. Notation: a, b, c, · · · ∈ F , α, β , γ , · · · ∈ C , u, v , w v ariables, m, n , p, q ∈ Z + . A system of m linear diﬀerentia l equ ations in the v ariables u i , i = 1 , . . . , ℓ , has th e form (A.2.1) M ( ∂ ) u = b , 98 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 where u =    u 1 . . . u ℓ    , b =    b 1 . . . b ℓ    , M ( ∂ ) =   L 11 ( ∂ ) . . . L 1 ℓ ( ∂ ) . . . L m 1 ( ∂ ) . . . L mℓ ( ∂ )   , with b i ∈ F and L ij ( ∂ ) ∈ F [ ∂ ]. In order to study this system of linear diﬀeren tial equ ations we will use the follo wing simple result: Lemma A.2.1. L et ( n ij ) b e an m × ℓ matr ix with entries i n Z , and let (A.2.2) N j = max i { n ij } , h i = min j { N j − n ij } . Then (A.2.3) n ij ≤ N j − h i , ∀ i = 1 , . . . , m, j = 1 , . . . , ℓ . Any other choic e N ′ j , h ′ i satisfying ( A.2.3 ) is such that N ′ j ≥ N j for al l j , and, if N ′ j = N j for al l j , then h ′ i ≤ h i for al l i . Pr o of. Clearly , ( A.2.3 ) holds. Giv en j , there exists i su c h th at N j = n ij . But, b y assumption, n ij ≤ N ′ j − h ′ i . Hence N j ≤ N ′ j . S u pp ose no w that N j = N ′ j for all j . Giv en i there exists j such that h i = N j − n ij = N ′ j − n ij . But n ij ≤ N ′ j − h ′ i , h en ce h i ≥ h ′ i .  Deﬁnition A.2.2. The c ol le ction of inte g ers { N j , j = 1 , . . . , ℓ ; h i , i = 1 , . . . . . . , m } satisfying ( A.2.3 ) is c al le d a ma joran t of the matrix ( n ij ) . Consider the system of equations ( A.2.1 ). A major ant { N j ; h i } of the m × ℓ matrix diﬀeren tial op erato r M ( ∂ ) is deﬁn ed as a ma joran t of its matrix of orders ( n ij ). Give n an arbitrary ma j orant { N j ; h i } of the matrix diﬀeren tial op erator M ( ∂ ), w e can write the i, j entry of M ( ∂ ) in the f orm L ij ( ∂ ) = N j − h i X n =0 a ij ; n ∂ n , a ij ; n ∈ F . W e deﬁne the corresp ond ing le ading matrix as the follo wing m × ℓ matrix whose entries are monomials in an ind eterminan t ξ with co eﬃcien ts in F : (A.2.4) ¯ M ( ξ ) =   a 11; N 1 − h 1 ξ N 1 − h 1 . . . a 1 ℓ ; N ℓ − h 1 ξ N ℓ − h 1 . . . a m 1; N 1 − h m ξ N 1 − h m . . . a mℓ ; N ℓ − h m ξ N ℓ − h m   . Clearly , if m = ℓ , w e h a ve (A.2.5) det( ¯ M ( ξ )) = det( ¯ M (1)) ξ d , where (A.2.6) d = ℓ X j =1 ( N j − h j ) . Note that this matrix dep ends on the c hoice of the ma joran t { N j ; h i } , THE V ARIA TIONAL POISSON COHOMOLOGY 99 P erm uting the equations in the system ( A.2.1 ) if n ecessary , w e can (and will) assume that h 1 ≥ · · · ≥ h ℓ . As in linear algebra, the set of solutions of the sy s tem ( A.2.1 ) do es n ot c hange if we exc hange t wo equations or if w e add to the i -th equ ation the j -th equation, with j 6 = i , to whic h we apply a diﬀerential op erator P ( ∂ ). Since we wan t to preserve the fact that ord( L ij ) ≤ N j − h i , we giv e the follo win g: Deﬁnition A.2.3. An elemen tary row op eration of the matrix diﬀer ential op er ator M ( ∂ ) is either a p ermutation of two r ows of it, or the op er ation T ( i, j ; P ) , wher e 1 ≤ i 6 = j ≤ m and P ( ∂ ) is a diﬀer ential op er ator, which r eplac es the j -th r ow by itself minus i -th r ow multiplie d on the left by P ( ∂ ) . Assuming that h 1 ≥ · · · ≥ h ℓ , we say that the elementary r ow op er ation T ( i, j ; P ) is ma jorant preserving i f i < j and P ( ∂ ) has or der less than or e qual to h i − h j . R emark A.2.4 . After a m a joran t p reserving ro w op eration T ( i, j ; P ), the leading m atrix ¯ M ( ξ ) in ( A.2.4 ) is unchanged unless P ( ∂ ) has order equal to h i − h j , and in this case it c hanges b y an elemen tary ro w op eration o v er F , n amely we add to the j -th row of ¯ M ( ξ ) the i -th ro w m ultiplied b y the leading coeﬃcient of P ( ∂ ). Using the usual Gauss elimination, w e can get the (w ell kno wn) analogues of standard linear algebra theorems for matrix diﬀerenti al op erators. I n particular, w e ha v e the follo wing Lemma A.2.5. Any m × ℓ matrix diﬀer ential op er ator M ( ∂ ) c an b e br ought by elementary r ow op er ations to a r ow e chelon form. Pr o of. Let j 1 b e the ﬁrst non zero column of M ( ∂ ). Among all matrices obtained from M ( ∂ ) by elemen tary ro w op erations, c hose one for which the ﬁ rst entry of column j 1 , L 1 j 1 ( ∂ ), is n on zero of minimal p ossible order (minimal among the orders of the (1 , j 1 ) ent ry in all these matrices). C learly , all the other en tries in column j 1 m ust b e divisible (on the left) b y L 1 j 1 ( ∂ ), and using elemen tary row op eratio ns we can mak e them zero. T hen, w e pro ceed b y indu ction on the submatrix w ith ﬁrst ro w deleted.  W e next discuss ma joran t p reserving Gauss eliminatio n for a matrix dif- feren tial op erator. Lemma A.2.6. Consider the m × ℓ matrix diﬀer ential op er ator M ( ∂ ) =  L ij ( ∂ )  with m ≤ ℓ , and let { N 1 , . . . , N ℓ ; h 1 ≥ · · · ≥ h m } b e a major ant of M ( ∂ ) . Supp ose, mor e over, that ord( L j j ) = N j − h j for 1 ≤ j ≤ m − 1 , and ord( L ij ) < N j − h j for 1 ≤ j < i ≤ m − 1 . Then, we c an p e rform major ant pr eserving elementary r ow op er ations on the m -th r ow of M ( ∂ ) so that its new m -th r ow e L mj ( ∂ ) , j = 1 , . . . , ℓ , satisﬁes: ord( e L mj ) < N j − h j for j < m , ord( e L mj ) ≤ N j − h m for j ≥ m . 100 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Pr o of. By assumption, the m -th ro w of the starting matrix M ( ∂ ) satisﬁes ord( L mj ) ≤ N j − h m for 1 ≤ j ≤ ℓ . Applying to M ( ∂ ) the elemen tary ro w op eration T (1 , m ; c∂ h 1 − h m ), where c = a m 1; N 1 − h m /a 11; N 1 − h 1 , w e get a new m atrix satisfying ord( e L m 1 ) ≤ N 1 − h m − 1 , ord( e L mj ) ≤ N j − h m for 2 ≤ j ≤ ℓ . Next, applying th e elemen tary ro w op eratio n T (2 , m ; c∂ h 2 − h m ), for suitable c ∈ F , w e get a new matrix satisfying ord( e L m 1 ) ≤ N 1 − h m − 1 , ord( e L m 2 ) ≤ N 2 − h m − 1 , ord( e L mj ) ≤ N j − h m for 3 ≤ j ≤ ℓ . Pro ceeding in th e same wa y , w e get after m − 1 steps, a new m atrix satisfying ord( e L mj ) ≤ N j − h m − 1 for 1 ≤ j ≤ m − 1 , ord( e L mj ) ≤ N j − h m for m ≤ j ≤ ℓ . If h 1 = h 2 = · · · = h m , w e are done. O therwise, h 1 > h m and w e apply to the last matrix the elemen tary row op eration T (1 , m ; c∂ h 1 − h m − 1 ), where c = e a m 1; N 1 − h m − 1 /a 11; N 1 − h 1 . W e th us get a new matrix satisfying ord( e L m 1 ) ≤ N 1 − h m − 2 , ord( e L mj ) ≤ N j − h m − 1 for 2 ≤ j ≤ m − 1 , ord( e L mj ) ≤ N j − h m for m ≤ j ≤ ℓ . Next, if h 2 > h m , w e apply the ro w operation T (2 , m ; c∂ h 2 − h m − 1 ), for suit- able c ∈ F , and we get a new m atrix satisfying ord( e L m 1 ) ≤ N 1 − h m − 2 , ord( e L m 2 ) ≤ N 2 − h m − 2 , ord( e L mj ) ≤ N j − h m − 1 for 3 ≤ j ≤ m − 1 , ord( e L mj ) ≤ N j − h m for m ≤ j ≤ ℓ . Let r ≤ m − 1 b e such that h r > h r +1 = h m . W e pro ceed in the same w ay and we get, after r steps, a new matrix satisfying ord( e L mj ) ≤ N j − h m − 2 for 1 ≤ j ≤ r , ord( e L mj ) ≤ N j − h j − 1 for r + 1 ≤ j ≤ m − 1 , ord( e L mj ) ≤ N j − h m for m ≤ j ≤ ℓ . If h r − h m ≥ 2, w e again apply consecutiv ely elemen tary row op erations T (1 , m ; c 1 ∂ h 1 − h m − 2 ) , T (2 , m ; c 2 ∂ h 2 − h m − 2 ) , . . . , T ( r, m ; c r ∂ h r − h m − 2 ), for ap- propriate c 1 , . . . , c r ∈ F . As a result w e get a new matrix satisfying ord( e L mj ) ≤ N j − h m − 3 for 1 ≤ j ≤ r , ord( e L mj ) ≤ N j − h j − 1 for r + 1 ≤ j ≤ m − 1 , ord( e L mj ) ≤ N j − h m for m ≤ j ≤ ℓ , and pr o ceeding as b efore h r − h m times, w e get a matrix satisfying ord( e L mj ) ≤ N j − h r − 1 for 1 ≤ j ≤ r , ord( e L mj ) ≤ N j − h j − 1 for r + 1 ≤ j ≤ m − 1 , ord( e L mj ) ≤ N j − h m for m ≤ j ≤ ℓ . THE V ARIA TIONAL POISSON COHOMOLOGY 101 If h 1 = · · · = h r , w e are done. Oth er w ise, let s ≤ r − 1 b e such that h s > h s +1 = · · · = h r . W e pro ceed in the same wa y as b efore to get, after a ﬁnite num b er of steps, a new matrix satisfying ord( e L mj ) ≤ N j − h s − 1 for 1 ≤ j ≤ s , ord( e L mj ) ≤ N j − h j − 1 for s + 1 ≤ j ≤ m − 1 , ord( e L mj ) ≤ N j − h m for m ≤ j ≤ ℓ . Con tin uing along these lines, one gets the desired result.  Prop osition A.2.7. L et M ( ∂ ) =  L ij ( ∂ )  b e an ℓ × ℓ mat rix diﬀer ential op- er ator with major ant { N 1 , . . . , N ℓ ; h 1 ≥ · · · ≥ h ℓ } . Assume that the le ading matrix ¯ M ( ξ ) asso ciate d to this major ant, deﬁne d in ( A.2.4 ) , is non de gener- ate. Then, after p ossibly p ermuting the c olumns of M ( ∂ ) , and after applying major ant pr eserving elementary r ow op er ations, we get a matrix of the form f M ( ∂ ) =  e L ij ( ∂ )  , wher e ord( e L j j ) = N j − h j for 1 ≤ j ≤ ℓ , ord( e L ij ) ≤ N j − h i for 1 ≤ i < j ≤ ℓ , ord( e L ij ) < N j − h j for 1 ≤ j < i ≤ ℓ . Pr o of. Since th e ﬁrst ro w of the leading m atrix ¯ M ( ξ ) is non zero, after p ossibly exc h anging the ﬁrst col umn of M ( ∂ ) with its j -th column , j > 1, w e can assume th at L 11 ( ∂ ) has order N 1 − h 1 . Applying Lemma A.2.6 to the ﬁrst t wo ro ws of the matrix M ( ∂ ), we get, after elemen tary row op eratio ns on the second row, a new matrix f M ( ∂ ) with e L 21 ( ∂ ) of order str ictly less than N 1 − h 1 , and e L 2 j ( ∂ ) of order less th an or equal to N j − h 2 for j ≥ 2. By Remark A.2.4 , the leading matrix ¯ f M ( ξ ) of th e n ew matrix f M ( ∂ ) is again n on degenerate, and it has zero in p osition (2 , 1). In p articular, the ﬁrst tw o ro ws of ¯ f M ( ξ ) are linearly indep enden t, and, after p ossibly exc hanging the second column with the j -th column with j > 2, w e can assume that e a 22; N 2 − h 2 6 = 0, i.e. ord( e L 22 ) = N 2 − h 2 . Applying Lemma A.2.6 to th e ﬁrst thr ee ro ws of the matrix f M ( ∂ ), we get, after eleme n tary row op erations on the thir d ro w , a new matrix with ord( e L 31 ) < N 1 − h 1 , ord( e L 32 ) < N 2 − h 2 , ord ( e L 3 j ) ≤ N j − h 3 , j ≥ 3. Rep eat ing the same pro cedure for eac h s u bsequent ro w, we get the desired result.  