Poisson vertex algebras in the theory of Hamiltonian equations

Transcription

1 Japan. J. Math. 4, (2009 DOI: /s y Posson vertex algebras n the theory of Hamltonan eqatons Alaa Barakat Alberto De Sole Vctor G. Kac Receved: 7 Jly 2009 / Revsed: 18 November 2009 / Accepted: 25 November 2009 Pblshed onlne: 25 December 2009 c The Mathematcal Socety of Japan and Sprnger 2009 Commncated by: Yasyk Kawahgash Abstract. We lay down the fondatons of the theory of Posson vertex algebras amed at ts applcatons to ntegrablty of Hamltonan partal dfferental eqatons. Sch an eqaton s called ntegrable f t can be nclded n an nfnte herarchy of compatble Hamltonan eqatons, whch admt an nfnte seqence of lnearly ndependent ntegrals of moton n nvolton. The constrcton of a herarchy and ts ntegrals of moton s acheved by makng se of the so called Lenard scheme. We fnd smple condtons whch garantee that the scheme prodces an nfnte seqence of closed 1-forms ω, Z +, of the varatonal complex Ω. If these forms are exact,.e., ω are varatonal dervatves of some local fnctonals h, then the latter are ntegrals of moton n nvolton of the herarchy formed by the correspondng Hamltonan vector felds. We show that the complex Ω s exact, provded that the algebra of fnctons V s normal ; n partclar, for arbtrary V, any closed form n Ω becomes exact f we add to V a fnte nmber of antdervatves. We demonstrate on the examples of the KdV, HD and CNW herarches how the Lenard scheme works. We also dscover a new ntegrable herarchy, whch we call the CNW herarchy of HD type. Developng the deas of Dorfman, we extend the Lenard scheme to arbtrary Drac strctres, and demonstrate ts applcablty on the examples of the NLS, pkdv and KN herarches. Keywords and phrases: evolton eqaton, evoltonary vector feld, local fnctonal, ntegral of moton, ntegrable herarchy, normal algebra of dfferental fnctons, Le conformal algebra, A. BARAKAT Department of Mathematcs, MIT, 77 Massachsetts Avene, Cambrdge, MA 02139, USA (e-mal: barakat@math.mt.ed A. DE SOLE Dpartmento d Matematca, Unverstà d Roma La Sapenza Cttà Unverstara, Roma, Italy (e-mal: desole@mat.nroma1.t V.G. KAC Department of Mathematcs, MIT, 77 Massachsetts Avene, Cambrdge, MA 02139, USA (e-mal: kac@math.mt.ed

2 142 A. Barakat, A. De Sole and V.G. Kac Posson vertex algebra, compatble λ-brackets, Lenard scheme, Beltram λ-bracket, varatonal dervatve, Fréchet dervatve, varatonal complex, Drac strctre, compatble Drac strctres Mathematcs Sbect Classfcaton (2000: 17B80 Contents 0. Introdcton Posson vertex algebras and Hamltonan operators Algebras of dfferental fnctons Normal algebras of dfferental fnctons Posson vertex algebras The Beltram λ-bracket Hamltonan operators and Hamltonan eqatons Center of the Le algebra V / V Posson vertex algebras and Le algebras Compatble PVA strctres and the Lenard scheme of ntegrablty Compatble PVA strctres The KdV herarchy The Harry Dym (HD herarchy The copled non-lnear wave (CNW system The CNW of HD type herarchy The varatonal complex g-complex and ts redcton The basc and the redced de Rham complexes Ω and Ω over V Cohomology of Ω and Ω The varatonal complex as a Le conformal algebra cohomology complex The varatonal complex as a complex of poly-λ-brackets or poly-dfferental operators Proecton operators Drac strctres and the correspondng Hamltonan eqatons Derved brackets and the Corant Dorfman prodct Defnton of a Drac strctre Hamltonan fnctonals and Hamltonan vector felds Pars of compatble Drac strctres Lenard scheme of ntegrablty for a par of Drac strctres The non-lnear Schrödnger (NLS herarchy The Hamltonan case The symplectc case The potental KdV herarchy The KN herarchy References...251