Prop osition A.2.8. L et M ( ∂ ) =  L ij ( ∂ )  b e an ℓ × ℓ matrix diﬀer ential op er ator as in the c onclusion of P r op ositio n A.2.7 , i. e. ord( L j j ) = N j − h j for 1 ≤ j ≤ ℓ , ord( L ij ) ≤ N j − h i for 1 ≤ i < j ≤ ℓ , and ord( L ij ) < N j − h j for 1 ≤ j < i ≤ ℓ , wher e N 1 , . . . , N ℓ and h i ≥ · · · ≥ h ℓ ar e non ne g ative inte gers. Then it has the fol lowing major ant { N ′ j ; h ′ i } : N ′ 1 = N 1 − h 1 , . . . , N ′ ℓ = N ℓ − h ℓ ; h ′ 1 = · · · = h ′ ℓ = 0 . The le ading matrix ¯ M ( ξ ) asso ciate d to this major ant of M ( ∂ ) is upp er tri- angular with non zer o diagonal entries, and the ( ij ) entry with i < j i s zer o unless h i = h j . 102 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Pr o of. Obvious.  The ring F [ ∂ ] of scalar diﬀerentia l op erators ov er F is n atur ally emb edded in the ring of pseudo diﬀerentia l op erators F [ ∂ ][[ ∂ − 1 ]], whic h is a sk ewﬁeld. All the ab o v e deﬁnitions and statemen ts ha v e an ob vious generaliza tion to pseudo diﬀeren tia l op erators. In p articular, w e deﬁne a ma jorant { N j ; h i } of an m × ℓ matrix p seudo diﬀerential op erato r M ( ∂ ) in th e same w a y as b efore, and the corresp onding leading matrix ¯ M ( ξ ) by the same equation ( A.2.4 ) (except that here we allo w negativ e p o wers of the v ariable ξ ). W e also deﬁn e elemen tary ro w op erations and m a joran t pr eservin g elemen tary ro w op erations as in Deﬁnition A.2.3 , except that w e allo w P ( ∂ ) to b e a pseudo d iﬀeren tial op er ator. Finally , in the case of pseudo diﬀeren tial op era- tors P rop ositions A.2.7 and A.2.8 still hold, and ha v e the follo wing stronger analogue: Prop osition A.2.9. L et M ( ∂ ) =  L ij ( ∂ )  b e an ℓ × ℓ matrix pseudo diﬀer- ential op er ator with major ant { N 1 , . . . , N ℓ ; h 1 ≥ · · · ≥ h ℓ } . Assume that the le ading matrix ¯ M ( ξ ) asso ciate d to this major ant is non de ge ner ate. Then, after p ossibly p ermuting the c olumns of M ( ∂ ) , and after applying major ant pr eserving elementary r ow op er atio ns, we get an upp er triangular matrix f M ( ∂ ) =  e L ij ( ∂ )  , with ord( e L j j ) = N j − h j for al l j = 1 , . . . , ℓ . The r esult- ing matrix f M ( ∂ ) has the fol lowing major ant { e N j ; e h i } : e N 1 = N 1 − h 1 , . . . , e N ℓ = N ℓ − h ℓ ; e h 1 = · · · = e h ℓ = 0 . Pr o of. First, f ollo wing the pro of of Prop osition A.2.7 , we can app ly ma jo- ran t preservin g elemen tary r o w op erations, and p ossibly p ermutat ions of columns, to reduce M ( ∂ ) to a matrix p seudo diﬀerential op erator satisfying the follo wing conditions: ord( e L ij ) < ord( L j j ) = N j − h j for all 1 ≤ j < i ≤ ℓ . Let then i > j , and recall that, by assu mption, h j ≥ h i , and, b y the ab ov e condition, P ( ∂ ) = L j j ( ∂ ) L ij ( ∂ ) − 1 has n egativ e order. Hence, the elemen tary ro w op eration T ( j, i ; P ) is ma joran t preservin g. Applying s uc h elemen tary ro w op erations a ﬁn ite num b er of times, we get the desired upp er tr iangular matrix. The last statemen t is ob vious.  Recall that an y ℓ × ℓ matrix p s eudo diﬀerentia l op erator M ( ∂ ) has the Dieudonn ´ e determinant of the form det( M ( ∂ )) = cξ d , wh er e c ∈ F , ξ is an ind eterminate, and d ∈ Z . In fact, the Dieudonn ´ e determinant is de- ﬁned f or squ are matrices o v er an arbitrary skewﬁeld K , and it tak es v al- ues in K × / ( K × , K × ) ∪ { 0 } , [ Die ], [ Art ]. By deﬁnition, det( M ( ∂ )) changes sign if we p ermute t w o ro ws or t w o column s of M ( ∂ ), and it is unchange d under an y elemen tary r o w op eration T ( i, j ; P ) in Deﬁnition A.2.3 , for ar- bitrary i 6 = j and a p seudo diﬀerential op erator P ( ∂ ). Also, if M ( ∂ ) is THE V ARIA TIONAL POISSON COHOMOLOGY 103 upp er triangular, with diagonal en tries L ii ( ∂ ) of order n i and leading co ef- ﬁcien t a i , then d et( M ( ∂ )) =  Q i a i  ξ P i n i . It is pro v ed in the ab o ve refer- ences that the Dieudonn´ e determinan t is w ell deﬁned and det( A ( ∂ ) B ( ∂ )) = det( A ( ∂ )) d et( B ( ∂ )) for ev ery ℓ × ℓ matrix pseud o diﬀerent ial op erators A ( ∂ ) and B ( ∂ ). Moreo v er, we h a v e the follo wing prop ositio n (cf. [ Huf ], [ SK ], [ Miy ]): Prop osition A.2.10. If M ( ∂ ) i s an ℓ × ℓ matrix pseudo diﬀer ential op er ator with non de ge ner ate le ading matrix ¯ M ( ξ ) (for a c ertain major ant { N j ; h i } of M ( ∂ ) ), then det( M ( ∂ )) = det( ¯ M ( ξ )) . In p articular, deg ξ det( M ( ∂ )) = P ℓ j =1 ( N j − h j ) . Pr o of. It follo ws Pr op osition A.2.9 sin ce, b y Remark A.2.4 , the determinant of th e leading matrix ¯ M ( ξ ) is u nc hanged b y ma j oran t preserving elemen tary ro w op eratio ns.  Example A.2.1 1 . If ¯ M ( ξ ) is d egenerate, we can still ha v e det( M ( ∂ )) 6 = 0. F or example, the matrix diﬀerentia l op erator M ( ∂ ) =  1 a ∂ a∂  , has the ma joran t N 1 = N 2 = 1 , h 1 = 1 , h 2 = 0, and the corresp onding leading matrix ¯ M ( ξ ) =  1 a ξ aξ  is degenerate. Ho wev er, M ( ∂ ) can b e brought, by elemen tary ro w op era- tions, to the matrix  1 a 0 − a ′  , whic h s h o ws that det( M ( ∂ )) = − a ′ . A.3. L inearly closed diﬀerential ﬁelds. Deﬁnition A.3.1. A diﬀer ential ﬁeld F is c al le d linearly closed if any line ar diﬀer ential e qu ation, 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 every b ∈ F , and it has a non zer o solution for b = 0 , pr ovide d that n ≥ 1 . R emark A.3.2 . F or a linearly closed diﬀerential ﬁeld F and a non zero dif- feren tial op erator L ( ∂ ) ∈ F [ ∂ ], the map L ( ∂ ) : F → F giv en by a 7→ L ( ∂ ) a is surj ectiv e. Indeed, b y deﬁn ition, the diﬀeren tial equation L ( ∂ ) u = b has a solution in F for ev ery b ∈ F . R emark A.3.3 . Any diﬀeren tial ﬁeld F can b e em b edded in a linearly closed one. Note also that a d iﬀeren tially closed ﬁeld is automatically linearly closed. 104 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 R emark A.3.4 . Let F b e a linearly closed diﬀeren tial ﬁeld. Letting x ∈ F b e a solution of ∂ x = 1 w e get th at F con tains the ﬁeld of rational functions o v er C in x . In particular, F is inﬁ nite d imensional o ver C . Theorem A.3.5. L et F b e a diﬀer ential ﬁeld. Consider a line ar diﬀer ential e quation of or der N over F in the variable u : L ( ∂ ) u = 0 , wher e L ( ∂ ) = a N ∂ N + a N − 1 ∂ N − 1 + · · · + a 1 ∂ + a 0 ∈ F [ ∂ ] , a N 6 = 0 . (a) The sp ac e of solutions of this e quation is a ve ctor sp ac e over C of di- mension at most N . (b) If F is line arly close d, then the sp ac e of solution has dimension e qual to N . Pr o of. W e pr ov e (a) b y indu ction on N . F or N = 0, it is clea r. F or N ≥ 1, if there are no non zero solutions, we are done (note that this do es not h ap p en if F is linearly closed). If a ∈ F is a non zero solution of L ( ∂ ) u = 0, we divide L ( ∂ ) b y ∂ − a ′ /a with remainder, to get L ( ∂ ) = L 1 ( ∂ )( ∂ − a ′ /a ) + R , where L 1 has order N − 1 in ∂ and R ∈ F . S ince L ( ∂ ) a = 0, it follo ws that R = 0. By ind uctiv e assum ption, the space of solutions of L 1 ( ∂ ) u = 0 has dimen s ion at most N − 1 o ver C . Consider the linear map ov er C , b 7→ ( ∂ − a ′ /a ) b, b ∈ F . It is immediate to c h ec k that it maps surjectiv ely the space of solutions f or L ( ∂ ) on to th e space of solutions of L 1 ( ∂ ), and its k ernel is C a . The s tatemen t (a) follo ws. F or part (b) we use the same argumen t.  Theorem A.3.6. (a) L et M ( ∂ ) b e an ℓ × ℓ matrix diﬀer ential op er ator over a diﬀer ential ﬁeld F . i) If det( M ( ∂ )) 6 = 0 , then dim C (Ker M ( ∂ )) ≤ deg ξ det( M ( ∂ )) . ii) If Im M ( ∂ ) ⊂ F ℓ has ﬁnite c o dimension over C , then det( M ( ∂ )) 6 = 0 , pr ovide d that C 6 = F . (b) Assuming that the diﬀer ential ﬁeld F is line arly c lose d, the fol lowing statements ar e e quivalent for an ℓ × ℓ matrix diﬀer ential op er ator M ( ∂ ) : i) det( M ( ∂ )) 6 = 0 , ii) d im C (Ker M ( ∂ )) < ∞ , iii) d et( M ( ∂ )) 6 = 0 and dim C (Ker M ( ∂ )) = deg ξ det( M ( ∂ )) , iv) co d im C Im M ( ∂ ) < ∞ , v) M ( ∂ ) : F ℓ → F ℓ is surje ctive. (c) L et M ( ∂ ) b e an m × ℓ matrix diﬀer ential op er ator over a line arly close d diﬀer ential ﬁeld F , such that Ker( M ( ∂ )) has ﬁnite dimension over C and Im( M ( ∂ )) has ﬁnite c o dimension over C . Then ne c essarily m = ℓ and d et( M ( ∂ )) 6 = 0 . Pr o of. Since the dimens ion (o v er C ) of Ker( M ( ∂ )) and the co d imension of Im( M ( ∂ )) are unc hanged b y elemen tary row op erations on M ( ∂ ), w e ma y assume, by Lemma A.2.5 , that M ( ∂ ) is in row ec helon form. Ass ume ﬁrst that M ( ∂ ) is an ℓ × ℓ m atrix. If det( M ( ∂ )) 6 = 0, it means that its THE V ARIA TIONAL POISSON COHOMOLOGY 105 diagonal en tries L ii ( ∂ ) are all n on zero, say of order n i . Hence the cor- resp ond ing homogeneous system M ( ∂ ) u = 0 is upp er triangular and, by Theorem A.3.5 , its space of solutions has dimension less than or equal to P i n i = deg ξ det( M ( ∂ )) (and equal to it, p ro vided that F is linearly closed). This pro v es part (a)(i) and, in (b), (i) implies (iii) (and hence it is equiv alen t to it). Similarly , if d et( M ( ∂ )) = 0, then the last row of M ( ∂ ) is zero, so that Im M ( ∂ ) is of inﬁnite codimen s ion o v er C . Here we are using the fact that F is inﬁnite dimensional o ver C , since any f ∈ F , su c h that f ′ 6 = 0, is n ot algebraic o v er C . (Ind eed, if f ∈ F is algebraic and P ( f ) = 0 is its mon ic minimal p olynomial o v er C , then 0 = ∂ P ( f ) = P ′ ( f ) f ′ , so that P ′ ( f ) = 0 if f ′ 6 = 0, a con tradiction.) T his p ro v es part (a)(ii) and, in (b), that (iv) implies (i). Since, in statemen t (b), condition (iii) ob viously implies (ii), and (v) ob- viously implies (iv), in order to pro v e p art (b) w e only need to pro ve that (ii) implies (v). Assume th at F is linearly clo sed (hence inﬁn ite dimensional o v er C , by Remark A.3.4 ), and that, by condition (ii), M ( ∂ ) has ﬁnite di- mensional kernel ov er C . Then its last diagonal en try is non zero, otherwise, b y Theorem A.3.5 , there is a solution of th e homogeneous system M ( ∂ ) u = 0 for ev ery c h oice of u ℓ . Therefore M ( ∂ ) is upp er triangular with non zero diagonal en tries, and then, again b y Theorem A.3.5 , the inhomogeneous sys- tem M ( ∂ ) u = b has a solution for every b ∈ F ℓ , i.e. M ( ∂ ) is s urjectiv e. Th is completes the pr o of of part (b). Finally , w e p ro v e part (c). Assu me, as b efore, that M ( ∂ ) is in r o w ec helon form. The homogeneous system M ( ∂ ) u = 0 admits a solution f or ev ery c hoice of a coord inate u i whic h d o es n ot corresp ond to a piv ot of M ( ∂ ). Hence, sin ce Ker( M ( ∂ )) is ﬁn ite dimensional o ver C , we m ust ha ve m ≥ ℓ . If m > ℓ , then the last m − ℓ r o w s of M ( ∂ ) are zero, and th en th e image of M ( ∂ ) has in ﬁnite codimen s ion.  Corollary A.3.7. (cf. [ Huf , Miy ] ) L et M ( ∂ ) b e an ℓ × ℓ matrix diﬀer ential op er ator with c o eﬃcients in a diﬀer ential ﬁeld F . Supp ose that the le ading matrix ¯ M ( ξ ) , asso ciate d to a major ant { N j ; h i } of M ( ∂ ) , is non de g e ner ate. Then the sp ac e of solutions for the homo gene ous system M ( ∂ ) u = 0 has dimension over C less than or e qual to d = ℓ X j =1 ( N j − h j )  = deg ξ det( M ( ∂ ))  . Mor e over, if F is a line arly close d diﬀer e ntial ﬁeld then the inhomo gene ous system M ( ∂ ) u = b has a solution for every b ∈ F ℓ , and the sp ac e of solutions for the homo gene ous system M ( ∂ ) u = 0 has dimension e qual to d . Pr o of. By Prop osition A.2.10 , if ¯ M ( ξ ) is non degenerate, th en det ( M ( ∂ )) = det( ¯ M ( ξ )), and deg ξ det( ¯ M ( ∂ )) = P j ( N j − h j ) = d . Hence, b y T h eorem A.3.6 , dim C (Ker M ( ∂ )) ≤ d and, if F is linearly closed, M ( ∂ ) is surjectiv e and dim C (Ker M ( ∂ )) = d .  106 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 In S ection A.5 w e will need the follo wing sligh t generaliz ation of Corollary A.3.7 . Corollary A.3.8. L et F b e a line arly close d diﬀer ential ﬁeld with sub ﬁeld of c onstants C . L et A ( ∂ ) b e an ℓ × ℓ matrix diﬀer ential op er ator such that det( A ( ∂ )) 6 = 0 . L et M ( ∂ ) = ( L ij ( ∂ )) b e an ℓ × ℓ matrix pseudo diﬀer ential op er ator with non de gener ate le ading matrix ¯ M ( ξ ) asso ciate d to a major ant { N j , j = 1 , . . . ℓ ; h i , i = 1 , . . . ℓ } . Assume, mor e over, that A ( ∂ ) M ( ∂ ) is a matrix diﬀer ential op er ator. Then the inhomo gene ous system of diﬀer ential e quations A ( ∂ ) M ( ∂ ) u = b has a solution for every b ∈ F ℓ , and the sp ac e of solutions of the c orr esp onding homo gene ous system A ( ∂ ) M ( ∂ ) u = 0 has dimension over C e qual to d = dim C (Ker A ( ∂ )) + ℓ X j =1 ( N j − h j ) . Pr o of. W e ha v e det( A ( ∂ ) M ( ∂ )) = det( A ( ∂ )) d et( M ( ∂ )) 6 = 0. Moreo v er, b y Prop osition A.2.10 , w e ha ve deg ξ det( M ( ∂ )) = P j ( N j − h j ), while, b y The- orem A.3.6 (b)(iii), we hav e d eg ξ det( A ( ∂ )) = d im C (Ker A ( ∂ )). Th erefore deg ξ det( A ( ∂ ) M ( ∂ )) = dim C (Ker A ( ∂ )) + X j ( N j − h j ) . The statemen t follo ws from Theorem A.3.6 (b) applied to the matrix d iﬀer- en tial op erator A ( ∂ ) M ( ∂ ).  A.4. Main results. A.4.1. The sc alar c ase. Theorem A.4.1. L et F b e a line arly close d diﬀer ential ﬁeld, and let K ( ∂ ) ∈ F [ ∂ ] b e a non zer o sc alar diﬀer ential op er ator . F or every skewadjoint dif- fer ential op er ator S ( ∂ ) , ther e exi sts a diﬀer ential op er ator P ( ∂ ) such that (A.4.1) K ( ∂ ) ◦ P ( ∂ ) − P ∗ ( ∂ ) ◦ K ∗ ( ∂ ) = S ( ∂ ) . Pr o of. Let K ( ∂ ) b e of order N with leading coeﬃcient k N 6 = 0. Note th at, replacing P ( ∂ ) b y k N P ( ∂ ), w e can r educe to the case when k N = 1. If S ( ∂ ) h as order n ≥ N (clearly n m ust b e o d d) with leading co eﬃcien t a ∈ F , letting P ( ∂ ) = a 2 ∂ n − N , we ha ve that S ( ∂ ) − K ( ∂ ) ◦ P ( ∂ ) + P ∗ ( ∂ ) ◦ K ∗ ( ∂ ) is a sk ew adjoin t diﬀerentia l op er ator of order strictly less than n . Hence, rep eat ing th e same argument a ﬁnite num b er of times, w e reduce to the case wh en S ( ∂ ) has order n ≤ N − 1. In particular, for N = 1 there is nothing to prov e since S is sk ew adjoin t, hence zero. In fact, we will consider the case when ord( S ) = n ≤ 2 N − 3, which, for N ≥ 2, co v ers all p ossibilities. W e will pro v e that in this case we can ﬁnd P ( ∂ ) solving ( A.4.1 ) of order less than or equal to N − 2. By Corollary THE V ARIA TIONAL POISSON COHOMOLOGY 107 A.1.3 and Lemma A.1.5 , the op erators K ( ∂ ) , P ( ∂ ) and S ( ∂ ) can b e written, uniquely , in the f orms (A.4.2) K ( ∂ ) = N X n =0 ∂ n ◦ k n , P ( ∂ ) = N − 2 X n =0 u n ∂ n , S ( ∂ ) = S + ( ∂ ) − S − ( ∂ ) , S + ( ∂ ) = N − 2 X m =0 ∂ m +1 ◦ s m ∂ m = S ∗ − ( ∂ ) . Clearly , equation ( A.4.1 ) is equiv alent to sa y that K ( ∂ ) ◦ P ( ∂ ) and S + ( ∂ ) diﬀer by a s elfadjoin t op erator. By L emm a A.1.2 K ( ∂ ) ◦ P ( ∂ ) is, up to adding a selfadjoin t op erator, equal to N X p =0 N − 2 X q =0 [ p + q − 1 2 ] X m =min( p,q ) γ p,q m ∂ m +1 ◦ ( k p u q ) ( p + q − 2 m − 1) ∂ m , where γ p,q m are integ ers and γ p,q m =  p − m − 1 m − q  for p > q . Exc h an ging the ord er of su mmation, the ab ov e expr ession can b e w r itten in the form (A.4.3) N − 2 X m =0 X p,q ∈D N,m γ p,q m ∂ m +1 ◦ ( k p u q ) ( p + q − 2 m − 1) ∂ m , where (A.4.4) D N ,m = n p, q ∈ Z +    p ≤ N , q ≤ N − 2 , min( p, q ) ≤ m, p + q ≥ 2 m + 1 o . Comparing ( A.4.3 ) w ith the expr ession ( A.4.2 ) for S + ( ∂ ), we conclude that equation ( A.4.1 ) is equiv alent to the follo wing system of N − 1 linear diﬀer- en tial equations in the N − 1 v ariables u i , i = 0 , . . . , N − 2: (A.4.5) X p,q ∈D N,m γ p,q m ( k p u q ) ( p + q − 2 m − 1) = s m , for m = 0 , . . . , N − 2. The system ( A.4.5 ) is of the form M ( ∂ ) u = s , wh ere u = ( u q ) N − 2 q =0 , s = ( s m ) N − 2 m =0 and M = ( L mq ( ∂ )) is the matrix diﬀeren tial op erator with ent ries L mq ( ∂ ) = X p : ( p, q ) ∈D N,m γ p,q m ∂ p + q − 2 m − 1 ◦ k p , 0 ≤ m, q ≤ N − 2 . Note that L mq ( ∂ ) has ord er less than or equal to N q − h m , where N q = N + q − 1 and h m = 2 m . The leading matrix asso ciated to this m a jorant, deﬁned b y ( A.2.4 ), has ¯ M ( ξ ) =  ¯ M mq ξ N + q − 2 m − 1  N − 2 m,q =0 , where ¯ M mq =  0 if 0 ≤ m < q ≤ N − 2  N − m − 1 m − q  if 0 ≤ q ≤ m ≤ N − 2 . In particular ¯ M (1) is upp er triangular with 1’s on the diagonal. Hence, by Corollary A.3.7 w e conclude that the s y s tem ( A.4.5 ) has solutions.  108 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Theorem A.4.2. L et K ( ∂ ) ∈ F [ ∂ ] b e a sc alar diﬀer ential op er ator of or der N over a diﬀer ential ﬁeld F . Then the set of diﬀer ential op er ators P ( ∂ ) of or der at most N − 1 such that K ( ∂ ) ◦ P ( ∂ ) is selfadjoint is a ve ctor sp ac e over C of dimension less than or e qual to  N 2  and, if F is line arly c lose d, it has dimension e qual to  N 2  . Pr o of. First w e note that, since K ( ∂ ) ◦ P ( ∂ ) is selfadjoin t of order less than or equal to 2 N − 1, it m ust hav e order at m ost 2 N − 2, i.e. P ( ∂ ) has order at most N − 2. The condition on P ( ∂ ) means that P ( ∂ ) is a solution of ( A.4.1 ) with S ( ∂ ) = 0. Hence, if w e expand K ( ∂ ) and P ( ∂ ) as in ( A.4.2 ), the condition on P ( ∂ ) reduces to the system of d iﬀeren tial equations ( A.4.5 ) with s m = 0. As observ ed in the pr o of of Th eorem A.4.1 , the matrix M ( ∂ ) of co eﬃcients has ma joran t { N j = N + j − 1; h i = 2 i } and the corresp ond in g leading matrix ¯ M ( ξ ) is non degenerate. Hence, b y Corollary A.3.7 , the space of solutions has dimension at m ost d = N − 2 X i =0 ( N i − h i ) = N − 2 X i =0 ( N − 1 − i ) =  N 2  , and equal to d if F is linearly closed.  