3 Posson vertex algebras n the theory of Hamltonan eqatons Introdcton An evolton eqaton s a system of partal dfferental eqatons of the form d (0.1 = P (,,,..., I = {1,...,l} (l may be nfnte, dt where = (t,x are fnctons on a 1-dmensonal manfold M, dependng on tme t and x M, andp are dfferentable fnctons n =( I and a fnte nmber of ts dervatves =( x I, =( 2 x 2 I,... One sally vews,,,... as generators of the algebra of polynomals (0.2 R = C[ (n I, n Z + ], (n f (n eqpped wth the dervaton,defnedby ( (n = (n+1, n Z +.Analgebra of dfferental fnctons V s an extenson of the algebra R, endowed wth commtng dervatons, I, n Z +, extendng the sal partal dervatves n R, and sch that, gven f V, = 0 for all bt fntely many I and n Z +.Then extends to a dervaton of V by (0.3 = (n+1 I, n Z + (n An element of V / V s called a local fnctonal, and the mage of f V n V / V s denoted by f, the reason beng that n the varatonal calcls local fnctonals are of the form M f (,,...dx, and takng V / V provdes the nversal space for whch ntegraton by parts holds. A local fnctonal f s called an ntegral of moton of (0.1, or s sad to be conserved by the evolton eqaton (0.1, f ts evolton n tme along (0.1 s constant, namely. (0.4 df dt = 0. The element f, defned modlo V, s then called a conserved densty. Bythe chan rle, eqaton (0.4 s eqvalent to (0.5 X P ( f =0, where, for P =(P I V l, X P s the followng dervaton of V : (0.6 X P = n Z + I ( n P (n.

4 144 A. Barakat, A. De Sole and V.G. Kac Note that [,X P ]=0. A dervaton of the algebra V wth ths property s called an evoltonary vector feld. It s easy to see that all of these are of the form (0.6 for some P V l. Here and frther V l denotes the space of all l 1 colmn vectors wth entres n V.Also,V l wll be sed to denote the sbspace of l 1 colmn vectors wth only fntely many non-zero entres. Thogh, n ths paper we do not consder any example wth nfnte l, we stll se ths dstncton as a book-keepng devce. We have the sal parng V l V l V,defnedbyF P = I F P.Note that ntegratng by parts transforms (0.5 to δ f P = 0, where δ f =(δ f I V l and (0.7 δ f = n Z + ( n f (n s the varatonal dervatve of f by. Gven a seqence of commtng evoltonary vector felds,.e., a seqence P n V l, n = 0,1,2,..., sch that the correspondng evoltonary vector felds commte (compatblty condton: (0.8 [X P m,x P n]=0, for all m,n Z +, one consders a herarchy of evolton eqatons d (0.9 = P n (,,,..., n Z +, dt n where d dt n =( d dt n I V l and = (t 1,t 2,... If the evoltonary vector felds X P n, n Z +, span an nfnte-dmensonal vector space, then each of the eqatons of the herarchy (0.9 (and the whole herarchy s called ntegrable. For example, the herarchy of lnear evolton eqatons d (0.10 = (n, I,n Z +, dt n s ntegrable. Note that the compatblty condton (0.8 can be eqvalently expressed by the followng denttes: dp m dt n = dpn dt m, m,n Z +. Note also that the problem of classfcaton of ntegrable evolton eqatons s eqvalent to the problem of classfcaton of maxmal nfnte-dmensonal abelan sbalgebras of the Le algebra of evoltonary vector felds.