A.4.2. The matrix c ase. Theorem A.4.3. L et K ( ∂ ) ∈ Mat ℓ × ℓ ( F [ ∂ ]) b e an ℓ × ℓ matrix diﬀer ential op er ator of or der N with invertible le ading c o eﬃcient, over a diﬀer ential ﬁeld F . (a) If F is line arly close d, then for ev ery skewadjoint ℓ × ℓ matrix diﬀer ential op er ator S ( ∂ ) , ther e exists an ℓ × ℓ matrix diﬀe r ential op er ator P ( ∂ ) such that (A.4.6) K ( ∂ ) ◦ P ( ∂ ) − P ∗ ( ∂ ) ◦ K ∗ ( ∂ ) = S ( ∂ ) . (b) The set of diﬀer ential op er ators P ( ∂ ) of or der at most N − 1 such that K ( ∂ ) ◦ P ( ∂ ) is selfadjoint is a ve ctor sp ac e over C of dimension less than or e qual to d =  N ℓ 2  , and e qual to d pr ovide d that F is line arly close d. Pr o of. W e follo w the same steps as in the pr o of of Th eorems A.4.1 an d A.4.2 . Let K N ∈ Mat ℓ × ℓ ( F ) b e the leading co eﬃcient of K ( ∂ ). Replacing K ( ∂ ) by K ( ∂ ) ◦ K − 1 N and P ( ∂ ) by K N P ( ∂ ), we can reduce to the case wh en K ( ∂ ) has leading co eﬃcien t K N = 1 I. Let S ( ∂ ) b e of order n , with leading co eﬃcient S n ∈ Mat ℓ × ℓ ( F ). S ince, b y assumption, S ( ∂ ) is s kew adjoint, we ha v e S T n = ( − 1) n +1 S n . If n ≥ N , letting P ( ∂ ) = 1 2 S n ∂ n − N + P 1 ( ∂ ), the equation for P ( ∂ ) b ecomes K ( ∂ ) ◦ P 1 ( ∂ ) − P ∗ 1 ( ∂ ) ◦ K ∗ ( ∂ ) = S ( ∂ ) − 1 2 K ( ∂ ) ◦ S n ∂ n − N + ( − 1) N 1 2 ∂ n − N ◦ S n K ∗ ( ∂ ) . Note that the RHS of the ab o ve equation is a s kew adjoint ℓ × ℓ matrix diﬀeren tial op erator of order strictly less than n . Hence, rep eating the same THE V ARIA TIONAL POISSON COHOMOLOGY 109 argumen t a ﬁn ite n um b er of times, w e reduce to the case when S ( ∂ ) h as order n ≤ N − 1. In fact, w e will consider the more general case wh en ord ( S ) = n ≤ 2 N − 1. W e will prov e that, in this case, we can ﬁ nd P ( ∂ ) solving ( A.4.6 ) of order less than or equal to N − 1. By Corollary A.1.3 and Lemma A.1.5 , the op erators K ( ∂ ) , P ( ∂ ) and S ( ∂ ) can b e w r itten, uniquely , in th e forms (A.4.7) K ( ∂ ) = N X n =0 ∂ n ◦ K n , P ( ∂ ) = N − 1 X n =0 U n ∂ n , S ( ∂ ) = N − 1 X m =0 ∂ m ◦  ∂ ◦ A m + A m ∂ + B m  ∂ m , where K n , U n , A n , B n ∈ Mat ℓ × ℓ ( F ) and A T m = A m , B T m = − B m . By Lemma A.1.2 w e hav e (A.4.8) K ( ∂ ) ◦ P ( ∂ ) = N X p =0 N − 1 X q =0 ∂ p ◦ ( K p U q ) ∂ q = N X p =0 N − 1 X q =0 N − 1 X m =0 γ p,q m ∂ m +1 ◦ ( K p U q ) ( p + q − 2 m − 1) ∂ m + N X p =0 N − 1 X q =0 N − 1 X m =0 δ p,q m ∂ m ◦ ( K p U q ) ( p + q − 2 m ) ∂ m , where γ p,q m and δ p,q m are in tege rs and γ p,q m = 0 unless 0 ≤ p ≤ N , 0 ≤ q ≤ N − 1 and min( p, q ) ≤ m ≤ h p + q − 1 2 i , δ p,q m = 0 unless 0 ≤ p ≤ N , 0 ≤ q ≤ N − 1 and min( p, q + 1) ≤ m ≤  p + q 2  , γ p,q m =  p − m − 1 m − q  if 0 ≤ q < p ≤ N , q ≤ m ≤ h p + q − 1 2 i , δ p,q m =  p − m − 1 m − q − 1  if 0 ≤ q < p ≤ N , q + 1 ≤ m ≤  p + q 2  . W e th us get fr om ( A.4.8 ) (A.4.9) K ( ∂ ) ◦ P ( ∂ ) − P ∗ ( ∂ ) ◦ K ∗ ( ∂ ) = N X p =0 N − 1 X q =0 N − 1 X m =0 ∂ m ◦  γ p,q m ∂ ◦ ( K p U q ) ( p + q − 2 m − 1) + γ p,q m ( U T q K T p ) ( p + q − 2 m − 1) ∂ + δ p,q m ( K p U q ) ( p + q − 2 m ) − δ p,q m ( U T q K T p ) ( p + q − 2 m )  ∂ m . 110 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Comparing ( A.4.9 ) and ( A.4 .7 ) w e get, from the un iqueness of the decom- p osition ( A.1.3 ), the follo wing system of equations ( m = 0 , . . . , N − 1): (A.4.10 ) 1 2 N X p =0 N − 1 X q =0 γ p,q m ( K p U q + U T q K T p ) ( p + q − 2 m − 1) = A m 1 2 N X p =0 N − 1 X q =0  γ p,q m + 2 δ p,q m  ( K p U q − U T q K T p ) ( p + q − 2 m ) = B m This can b e viewed as a sys tem of linear diﬀerentia l equations in the ℓ 2 N en tries u q ij of the ℓ × ℓ matrices U q , q = 0 , . . . , N − 1. The n um b er of indep end en t equations is also ℓ 2 N , since b oth sid es of the ﬁrs t equation are manifestly symmetric and b oth sides of the second equation are manifestly sk ewsymmetric. W e m ak e a c hange of v ariables X q = 1 2 ( U q + U T q ) =  x q ij  ℓ i,j =1 and Y q = 1 2 ( U q − U T q ) =  y q ij  ℓ i,j =1 . In these v ariables, the system ( A.4.10 ) has the form X q ,i ′ ,j ′ L mij +; q i ′ j ′ + ( ∂ ) x q i ′ j ′ + X q ,i ′ ,j ′ L mij +; q i ′ j ′ − ( ∂ ) y q i ′ j ′ = a mij X q ,i ′ ,j ′ L mij − ; q i ′ j ′ + ( ∂ ) x q i ′ j ′ + X q ,i ′ ,j ′ L mij − ; q i ′ j ′ − ( ∂ ) y q i ′ j ′ = b mij where A m =  a mij  ℓ i,j =1 and B m =  b mij  ℓ i,j =1 , and , for m, q = 0 , . . . , N − 1 and i, j, i ′ , j ′ = 1 , . . . ℓ , L mij + , q i ′ j ′ + ( ∂ ) = 1 2 N X p =0 γ p,q m ∂ p + q − 2 m − 1  δ j,j ′ k pii ′ + δ i,i ′ k pj j ′  , L mij + , q i ′ j ′ − ( ∂ ) = 1 2 N X p =0 γ p,q m ∂ p + q − 2 m − 1  δ j,j ′ k pii ′ − δ i,i ′ k pj j ′  , L mij − , q i ′ j ′ + ( ∂ ) = 1 2 N X p =0 ( γ p,q m + 2 δ p,q m ) ∂ p + q − 2 m  δ j,j ′ k pii ′ − δ i,i ′ k pj j ′  , L mij − , q i ′ j ′ − ( ∂ ) = 1 2 N X p =0 ( γ p,q m + 2 δ p,q m ) ∂ p + q − 2 m  δ j,j ′ k pii ′ + δ i,i ′ k pj j ′  , where k pij denotes the ( i, j ) entry of the matrix K p . Hence, th e system ( A.4.10 ) is associated to the ℓ 2 N × ℓ 2 N matrix diﬀeren tial op erator M ( ∂ ) =  L mij ε ; q i ′ j ′ ε ′ ( ∂ )  with ro ws and columns ind exed by quadruples ( m, i, j, ε ), where m = 0 , . . . , N − 1 , ε = ± , i, j = 1 , . . . , ℓ w ith i ≤ j if ε = + and w ith i < j if ε = − . In particular, ord( L mij ε ; q i ′ j ′ ε ′ ( ∂ )) ≤ N + q − 2 m − 1+ ε 1 2 = N q i ′ j ′ ε ′ − h mij ε , wh ere (A.4.11 ) N q ij ε = N + q , h mij ε = 2 m + 1 + ε 1 2 . THE V ARIA TIONAL POISSON COHOMOLOGY 111 In ord er to apply Corollary A.3.7 , we need to c hec k th at the leading matrix ¯ M ( ξ ) (deﬁned by ( A.2.4 )) asso ciated to the ma joran t ( A.4.11 ) is non degenerate (or equiv alen tly , b y equation ( A.2.5 ), we need to chec k that ¯ M (1) is non d egenerate). Recalling that K N = 1 I, we ha ve the follo wing form ulas for the en tries l mij ε ,q i ′ j ′ ε ′ of the matrix ¯ M (1): l mij ε ,q i ′ j ′ ε ′ =    γ N ,q m δ i,i ′ δ j,j ′ for ε = ε ′ = + , γ N ,q m + 2 δ N ,q m δ i,i ′ δ j,j ′ for ε = ε ′ = − , 0 for ε 6 = ε ′ . In p articular, ¯ M (1) is a blo c k d iagonal matrix, with u pp er triangular blo cks, ha ving 1’s on the d iagonal, hence it is non degenerate. By Corollary A.3.7 w e conclude that the system ( A.4.1 0 ) alw a ys has solutions if F is linearly closed, pr o ving part (a). Moreo v er, by Corollary A.3.7 , the sp ace of solutions of the corresp onding homogeneous system h as dimens ion o ver C less than or equal to (and equal if F is linearly closed) d = N − 1 X m =0 ℓ X 1 ≤ i ≤ j ≤ ℓ ( N mij + − h mij + ) + N − 1 X m =0 ℓ X 1 ≤ i ν 1 , then c n 0 ,...,n k m 0 ,...,m k is zer o unless µ 0 ≤ ν 0 ; (vi) if n α = ν 0 , then c n 0 ,...,n k m 0 ,...,m k is zer o unless m α ≥ max( µ 1 , ν 1 ) ; (vii) if n β ≤ ν 1 , then c n 0 ,...,n k m 0 ,...,m k is zer o unless m β ≥ n β . Pr o of. The uniqu eness of the decomp osition ( A.5.20 ) follo ws fr om Lemma A.5.1 in the case V = F [ ∂ ± 1 ], with ∂ acting b y left multiplica tion, and we w an t to p ro v e the existence . By Lemma A.5.3 the monomial λ n 0 0 . . . λ n k k is a linear com b in ation o v er F [ ∂ ] of monomials λ m 0 0 . . . λ m k k with µ 0 − µ 1 = 0 or 1. Hence, w e are left to consider the monomials w ith ν 0 = ν 1 . In this case, m ultiplying the monomial λ n 0 0 λ n 1 1 . . . λ n k k b y 1 = − λ 0 ∂ − 1 − λ 1 ∂ − 1 − · · · − λ k ∂ − 1 , w e get (A.5.23 ) λ n 0 0 λ n 1 1 . . . λ n k k = − k X α =0 λ n 0 0 . . . λ n α +1 α . . . λ n k k ∂ − 1 . All the monomials wh ic h app ear in the RHS h a v e the diﬀerence b et ween maximal and second maximal upp er ind ices equal to 0 or 1. Hence, ( A.5.20 ) holds by in d uction on P i ( ν 0 − n i ). W e next pro v e part (b). Prop erty (i) is clear. Given a p ermutation σ ∈ S k +1 and n 0 , . . . , n k ∈ Z + , w e ha v e b y part (a) (after c hanging the indices of su mmation), λ n σ (0) 0 . . . λ n σ ( k ) k = X m 0 ,...,m k ∈ Z + ( µ 0 − µ 1 =1) c n σ (0) ,...,n σ ( k ) m σ (0) ,...,m σ ( k ) λ m σ (0) 0 . . . λ m σ ( k ) k ∂ P i ( n i − m i ) . On the other hand, by p erm u ting the v ariables λ 0 , . . . , λ k , since the condition λ 0 + · · · + λ k + ∂ = 0 is inv arian t, part (a) sa ys that w e ha v e a un ique 118 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 decomp osition λ n 0 σ − 1 (0) . . . λ n k σ − 1 ( k ) = X m 0 ,...,m k ∈ Z + ( µ 0 − µ 1 =1) c n 0 ,...,n k m 0 ,...,m k λ m 0 σ − 1 (0) . . . λ m k σ − 1 ( k ) ∂ P i ( n i − m i ) . Comparing the ab o ve tw o identitie s we conclude, by the uniqueness of th e decomp osition ( A.5.20 ), that c n σ (0) ,...,n σ ( k ) m σ (0) ,...,m σ ( k ) = c n 0 ,...,n k m 0 ,...,m k , proving p rop erty (ii). The t wo id en tities ( A.5.21 ) and ( A.5.22 ) follo w im m ediately b y part (a) and the equations ( A.5.23 ) and ( A.5.19 ) resp ectiv ely . Next, we p r o v e prop erties (iv)–(vii). Let n 0 , . . . , n k ∈ Z + . If ν 0 = ν 1 + 1, then b y (i) we hav e c n 0 ,...,n k m 0 ,...,m k = δ m 0 ,n 0 . . . δ m k ,n k , hence this co eﬃcien t is zero unless m 0 = n 0 , . . . m k = n k . Prop erties (iv)–(vii), in this case, trivially hold. Supp ose n ext that n 0 , . . . , n k ∈ Z + are su ch that ν 0 = ν 1 . W e p r o v e, by induction on P i ( ν 0 − n i ), th at prop erties (iv)—(vii) hold, i.e. c n 0 ,...,n k m 0 ,...,m k is zero u nless, resp ectiv ely: (iv) µ 0 ≤ ν 0 + 1, (vi) m α ≥ µ 1 and ν 1 , for α such that n α = ν 0 , (vii) m β ≥ n β for all β = 0 , . . . , k . By equation ( A.5.21 ) w e h av e (A.5.24 ) c n 0 ,...,n k m 0 ,...,m k = − X β | n β = ν 0 δ m 0 ,n 0 . . . δ m β ,n β +1 . . . δ m k ,n k − X γ | n γ ≤ ν 0 − 1 c n 0 ,...,n γ +1 ,...,n k m 0 ,...,m k . Giv en β suc h that n β = ν 0 , the corresp onding term in the ﬁrst sum of the RHS is zero unless m 0 = n 0 , . . . , m β = n β + 1 , . . . , m k = n k . In particular one easily c hec ks that it is zero unless all cond itions (iv), (vi) and (vii) ab o v e hold. Next, let γ b e suc h that n γ ≤ ν 0 − 1, and consider the corr esp onding term in the second su mmand of the RHS of ( A.5.24 ). It has maximal and second maximal upp er ind ices b oth equal to ν 0 . Hence, we can apply the inductiv e assumption to deduce that it is zero unless all conditions (iv), (vi), and (vii) hold. Finally , s u pp ose that n α = ν 0 ≥ ν 1 + 2. W e pro v e by indu ction on ν 0 − ν 1 that prop erties (iv)–(vii) hold, i.e. c n 0 ,...,n k m 0 ,...,m k is zero u nless (v) µ 0 ≤ ν 0 , (vi) m α ≥ µ 1 and ν 1 , (vii) m β ≥ n β for all β 6 = α , Consider equation ( A.5.22 ). I n all terms in the RHS the maximal upp er index is n α − 1 = ν 0 − 1, wh ile the second maximal upp er in dex is either ν 1 or ν 1 + 1. Hence, w e can use the results in the previous case an d the inductiv e assumption to deduce that all terms in the RHS of ( A.5.22 ) are zero u nless all conditions (v), (vi) and (vii) ab o v e are met.  