5 Posson vertex algebras n the theory of Hamltonan eqatons 145 Remark 0.1. The above setp can be generalzed to the case of any fnte-dmensonal manfold M, replacng by partal dervatves = x and V / V by V / ( V. However, we shall be concerned only wth the 1-dmensonal case. A specal case of an evolton eqaton s a system of eqatons of the form d (0.11 = H( δh dt, where H( =(H (,,...;, I s an l l matrx, whose entres are fnte order dfferental operators n wth coeffcents n V,and h V / V s a δ local fnctonal (note that, snce, n vew of (0.3, = 0forall I, the rght hand sde s well-defned. In ths case one consders the local bracket [TF] (, I: (0.12 { (x, (y} = H ((y, (y,...; y δ(x y, x,y M, where δ(x y s the δ-fncton: M f (xδ(x ydx = f (y. Ths bracket extends, by the Lebnz rle and blnearty, to arbtrary f,g V : f (x g(y (0.13 { f (x,g(y} =, I (m (n x m y n { (x, (y}. m,n Z + Usng blnearty and ntegraton by parts, eqaton (0.13 sggests a way to defne a bracket on the space of local fnctonals V / V. Indeed we have { } δ f (x δg(y f (xdx, g(ydy = { (x, (y}dxdy M M, I M M δg(y = H ((y, (y,..., y δ f (y dy,, I M where, for the last dentty, we performed ntegraton by parts. Hence, gven two local fnctonals f, g V / V, we defne ther bracket assocated to the operator H(, as the local fnctonal (0.14 { f, } g =, I δg H (,,...; δ f = δg ( H( δ f. Lkewse, gven a local fnctonal f V / V and a fncton g V, we can defne ther bracket, whch wll now be an element of V, by the formla { } g (0.15 f,g =, I (n n H (,,...; δ f. n Z +

6 146 A. Barakat, A. De Sole and V.G. Kac Of corse, by ntegraton by parts, we get that the brackets (0.14 and (0.15 are compatble n the sense that { } { f,g = We can then rewrte the evolton eqaton (0.11, sng the above notaton, n the Hamltonan form: d { } (0.16 dt = h,. f, } g. Here and frther, for =( I V l, { h,} stands for ({ h, } I V l. The bracket (0.13 s called a Posson bracket f the bracket (0.14 on V / V satsfes the Le algebra axoms (ths s the basc ntegrablty condtons of the evolton eqaton (0.11. The skew-commtatvty of the bracket (0.14 smply means that the dfferental operator H( s skew-adont. The Jacob dentty s more nvolved, bt can be easly nderstood sng the langage of λ-brackets, whch, we beleve, greatly smplfes the theory. The dea s to apply the Forer transform F(x,y M dxeλ (x y F(x,y to both sdes of (0.13. Denotng { f (y λ g(y} = e λ (x y { f (x,g(y}dx, M a straghtforward calclaton gves the followng mportant formla [DK]: (0.17 { f λ g} =, I m,n Z + g (n ( + λ n { +λ } ( λ m f (m where { λ } = H (,,...;λ, and{ +λ } means that s moved to the rght. In terms of the λ-bracket (0.17, the bracket (0.14 on the space of local fnctonals V / V becomes (0.18 { f, } g = { f λ g} λ =0. Lkewse the bracket (0.15 between a local fnctonal f V / V and a fncton g V can be expressed as { } f,g = { f λ g} λ =0, so that the evolton eqaton (0.11 becomes, (0.19 d dt = {h λ} λ =0.

7 Posson vertex algebras n the theory of Hamltonan eqatons 147 It s easy to see that the λ-bracket (0.17 satsfes the followng sesqlnearty propertes (0.20 { f λ g} = λ{ f λ g}, { f λ g} =( + λ{ f λ g}, and the followng left and rght Lebnz rles: (0.21 { f λ gh} = { f λ g}h + g{ f λ h}, { fg λ h} = { f λ + h} g + {g λ + h} f. Frthermore, the skew-commtatvty of the bracket (0.14 s eqvalent to (0.22 { f λ g} = {g λ f }, where now s moved to the left