THE V ARIA TIONAL POISSON COHOMOLOGY 119 R emark A.5.5 . In the sp ecial case k = 1, Lemma A.5.4 follo ws from L emm a A.1.2 . In fact, in this case w e can use Lemma A.1.2 to compute explic- itly th e co eﬃcients c p,q m +1 ,m and c p,q m,m +1 deﬁned in Lemm a A.5.4 . F or p = q w e ha v e c p,p m +1 ,m = c p,p m,m +1 = − δ m,p . F or p > q w e ha ve c p,q m +1 ,m = ( − 1) m + p +1  p − m m − q  when q ≤ m ≤  ( p + q ) / 2  and zero otherwise, and c p,q m,m +1 = ( − 1) m + p +1  p − m − 1 m − q − 1  when q + 1 ≤ m ≤  ( p + q ) / 2  and zero otherwise. F or p < q they are obtained using the symmetry cond ition (ii) in Lemma A.5.4 . R emark A.5.6 . The p olynomial s (1 + x 1 + · · · + x k ) m 0 x m 1 1 . . . x m k k , w ith m 0 , m 1 , . . . , m k ∈ Z + suc h that µ 0 − µ 1 = 1, form a basis (ov er Z ) of the ring Z [ x 1 , . . . , x k ]. Ind eed, th e linear indep enden ce of these elements follo w s from Lemma A.5.1 in the sp ecial case when V = Q and ∂ = 1, while the fact th at th ey sp an the wh ole p olynomial ring follo ws fr om Lemma A.5.4 . The follo wing is a simp le com binatorial result that will b e used in the pro of of the subsequent Lemma A.5.8 . Lemma A.5.7. L et m 1 , . . . , m k , n 1 , . . . , n k ∈ Z + b e such that ( µ 1 , . . . , µ k ) < ( ν 1 , . . . , ν k ) in the lexic o gr aphic or der. Then: (a) m α < n α for some α ∈ { 1 , . . . , k } ; (b) supp ose, mor e over, that m α < n α for exactly one index α ∈ { 1 , . . . , k } , and m β ≥ n β for every other β 6 = α , and let n α = ν i , then µ 1 = ν 1 , . . . , µ i − 1 = ν i − 1 , µ i ≤ ν i . Pr o of. Let { α 1 , . . . , α k } and { β 1 , . . . , β k } b e p er mutations of { 1 , . . . , k } suc h that m α 1 ≥ m α 2 ≥ · · · ≥ m α k and n β 1 ≥ n β 2 ≥ · · · ≥ n β k . In particular, µ i = m α i and ν i = n β i for every i . F or part (a), sup p ose, by con tradiction, that m α ≥ n α for ev ery α = 1 , . . . , k . Then, clearly , µ 1 ≥ m β 1 ≥ n β 1 = ν 1 . Since, by assump tion, µ 1 ≤ ν 1 , we conclude that µ 1 = ν 1 . Supp ose, b y ind u ction, that µ 1 = ν 1 , . . . , µ i − 1 = ν i − 1 for i ≥ 2. If { α 1 , . . . , α i − 1 } = { β 1 , . . . , β i − 1 } , w e h a v e β i 6∈ { α 1 , . . . , α i − 1 } , and therefore µ i ≥ m β i ≥ n β i = ν i . Similarly , if { α 1 , . . . , α i − 1 } 6 = { β 1 , . . . , β i − 1 } , w e h a v e β j 6∈ { α 1 , . . . , α i − 1 } for s ome j ≤ i − 1, and therefore µ i ≥ m β j ≥ n β j = ν j ≥ ν i . In b oth cases we ha v e µ i ≥ ν i and, sin ce, by assumption, µ i ≤ ν i , we conclude that µ i = ν i . It follo w s th at ( µ 1 , . . . , µ k ) = ( ν 1 , . . . , ν k ), a contradict ion. F or part (b) w e use a similar argument. In our notation, α = β i for some i ≥ 1. If i = 1, there is nothing to p ro v e. If i ≥ 2, we h a v e µ 1 ≥ m β 1 ≥ n β 1 = ν 1 , and therefore µ 1 = ν 1 . Hence, the claim is pro v ed for i = 2. Supp ose next that i ≥ 3 and assume, by in duction, that µ 1 = ν 1 , . . . , µ j = ν j , where j ≤ i − 2. If { α 1 , . . . , α j } = { β 1 , . . . , β j } , we ha v e β j +1 6∈ { α 1 , . . . , α j } , and therefore µ j +1 ≥ m β j +1 ≥ n β j +1 = ν j +1 . Similarly , if { α 1 , . . . , α j } 6 = { β 1 , . . . , β j } , w e h a v e β h 6∈ { α 1 , . . . , α j } for s ome h ≤ j , and therefore µ j +1 ≥ m β h ≥ n β h = ν h ≥ ν j +1 . In b oth cases w e ha v e µ j +1 ≥ ν j +1 and, therefore, µ j +1 = ν j +1 . In conclusion, µ 1 = ν 1 , . . . , µ i − 1 = ν i − 1 , proving the claim.  120 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Lemma A.5.8. L et m 1 , . . . , m k , n 1 , . . . , n k ∈ { 0 , . . . , N − 1 } b e such that ( µ 1 , . . . , µ k ) < ( ν 1 , . . . , ν k ) in the lexic o gr aphic or der. Then: (a) c N ,n 1 ,...,n k µ 1 +1 ,m 1 ,...,m k = 0 ; (b) c n α ,n 1 ,..., α ˇ N ,. ..,n k µ 1 +1 ,m 1 ,...,m k = 0 for every α = 1 , . . . , k ; (c) c N ,n 1 ,...,n k ν 1 +1 ,n 1 ,...,n k = ( − 1) N − ν 1 − 1 ; (d) c n α ,n 1 ,..., α ˇ N ,. ..,n k ν 1 +1 ,n 1 ,...,n k = 0 for every α = 1 , . . . , k . Pr o of. By Lemm a A.5.7 (a), m γ < n γ for some γ 6 = 0. Then, by prop ert y (vii) of Lemma A.5.4 (b), w e hav e c N ,n 1 ,...,n k µ 1 +1 ,m 1 ,...,m k = 0, pro ving part (a). F or part (b), w e ﬁr st observe that, by prop erty (vi) of Lemma A.5.4 (b), if α 6 = 0 then c n α ,n 1 ,..., α ˇ N ,. ..,n k µ 1 +1 ,m 1 ,...,m k = 0 unless m α = µ 1 . Moreo v er , if m γ < n γ for γ 6 = α , again by p rop erty (vii) of Lemma A.5.4 (b), w e h a v e c n α ,n 1 ,..., α ˇ N ,...,n k µ 1 +1 ,m 1 ,...,m k = 0. Hence, b y Lemma A.5.7 (a), w e only h av e to pr o v e (b) in the case when µ 1 = m α < n α , and m β ≥ n β for every β 6 = α . Sup p ose, in this case, that n α = ν i , for i ≥ 2. Then, by Lemma A.5.7 (b), w e ha v e µ 1 < m α < n α = ν i ≤ ν 1 = µ 1 , wh ic h is imp ossible. Hence, we are left to consider the case when m α = µ 1 < n α = ν 1 and m β ≥ n β for ev ery β 6 = α . By prop er ty (vii) of Lemma A.5.4 (b), we ha v e that c n α ,n 1 ,..., α ˇ N ,. ..,n k µ 1 +1 ,m 1 ,...,m k = 0 unless µ 1 + 1 ≥ n α . Hence, w e only need to consider the case when µ 1 + 1 = ν 1 . But in this case m α = µ 1 < µ 1 + 1 = ν 1 , and hence c n α ,n 1 ,..., α ˇ N ,. ..,n k µ 1 +1 ,m 1 ,...,m k = 0 by p rop erty (vi) of Lemma A.5.4 (b). This completes the pro of of part (b). F or N = n 1 + 1, we ha ve c N ,n 1 ,...,n k ν 1 +1 ,n 1 ,...,n k = 1 by p rop erty (i) of Lemma A.5.4 (b). F or N ≥ n 1 + 2, we h a v e, by the recursive formula ( A.5.22 ), c N ,n 1 ,...,n k ν 1 +1 ,n 1 ,...,n k = − X β 6 =0 c N − 1 ,n 1 ,...n β +1 ,...,n k ν 1 +1 ,n 1 ,...,n k − c N − 1 ,n 1 ,...,n k ν 1 +1 ,n 1 ,...,n k . Since N − 1 is the maximal upp er index in all terms of the RHS, the ﬁrst term in the RHS is zero by prop ert y (vii) of Lemma A.5.4 (b). Hence, we get c N ,n 1 ,...,n k ν 1 +1 ,n 1 ,...,n k = − c N − 1 ,n 1 ,...,n k ν 1 +1 ,n 1 ,...,n k , wh ic h, by ind uction, imp lies part (c). W e are left to p ro v e p art (d). If N = ν 1 + 1, by prop ert y (i) of Lemma A.5.4 (b), we h av e that c n α ,n 1 ,..., α ˇ N ,. ..,n k ν 1 +1 ,n 1 ,...,n k = 0 for ev ery α 6 = 0, since n α 6 = N . F or N ≥ n 1 + 2, w e h a v e, by the recursive form ula ( A.5.22 ), c n α ,n 1 ,..., α ˇ N ,...,n k ν 1 +1 ,n 1 ,...,n k = − X β 6 =0 ,α c n α ,n 1 ,...,n β +1 ,..., α ˇ N − 1 ,. ..,n k ν 1 +1 ,n 1 ,...,n k − c n α +1 ,n 1 ,..., α ˇ N − 1 ,. ..,n k ν 1 +1 ,n 1 ,...,n k − c n α ,n 1 ,..., α ˇ N − 1 ,...,n k ν 1 +1 ,n 1 ,...,n k . Note that N − 1 is the maximal u pp er ind ex in all terms of the RHS. Hence, the ﬁrs t term in the RHS is zero by prop ert y (vii) of Lemma A.5.4 (b), THE V ARIA TIONAL POISSON COHOMOLOGY 121 since n β < n β + 1. Moreo v er, the second term in the RHS is zero b y prop erty (vi) of Lemma A.5.4 (b) since n α < n α + 1 ≤ ν 1 . Hence, we get c n α ,n 1 ,..., α ˇ N ,...,n k ν 1 +1 ,n 1 ,...,n k = − c n α ,n 1 ,..., α ˇ N − 1 ,. ..,n k ν 1 +1 ,n 1 ,...,n k , whic h, by induction, implies that c n α ,n 1 ,..., α ˇ N ,. ..,n k ν 1 +1 ,n 1 ,...,n k = 0, pro ving (d).  Corollary A.5.9. If ∂ : F → F is surje c tiv e, then any p olynomial S ∈ F [ λ 1 , . . . , λ k ] ⊗ F admits a de c omp osition of the fol lowing form: S ( λ 1 , . . . , λ k ) = N X m 0 ,...,m k =0 ( µ 0 − µ 1 =1) λ m 0 0 . . . λ m k k s m 0 ,...,m k , wher e λ 0 is as in ( A.5.9 ) (and it acts on the c o eﬃcients s m 0 ,...,m k ). Pr o of. Expand in g S and subs tituting eac h monomial λ n 1 1 . . . λ n k k with the RHS of ( A.5.20 ) (for n 0 = 0), w e get the desired expansion, using the assumption that ∂ is sur jectiv e.  A.5.2. The main r esults for p olydiﬀer ential op er ators. Corollary A.5.10. (a) L et S =  S i 0 ,...,i k ( λ 1 , . . . , λ k )  i 0 ,...,i k ∈ I b e a total ly skewsymmetric k - diﬀer ential op er ator on F ℓ . Assuming that ∂ : F → F is surje ctive, S admits a de c omp osition (A.5.25 ) S i 0 ,...,i k ( λ 1 , . . . , λ k ) = M X m 0 ,...,m k =0 ( µ 0 − µ 1 =1) λ m 0 0 . . . λ m k k s m 0 ,...,m k i 0 ,...,i k , with λ 0 is as in ( A.5.9 ) , wher e the c o eﬃcients s m 0 ,...,m k i 0 ,...,i k ∈ F ar e skewsym- metric with r esp e ct to simultane ous p ermutations of upp er and lower indic es: (A.5.26 ) s m σ (0) ,...,m σ ( k ) i σ (0) ,...,i σ ( k ) = sign( σ ) s m 0 ,...,m k i 0 ,...,i k ∀ σ ∈ S k +1 . (b) Assuming that F is line arly close d, the sp ac e of ve ctors { s m 0 ,...,m k i 0 ,...,i k ∈ F } , lab ele d by i 0 , . . . , i k ∈ { 1 , . . . , ℓ } and m 0 , . . . , m k ∈ { 0 , . . . , N } such that µ 0 − µ 1 = 1 , which ar e skewsymmetric w ith r esp e ct to simultane ous p ermutations of upp er and lower indic es, and which solve (A.5.27 ) N X m 0 ,...,m k =0 ( µ 0 − µ 1 =1) λ m 0 0 . . . λ m k k s m 0 ,...,m k i 0 ,...,i k = 0 , has dimension over C e qual to (A.5.28 ) D = N − 1 X n =0  ( n + 1) ℓ k + 1  −  nℓ k + 1  − ℓ  nℓ k  122 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Pr o of. By Corollary ( A.5.9 ), eac h p olynomial S i 0 ,...,i k ( λ 1 , . . . , λ k ) admits a decomp osition as in ( A.5.25 ), for some M ∈ Z + and some coeﬃcien ts s m 0 ,...,m k i 0 ,...,i k ∈ F . Applying total skewsymmetrizatio n to b oth sides of ( A.5.25 ), w e can replace the co eﬃcient s s m 0 ,...,m k i 0 ,...,i k b y totally ske wsymmetric ones, p r o v- ing p art (a). All the solutions of equatio n ( A.5.27 ) are giv en by elemen ts s m 0 ,...,m k i 0 ,...,i k ∈ F satisfying conditions (i) and (ii) of Lemma A.5.1 . Therefore, the space of arra ys { s m 0 ,...,m k i 0 ,...,i k ∈ F } as in part (b) is in bijectiv e corr esp ondence with the space of arrays T = { t n 0 ,...,n k i 0 ,...,i k ∈ F } , lab eled by i 0 , . . . , i k ∈ { 1 , . . . , ℓ } and n 0 , . . . , n k ∈ { 0 , . . . , N − 1 } s uc h that ν 0 = ν 1 , whic h are sk ewsy m metric with resp ect to sim ultaneous p erm u tations of upp er and lo wer indices, and whic h s olve the system of equ ations ∂ t n 0 ,...,n k i 0 ,...,i k + k X h =0 t n 0 ,...,n h − 1 ,...,n k i 0 ,...,i k = 0 . This is a system of linear diﬀeren tial equations of the form ∂ T + AT = 0, hence, since by assu mption F is linearly closed, the space of solutions has dimension equal to the n um b er of un kno wns. The functions t n 0 ,...,n k i 0 ,...,i k are lab eled by the index set e C = { 1 , . . . , ℓ } k +1 × n ( n 0 , . . . , n k ) ∈ { 0 , . . . , N − 1 } k +1    ν 0 = ν 1 o , and since they are skewsymmetric with resp ect to simulta neous p erm uta- tions of indices n 0 , . . . , n k and i 0 , . . . , i k , we can say that the en tries of the arra y T are lab eled by the S k +1 -orbits in e C with trivial stabilizer. Therefore D = #( C ), w here C = n ω ∈ e C /S k +1    Stab( ω ) = { 1 } o . W e can decomp ose the index set e C as disjoin t union of the su bsets e C s,n , s = 2 , . . . , k + 1 , n = 0 , . . . , N − 1, deﬁned by e C s,n = { 1 , . . . , ℓ } k +1 × n ( n 0 , . . . , n k ) ∈ { 0 , . . . , N − 1 } k +1    n = ν 0 = ν s − 1 > ν s o and the action of the p erm utation group S k +1 preserve s eac h of these subsets. Hence, D = P N − 1 n =0 P k +1 s =2 #( C s,n ), where C s,n = n ω ∈ e C s,n /S k +1    Stab( ω ) = { 1 } o . It is easy to see, by pu tting all maximal elements in the ﬁrs t p ositions, that the s et C s,n is in bijection with th e cartesian pro duct of th e set of S s - orbits with trivial stabilizer in { 1 , . . . , ℓ } k +1 × { 0 , . . . , N − 1 } k +1 of th e form (( i 0 , . . . , i s ) , ( n, . . . , n )), and the set of S k +1 − s -orbits with trivial stabilizer in { 1 , . . . , ℓ } k +1 × { 0 , . . . , N − 1 } k +1 of the f orm (( i s , . . . , i k ) , ( n s , . . . , n k )) with n s , . . . , n k < n . Clearly , the cardinalit y of the ﬁrst set is  ℓ s  , and the THE V ARIA TIONAL POISSON COHOMOLOGY 123 cardinalit y of the second set is  nℓ k +1 − s  . Hence, D = N − 1 X n =0 k +1 X s =2  ℓ s  nℓ k + 1 − s  . This formula implies equation ( A.5.28 ), since  ( n +1) ℓ k +1  = P k +1 s =0  ℓ s  nℓ k +1 − s  .  Theorem A.5.11. L et k ∈ Z + , and let K ( ∂ ) ∈ Mat ℓ × ℓ ( F [ ∂ ]) b e an ℓ × ℓ ma- trix diﬀer ential op er ator of or der N with invertible le ading c o eﬃcient, over a line arly close d diﬀer ential ﬁeld F . Then for every total ly skewsymmetric k -diﬀer ential op er ator S on F ℓ ther e exists a skewsymmetric k -diﬀer ential op er ator P on F ℓ such that (A.5.29 ) h K ◦ P i − = S . Pr o of. F or k = 0 w e hav e P ∈ F ℓ and h K ◦ P i − = K ( ∂ ) P . Hence, the claim follo ws from Corollary A.3.7 b y taking the ma j oran t N j = N ∀ j, h i = 0 ∀ i of the matrix diﬀeren tial op erator K ( ∂ ). Next let k ≥ 1. Let K ( ∂ ) = P N n =0 ( − ∂ ) n ◦ K n , where ( − 1) N K N 6 = 0 is the leading co eﬃcient. If we let K 1 ( ∂ ) = K ( ∂ ) ◦ K − 1 N and P 1 ( λ 1 , . . . , λ k ) = K N P 1 ( λ 1 , . . . , λ k ), we ha ve, by ( A.5.8 ), K ◦ P = K 1 ◦ P 1 . Hence, w e ma y assume that K N = 1 I. Let S b e a totall y sk ewsymmetric k -diﬀerential op er ator on F ℓ . By Corol- lary A.5.10 , S admits a d ecomp osition as in ( A.5.25 ). The ﬁrst part of the pro of w ill consist in reducing to the case when M = N . Let M 0 ≥ M 1 ≥ · · · ≥ M k b e the m aximal, in the lexicographic order, among all non increasing ( k + 1)-tuples µ 0 ≥ µ 1 ≥ · · · ≥ µ k suc h that s µ 0 ,...,µ k i 0 ,...,i k 6 = 0 f or some i 0 , . . . , i k ∈ I . Clearly , by the sk ewsymmetry con- dition ( A.5.26 ), for m 0 , . . . , m k ∈ Z + w e ha v e that s m 0 ,...,m k i 0 ,...,i k is zero un less ( µ 0 , . . . , µ k ) ≤ ( M 0 , . . . , M k ). Hence, the decomp osition ( A.5.25 ) of S can b e r ewr itten as follo ws (A.5.30 ) S i 0 ,...,i k ( λ 1 , . . . , λ k ) = X m 0 ,...,m k ∈ Z + µ 0 − µ 1 =0 or 1 ( µ 0 ,...,µ k ) ≤ ( M 0 ,...,M k ) λ m 0 0 . . . λ m k k s m 0 ,...,m k i 0 ,...,i k . Notice that in ( A.5.30 ), for reasons that will b ecome clear later, we allo w terms with µ 0 − µ 1 equal 0 or 1 (ev en though, by Corollary A.5.10 , we could restrict to the term s with µ 0 − µ 1 = 1). If M 0 ≥ N , let P 0 b e the follo wing k -diﬀeren tial op erator on F ℓ : P 0 i 0 ,...,i k = C λ M 0 − N 0 λ M 1 1 . . . λ M k k s M 0 ,...,M k i 0 ,...,i k . where C d enotes the cardinalit y of the orbit of ( M 0 , . . . , M k ) und er the action of the symmetric group S k +1 . By ( A.5.8 ) and the assumption that 124 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 K N = 1 I, we ha v e (A.5.31 ) ( K ◦ P 0 ) i 0 ,...,i k = C λ M 0 0 λ M 1 1 . . . λ M k k s M 0 ,...,M k i 0 ,...,i k + C ℓ X j =1 N − 1 X n =0 M 0 − N X h =0  M 0 − N h  λ M 0 − N + n − h 0 λ M 1 1 . . . λ M k k ( ∂ h K n ) i 0 ,j s M 0 ,M 1 ,...,M k j,i 1 ,...,i k . Let P b e the sk ewsymmetrizatio n (o v er S k ) of P 0 . Clearly , the sk ewsym- metrization (o ver S k ) of K ◦ P 0 is equal to K ◦ P , and therefore h K ◦ P i − = h K ◦ P 0 i − . Hence, taking the tota l skewsymmetrizat ion of b oth sides of ( A.5.31 ), w e get (A.5.32 ) h K ◦ P i − i 0 ,...,i k ( λ 1 , . . . , λ k ) = C ( k + 1)! X σ ∈ S k +1 sign( σ ) λ M 0 σ − 1 (0) λ M 1 σ − 1 (1) . . . λ M k σ − 1 ( k ) s M 0 ,...,M k i σ − 1 (0) ,...,i σ − 1 ( k ) + ℓ X j =1 N − 1 X n =0 M 0 − N X h =0  M 0 − N h  C ( k + 1)! X σ ∈ S k +1 sign( σ ) × λ M 0 − N + n − h σ − 1 (0) λ M 1 σ − 1 (1) . . . λ M k σ − 1 ( k ) ( ∂ h K n ) i σ − 1 (0) ,j s M 0 ,M 1 ,...,M k j,i σ − 1 (1) ,...,i σ − 1 ( k ) . By the skewsymmetry condition ( A.5.26 ) on the coeﬃcien ts s m 0 ,...,m k i 0 ,...,i k , and since ( k +1)! C is the cardinalit y of the stabilizer of ( M 0 , . . . , M k ) under the action of S k +1 , the ﬁrst term in the RHS of ( A.5.32 ) is equal to X m 0 ,...,m k ∈ Z + ( µ 0 ,...,µ k )=( M 0 ,...,M k ) λ m 0 0 . . . λ m k k s m 0 ,...,m k i 0 ,...,i k . Moreo ver, eac h monomial λ M 0 − N + n − h σ − 1 (0) λ M 1 σ − 1 (1) . . . λ M k σ − 1 ( k ) whic h ent ers in the second term of the RHS of ( A.5.32 ) can b e expand ed, using Lemma A.5.3 , as λ M 0 − N + n − h σ − 1 (0) λ M 1 σ − 1 (1) . . . λ M k σ − 1 ( k ) = X m 0 ,...,m k ∈ Z + ( µ 0 − µ 1 =0 or 1) b M σ (0) ,..., σ − 1 (0) ˇ M 0 − N + n − h,...,M σ ( K ) m 0 ,...,m k λ m 0 0 λ m 1 1 . . . λ m k k ∂ P i ( M i − m i ) − N + n − h and, by the last statement of Lemma A.5.3 , b M σ (0) ,..., σ − 1 (0) ˇ M 0 − N + n − h,...,M σ ( K ) m 0 ,...,m k is zero unless µ 0 ≤ N 0 and, for µ 0 = N 0 , it is zero u nless m α = M σ ( α ) − δ α,σ − 1 (0) ( N − n + h ), for ev ery α = 0 , . . . , k . In particular, since n ≤ N − 1, this co eﬃcien t is zero unless ( µ 0 , . . . , µ k ) < ( M 0 , . . . , M k ). Putting together THE V ARIA TIONAL POISSON COHOMOLOGY 125 the ab o ve obs er v ations, we can write h K ◦ P i − as (A.5.33 ) h K ◦ P i − i 0 ,...,i k ( λ 1 , . . . , λ k ) = X m 0 ,...,m k ∈ Z + ( µ 0 ,...,µ k )=( M 0 ,...,M k ) λ m 0 0 . . . λ m k k s m 0 ,...,m k i 0 ,...,i k + X m 0 ,...,m k ∈ Z + µ 0 − µ 1 =0 or 1 ( µ 0 ,...,µ k ) < ( M 0 ,...,M k ) λ m 0 0 . . . λ m k k t m 0 ,...,m k i 0 ,...,i k , for some co eﬃcien ts t m 0 ,...,m k i 0 ,...,i k ∈ F . It follo w s that S − h K ◦ P i − has a decom- p osition as in ( A.5.30 ) where only terms with ( µ 0 , . . . , µ k ) < ( M 0 , . . . , M k ) app ear. Hence, b y induction, w e can reduce to the case when S has a de- comp osition as in ( A.5.30 ) with M 0 ≤ N − 1. T o conclude, w e further reduce the monomials λ m 0 0 . . . λ m k k in the expan- sion ( A.5.30 ) of S with µ 0 = µ 1 using Lemma A.5.4 . By p rop erty (iv) of Lemma A.5.4 (b), the only monomials which en ter in the obtained de- comp osition are such that µ 0 ≤ M 0 + 1 ≤ N . Therefore, this S admits a decomp osition as in ( A.5.25 ) with M = N , completing the ﬁrst step of the pro of. Let then S ha ve the follo wing decomp osition: (A.5.34 ) S i 0 ,...,i k ( λ 1 , . . . , λ k ) = N X m 0 ,...,m k =0 ( µ 0 − µ 1 =1) λ m 0 0 . . . λ m k k s m 0 ,...,m k i 0 ,...,i k , with co eﬃcien ts s m 0 ,...,m k i 0 ,...,i k sk ewsymmetric with resp ect to sim ultaneous p er- m utations of upp er and low er indices. W e w an t to pro v e that there exists a sk ewsymmetric k -diﬀeren tial op erator P of d egree at most N − 1 in eac h v ariable, (A.5.35 ) P i 0 ,...,i k ( λ 1 , . . . , λ k ) = N − 1 X n 1 ,...,n k =0 λ n 1 1 . . . λ n k k p n 1 ,...,n k i 0 ,i 1 ,...,i k , with co eﬃcien ts p m 1 ,...,m k i 0 ,...,i k ∈ F skewsymmetric w ith r esp ect to the action of S k = P er m (1 , . . . , k ) on upp er an d lo wer indices simulta neously , satisfying equation ( A.5.29 ). By ( A.5.8 ) w e ha ve ( K ◦ P ) i 0 ,...,i k ( λ 1 , . . . , λ k ) = N X n 0 =0 N − 1 X n 1 ,...,n k =0 ℓ X j =1 λ n 0 0 λ n 1 1 . . . λ n k k ( K n 0 ) i 0 ,j p n 1 ,...,n k j,i 1 ,...,i k , 126 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 and, b y Lemma A.5.4 , w e get (A.5.36 ) ( K ◦ P ) i 0 ,...,i k ( λ 1 , . . . , λ k ) = N X m 0 ,...,m k =0 ( µ 0 − µ 1 =1) N X n 0 =0 N − 1 X n 1 ,...,n k =0 ℓ X j =1 c n 0 ,...,n k m 0 ,...,m k × λ m 0 0 λ m 1 1 . . . λ m k k ∂ P i ( n i − m i ) ( K n 0 ) i 0 ,j p n 1 ,...,n k j,i 1 ,...,i k . In the RHS ab ov e we can take the su m o v er m 0 , . . . , m k ≤ N for th e follo wing reason. If n 0 = N ( > n 1 , . . . , n k ), by p rop erty (v) of Lemma A.5.4 (b), c n 0 ,...,n k m 0 ,...,m k is zero un less µ 0 ≤ ν 0 = N , wh ile, if n 0 ≤ N − 1 we hav e, by prop erty (iv) of L emm a A.5.4 (b), that c n 0 ,...,n k m 0 ,...,m k is zero unless µ 0 ≤ ν 0 + 1 ≤ N − 1 + 1 = N . T aking the sk ewsymmetrization of b oth sides of equation ( A.5.36 ), w e ha v e, b y ( A.5.7 ) and b y the symmetry prop erty (ii) of Lemma A.5.4 (b), (A.5.37 ) h K ◦ P i − i 0 ,...,i k ( λ 1 , . . . , λ k ) = 1 k + 1 N X m 0 ,...,m k =0 ( µ 0 − µ 1 =1) N X n 0 =0 N − 1 X n 1 ,...,n k =0 ℓ X j =1 λ m 0 0 . . . λ m k k ∂ P i ( n i − m i )  c n 0 ,...,n k m 0 ,...,m k ( K n 0 ) i 0 ,j p n 1 ,...,n k j,i 1 ,...,i k − k X α =1 c n α ,n 1 ,..., α ˇ n 0 ,...,n k m 0 ,...,m k ( K n 0 ) i α ,j p n 1 ,...,n k j,i 1 ,..., α ˇ i 0 ,...,i k  . Comparing equations ( A.5.34 ) and ( A.5.3 7 ) , w e get the follo win g equation in F [ λ 1 , . . . , λ k ] ⊗ F : (A.5.38 ) N X m 0 ,...,m k =0 ( µ 0 − µ 1 =1) λ m 0 0 . . . λ m k k 1 k + 1 N X n 0 =0 N − 1 X n 1 ,...,n k =0 ℓ X j =1 ∂ P i ( n i − m i )  c n 0 ,...,n k m 0 ,...,m k ( K n 0 ) i 0 ,j × p n 1 ,...,n k j,i 1 ,...,i k − k X α =1 c n α ,n 1 ,..., α ˇ n 0 ,...,n k m 0 ,...,m k ( K n 0 ) i α ,j p n 1 ,...,n k j,i 1 ,..., α ˇ i 0 ,...,i k  − s m 0 ,...,m k i 0 ,...,i k ! = 0 . The ab ov e equation sh ould b e read as an equation in the unkno wn v ariables (A.5.39 ) X =  p n 1 ,...,n k i 0 ,i 1 ,...,i k ∈ F  1 ≤ i 0 ,...,i k ≤ ℓ 0 ≤ n 1 ,...,n k ≤ N − 1 , suc h that p n σ (1) ,...,n σ ( k ) i 0 ,i σ ( k ) ,...,i σ ( k ) = sign( σ ) p n 1 ,...,n k i 0 ,i 1 ,...,i k for ev ery σ ∈ S k , and the elemen t (A.5.40 ) B =  s m 0 ,...,m k i 0 ,...,i k ∈ F  1 ≤ i 0 ,...,i k ≤ ℓ 0 ≤ m 0 ,m 1 ,...,m k ≤ N ( µ 0 − µ 1 =1) , sk ewsymmetric with resp ect to the action of S k +1 , is giv en. T o complete the pro of of the theorem, w e only need to sho w th at this equ ation admits a solution. THE V ARIA TIONAL POISSON COHOMOLOGY 127 Note that the co eﬃcient of λ m 0 0 . . . λ m k k in the LHS of ( A.5.38 ) can b e rewritten, u p to the summand − s m 0 ,...,m k i 0 ,...,i k , as (A.5.41 ) 1 k + 1 N X n 0 ,...,n k =0 ℓ X j =1 ∂ P i ( n i − m i ) k X α =0 ( − 1) α c n 0 ,...,n k m 0 ,...,m k ( K n α ) i α ,j p n 0 ,n 1 , α ˇ ...,n k j,i 0 , α ˇ ...,i k , and, in this form , it is manifestly ske wsymmetric with resp ect to sim ultane- ous p erm utations of i 0 , . . . , i k and m 0 , . . . , m k . The v ariables p n 1 ,...,n k j 0 ,j 1 ,...,j k are lab eled by the index set e A = { 1 , . . . , ℓ } k +1 × { 0 , . . . , N − 1 } k . On the other hand, since , b y assumption, they are skewsymmetric with resp ect to sim ultaneous p ermutat ions of indices n 1 , . . . , n k and j 1 , . . . , j k , w e can sa y that the ent ries of the v ariable X are lab eled by the S k -orbits in e A with trivial stabilizer, wher e S k = P erm (1 , . . . , k ) acts on the elemen t (( j 0 , j 1 , . . . , j k ) , ( n 1 , . . . , n k )) by ﬁ xing j 0 and p erm u ting, simultaneously , the other entries. W e therefore write X =  p a  a ∈A , where (A.5.42 ) A = n ω ∈ e A /S k    Stab( ω ) = { 1 } o , and, for (A.5.43 ) a = S k · (( j 0 , j 1 , . . . , j k ) , ( n 1 , . . . , n k )) ∈ A , w e let p a = ± p n 1 ,...,n k j 0 ,j 1 ,...,j k . T he sign of p a is ﬁ xed by taking + for the u nique represent ativ e of a with n 1 ≥ n 2 ≥ · · · ≥ n k and j s > j s +1 if n s = n s +1 . Similarly , the functions s m 0 ,...,m k i 0 ,...,i k are lab eled by the index set e B = { 1 , . . . , ℓ } k +1 × n ( m 0 , . . . , m k ) ∈ { 0 , . . . , N } k +1    µ 0 − µ 1 = 1 o , and since, b y assum p tion, they are skewsymmetric with resp ect to sim ulta- neous p ermutat ions of ind ices m 0 , . . . , m k and i 0 , . . . , i k , we can sa y that the en tries of the giv en arr a y B are lab eled by the S k +1 -orbits in e B with triv- ial stabilizer, where S k +1 = P er m (0 , . . . , k ) acts diagonally on the element (( i 0 , . . . , i k ) , ( m 0 , . . . , m k )). W e therefore write B =  s b  b ∈B , wh ere (A.5.44 ) B = n ω ∈ e B /S k +1    Stab( ω ) = { 1 } o , and, for (A.5.45 ) b = S k +1 · (( i 0 , . . . , i k ) , ( m 0 , . . . , m k )) ∈ B , w e let s b = ± s m 0 ,...,m k i 0 ,...,i k . As b efore, the s ign of s b is ﬁxed b y taking + for the unique representa tiv e of b with m 0 = m 1 + 1 > m 1 ≥ · · · ≥ m k and i s > i s +1 if m s = m s +1 . F or a as in ( A.5.43 ), w e let (A.5.46 ) ϕ ( a ) = S k +1 · (( j 0 , j 1 , . . . , j k ) , (max( n 1 , . . . , n k ) + 1 , n 1 , . . . , n k )) . 128 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 It is n ot h ard to c heck that ϕ is a we ll-deﬁned bijectiv e map A ∼ → B . In particular, the vec tors X and B ha v e the same n um b er of entries. I n fact, (A.5.47 ) #( A ) = # ( B ) = ℓ  N ℓ k  . Equation ( A.5.38 ) is equiv alent to the follo wing equation (A.5.48 ) A ( ∂ )( M ( ∂ ) X − B ) = 0 , where X and B are as in ( A.5.39 ) and ( A.5.40 ) resp ectiv ely , and M ( ∂ ) and A ( ∂ ) are deﬁned as f ollo ws. First, M ( ∂ ) =  L b,a ( ∂ )  b ∈B , a ∈A , giv en by ( A.5.41 ), is the square matrix pseudo diﬀerential op erator with entries (A.5.49 ) L b,a ( ∂ ) = 1 k + 1 N X n 0 =0 ℓ X j =1 ∂ P i ( n i − m i )  c n 0 ,...,n k m 0 ,...,m k ( K n 0 ) i 0 ,j δ j,j 0 δ j 1 ,i 1 . . . δ j k ,i k − k X α =1 c n α ,n 1 ,..., α ˇ n 0 ,...,n k m 0 ,...,m k ( K n 0 ) i α ,j δ j,j 0 δ j α ,i 0 δ j 1 ,i 1 α ˇ . . . δ j k ,i k  , for a and b as in ( A.5.43 ) and ( A.5.4 5 ) resp ecti v ely . Note th at in order to sa y that the en tries of the matrix M ( ∂ ) can b e lab eled by the set B × A (and not b y the set e B × A ) w e are using the f act that M ( ∂ ) X , given b y ( A.5.41 ), is m an if estly skewsymmetric w ith resp ect to the action of S k +1 . T o deﬁne A ( ∂ ), consider ﬁ r st the map e A from F e B to the space D of k - diﬀeren tial op erators on F ℓ of d egree at most 2 N − 1 in eac h v ariable, giv en b y e B = { s m 0 ,...,m k i 0 ,...,i k } 7→ e A ( e B ), w here e A ( e B ) i 1 ,...,i k ( λ 1 , . . . , λ k ) = X e B ( − λ 1 − · · · − λ k − ∂ ) m 0 λ m 1 1 . . . λ m k k s m 0 ,...,m k i 0 ,...,i k . Note that e A is a C -linear, not an F -linear map, b ut the space D is in fact a ﬁ- nite d imensional v ector sp ace o v er F , sa y of dimension d , and , for any c hoice of basis of it, e A b ecomes a d × #( e B ) matrix diﬀeren tial op erator. W e next consider F B as the su bspace of F e B consisting of the elemen ts e B = { s m 0 ,...,m k i 0 ,...,i k } sk ewsymmetric with resp ect to the action of S k +1 . The restriction of e A to the subspace F B ⊂ F e B is then a d × #( B ) matrix d iﬀeren tial op erator from F B to D , f or any choic e of basis of D ov er F (once w e ﬁx representati v es, w e can consid er B as a sub set of e B , and the matrix of e A | F B consists of the ro ws of the matrix of e A corresp ond ing to the indices in B ). By Lemma A.2.5 , w e can c hoose a basis of D suc h that the matrix for e A | F B : F B → D is in r o w ec helon f orm, sa y with k piv ots and d − k zero ro ws. W e then let A ( ∂ ) : F B → F k the k × #( B ) matrix diﬀerentia l op erator giv en by the ﬁrst k ro ws of this matrix. It is then clear from the ab o v e constru ction that equation ( A.5.38 ) is equiv alent to equation ( A.5 .48 ). Therefore, to pro v e the theorem, w e need to sho w that, for ev ery B ∈ F B there exists X ∈ F A solving equation ( A.5.48 ). THE V ARIA TIONAL POISSON COHOMOLOGY 129 Notice th at, even though in the expression of the L HS of ( A.5.38 ) there app ear negativ e p o wers of ∂ , w e kno w that all su c h n egativ e p ow ers can- cel out when w e compute all the su ms in ( A.5.38 ), sin ce th is equation is the same as ( A.5.29 ), whic h do es n ot inv olv e any negativ e p o w er of ∂ . In terms of equation ( A.5.48 ) this means that, ev en th ough M ( ∂ ) is a matrix pseudo d iﬀeren tial op erator, the pro d uct A ( ∂ ) M ( ∂ ) is a matrix diﬀeren tial op erator. By construction, A ( ∂ ) is surjectiv e, and its k ernel consists of elemen ts B = { s m 0 ,...,m k i 0 ,...,i k } solving equation ( A.5.27 ). Hence, b y Corollary A.5.10 (b), it is of ﬁnite d imension o v er C . Therefore, b y Theorem A.3.6 (c), it is a square matrix diﬀerentia l op erator with non zero determinant. Next, note that L b,a ( ∂ ) has order less than or equal to N a − h b , where, for a and b as in ( A.5.43 ) and ( A.5.45 ) resp ectiv ely , (A.5.50 ) N a = N + k X i =1 n i , h b = k X i =0 m i . The leading matrix asso ciated to this ma joran t (see equation ( A.2.4 )) is ¯ M ( ξ ) =  m b,a ξ N a − h b  b ∈B , a ∈A , wh ere m b,a = 1 k + 1  c N ,n 1 ,...,n k m 0 ,...,m k δ i 0 ,j 0 δ j 1 ,i 1 . . . δ j k ,i k − k X α =1 c n α ,n 1 ,..., α ˇ N ,...,n k m 0 ,...,m k δ j 0 ,i α δ j α ,i 0 δ j 1 ,i 1 α ˇ . . . δ j k ,i k  . W e w an t to pro v e that the leading matrix ¯ M ( ξ ) or, equiv alen tly , ¯ M (1) (see ( A.2.5 )), is non degenerate. In order to pro v e this, w e ﬁx a tota l ordering of the sets A ≃ B (identiﬁed via ϕ ), and we p ro v e that, w ith resp ect to this ord er in g, the matrix ¯ M (1) is lo wer triangular w ith n on zero diagonal en tries. Giv en elemen ts b = S k +1 · (( i 0 , . . . , i k ) , ( m 0 , . . . , m k )) ∈ B and b ′ = S k +1 · (( i ′ 0 , . . . , i ′ k ) , ( m ′ 0 , . . . , m ′ k )) ∈ B , we sa y that b > b ′ if ( µ 0 , . . . , µ k ) > ( µ ′ 0 , . . . , µ ′ k ) in the lexicographic order, or ( µ 0 , . . . , µ k ) = ( µ ′ 0 , . . . , µ ′ k ) and b > b ′ in some total ordering of the remaining indices (w h ic h will p la y no role). Therefore, for a ∈ A an d b ∈ B as in ( A.5.43 ) and ( A.5.45 ) resp ectiv ely , usin g the map ϕ we hav e that b > a if ( µ 0 , µ 1 , . . . , µ k ) > ( ν 1 + 1 , ν 1 , . . . , ν k ) in the lexico graphic order , or ( µ 0 , µ 1 , . . . , µ k ) = ( ν 1 + 1 , ν 1 , . . . , ν k ) and b > ϕ ( a ) in some tota l ord ering of the remaining in dices. W e wan t to prov e that, with resp ect to this ordering, for a ∈ A and b ∈ B , w e hav e (A.5.51 ) m b,a = 0 if b < a , and m b,a 6 = 0 if b = ϕ ( a ) . This follo ws from Lemma A.5.8 . Ind eed, for b ≤ a , w e h a ve in particular that ( µ 0 , µ 1 , . . . , µ k ) ≤ ( ν 1 + 1 , ν 1 , . . . , ν k ). Hence, by Lemma A.5.8 (a) and (b) w e hav e m b,a = 0 unless ( µ 0 , µ 1 , . . . , µ k ) = ( ν 1 + 1 , ν 1 , . . . , ν k ), an d , in this case, by Lemma A.5.8 (c) an d (d), m b,a = ( − 1) N − ν 1 − 1 k +1 δ i 0 ,j 0 δ j 1 ,i 1 . . . δ j k ,i k . 130 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 Summarizing the ab o ve results, M ( ∂ ) is a square matrix p seudo diﬀer- en tial op erator of size #( A ) = #( B ), with non degenerate leading matrix asso ciated to the ma joran t ( A.5.50 ), A ( ∂ ) is a square matrix diﬀerentia l op erator of the same size, with non zero determinan t, and A ( ∂ ) M ( ∂ ) is a matrix d iﬀeren tial op er ator. By Corollary A.3.8 , it follo ws that equation ( A.5.48 ) has a solution f or eve ry B , completing the pro of of the theorem.  Theorem A.5.12. L et F b e a line arly close d diﬀer ential ﬁeld with subﬁeld of c onstants C ⊂ F . L et k ∈ Z + , and let K ( ∂ ) ∈ Mat ℓ × ℓ ( F [ ∂ ]) b e an ℓ × ℓ matrix diﬀer ential op er ator of or der N with i nv e rtible le ading c o eﬃci e nt, over F . Then, the set of ske wsymmetric k -diﬀer ential op er ator s P on F ℓ of de gr e e at most N − 1 in e ach variable such that (A.5.52 ) h K ◦ P i − = 0 , is a ve ctor sp ac e over C of dimension d =  N ℓ k + 1  . Pr o of. As in the pro of of Theorem A.5.11 , for k = 0 w e h a v e P ∈ F ℓ and h K ◦ P i − = K ( ∂ ) P . Hence, the state men t follo ws from Corollary A.3.7 b y taking the ma jorant N j = N ∀ j, h i = 0 ∀ i of the m atrix diﬀerentia l op erator K ( ∂ ). Let then k ≥ 1. By the discussion in the pro of of Th eorem A.5.11 , equa- tion ( A.5.52 ) is the same as equation ( A.5 .48 ) with B = 0, an d , moreo ver, b y Corollary A.3.8 , the sp ace of solutions has d imension o ver C equal to (A.5.53 ) d = dim C (Ker A ( ∂ )) + X a ∈A ( N a − h ϕ ( a ) ) . By the construction of the matrix diﬀeren tial op erator A ( ∂ ), th e equation A ( ∂ ) B = 0, for B = { s m 0 ,...,m k i 0 ,...,i k } ∈ F B , is equiv alen t to equation ( A.5.27 ). Hence, by C orollary A.5.10 (b), Ker( A ( ∂ )) has dimension ov er C equal to ( A.5.28 ). Moreo v er, w e hav e, recall ing ( A.5.42 ) , ( A.5.46 ) and ( A.5.50 ) (let- ting a ∈ A as in ( A.5.43 )), (A.5.54 ) X a ∈A ( N a − h ϕ ( a ) ) = X a ∈A  N − 1 − max( n 1 , . . . , n k )  = N − 1 X n =0 ( N − n − 1)# n a ∈ A    ν 1 = n o = N − 1 X n =0 ( N − n − 1)  # n a ∈ A    ν 1 ≤ n o − # n a ∈ A    ν 1 ≤ n − 1 o = ℓ N − 1 X n =0 ( N − 1 − n )   ( n + 1) ℓ k  −  nℓ k   . THE V ARIA TIONAL POISSON COHOMOLOGY 131 In the last identit y w e us ed equation ( A.5.47 ), with N − 1 rep laced by n or n − 1. Putting together ( A.5.28 ) and ( A.5.54 ), we get d = N − 1 X n =0  ( n + 1) ℓ k + 1  −  nℓ k + 1  + ℓ ( N − 1 − n )  ( n + 1) ℓ k  − ℓ ( N − n )  nℓ k  , whic h is a telescopic s u m equal to  N ℓ k +1  , p r o ving the theorem.  Referen ces [Art] E. Artin, Ge ometric algebr a , In terscience Publishers, Inc., New Y ork-London, 1957. [BD AK] B. Bak alo v , A. D’A n drea, and V.G. Kac, The ory of ﬁnite pseudo algebr as , Adv. Math. 162 , (2001 ) 1–1 40. [BKV] B. Bak alo v, V.G. Kac, and A.A. V oronov, Cohomolo gy of c onformal al gebr as , Com- mun. Math. Ph ys. 200 , (1999) 561 –598. [BDSK] A. Barak at, A . De Sole, and V.G. Kac, Poisson vertex algebr as in the the ory of Hamiltonian e quations , Japan. J. Math. 4 , (2009) 141-252. [Bou] N. Bourbaki, Elements de mathemat ique. F asc. XXVI. Chapitr e I: Algebr es de Lie. (F renc h) Second edition (Hermann, Pa ris 1971) 146 pp. [CK] N. Cantarini, and V.G. Kac, Classiﬁc ation of line arly c omp act rigid simple sup er al- gebr as , IMRN 2010 no 17, (2010) 3341–339 3. [DSHK] A. De Sole, P . Hekmati and V.G. Kac, C al culus structur e on the Lie c onformal algebr a c omplex and the variational c omplex , J. Math. Phys. 52 no 5 (2011). [DSK1] A. De S ole, and V.G. Kac, Finite vs. aﬃne W -algebr as , Japan. J. Math. 1 , (2006) 137-261. [DSK2] A. De Sole, and V.G. K ac, Lie c onformal algebr a c ohomolo gy and the variational c omplex , Comm un. Math. Phys. 292 , (2009) 667– 719. [DSKW] A. De S ole, V.G. Kac, and M. W akimoto, On classiﬁc ation of Poisson vertex algebr as , T ransfo rm. Group s 15 (2010), no. 4, 883–907. [Die] J. Dieudonn´ e, L es d ´ eterminants sur un c orps non c ommutatif , Bull. Soc. Math. F rance 71 , (1943), 27–45. [Dor] I.Y a. Dorfman, Dir ac struct ur es and inte gr ability of nonline ar evolution e quations , Nonlinear Sci. Theory Appl. (John Wiley & Sons, 1993) 176 pp. [FZ] L.D. F addeev, V.E. Zakharo v, Kortewe g-de V ries e quation i s a c ompletely inte gr able Hamiltonian syste m , F unc. Anal. App l. 5 , (1971) 280–287. [Gar] C.S. Gardner, Kortewe g-de V ries e quation and gener al i zations IV. The Kortewe g-de V ries e quation as a Hamiltonian syst em , J. Math. Phys. 12 , (1971) 1548–1551. [GGKM] C.S. Gardner, J.M. Greene, M.D. Krusk al, and R.M. Miura, Kortewe g-de V ries e quations and gener ali zations. VI Metho ds for exact solution , Comm. Pure Ap pl. Math. 27 , (1974) 97–133. [Get] E. Getzler, A Darb oux the or em for Hamiltonian op er ator s i n the f ormal c alculus of variations , Duke Math. J. 111 , (2002) no. 3, 535 –560. [Huf ] G. Huﬀord, On the char acteristic matrix of a matrix of diﬀer ential op er ators , J. Diﬀerential Eq u ations 1 , (1965) 27–3 8. [IVV] S. Igonin, A. V erb ov etsky , and R. Vitolo, V ariational multive ctors and br ackets i n the ge ometry of jet sp ac es , Sy mmetry in nonlinear mathematical physics . P art 1, 2, 3, (2004) 1335–1342. [K] V.G. Kac, V erte x algebr as for b e ginners , Univ. Lecture Ser., vol 10, AMS, 1996. Second edition, 1998. [Kra] I.S. Krasilshc h ik, Schouten br ackets and c anonic al algebr as , Lecture Notes in Math. 1334 , (Sp ringer V erlag, New Y ork 1988). 132 ALBER TO DE SOLE 1 AND VICTOR G. KA C 2 [Kup] B.A. Kup ershmidt, Ge ometry of jet bund l es and the structur e of L agr angian and Hamiltonian formalisms , in Ge ometric metho ds in Mathematic al Physics , Lecture Notes in Math. 775 , (Springer V erlag, New Y ork 1980) 162–2 18. [Lax] P .D. Lax, A lmost p erio dic solutions of the KdV e quation , SIAM review 18 , (1976) 351–375 . [Ler] J. Leray , Hyp erb olic diﬀer ential e quations , The Institute for Adv anced Stu dy , Prince- ton, N. J., (1953) 240 pp . [Lic] A. Lichnero wicz, L es varietes de Poisson et leur algebr es de Lie asso cie es , J. Diﬀ. Geom. 12 , (1977) 253–3 00. [Mag] F. Magri, A simpl e mo del of the inte gr able Hamiltonian e quation , J. Math. Phys. 19 , (1987) 129–134. [Miy] M. Miy ak e, O n the determinant of m atric es of or dinary diﬀer ential op er ators and an index the or em , F unkc ial. Ekv ac. 26 , (1983) no. 2, 155– 171. [NR] A. Nijenhuis, an d R. Ric hardson, Deformations of Lie algebr a structur es , J. Math. Mec h. 17 , (1967) 89–10 5. [Ol1] P .J. Olver, BiHamil tonian systems , in Or dinary and p artial diﬀer ential e quat ions , Pitman R esearc h N otes in Mathematics S eries 157 , (Longman Scien tiﬁc and T echnical, New Y ork 1987) 176– 193. [Ol2] P .J. Olver, Applic ations of Lie gr oups to diﬀer ential e quations , Second Edition, Graduate T exts in Mathematics 107 , (Sprin ger V erlag, New Y ork 1993) 513 pp . [PS] J. Praught, and G. Smirnov, Andr ew L enar d: a m ystery unr avele d , SIGMA Symme- try Integrabilit y Geom. Metho ds Appl. 1 , (2005) Paper 005, 7 pp. (electronic). [SK] M. Sato, M. Kashiw ara, The determinant of matric es of pseudo-diﬀer ential op er ators , Proc. Japan Acad. 51 , (1975) 17–19. [V ol] L.R. V olevich, On gener al systems of diﬀer ential e quations , Dokl. Ak ad. Nauk SSSR 132 (1960) 20–23 (Russian), translated as So viet Math. D okl. 1 , (1960) 458–4 61.

The variational Poisson cohomology

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment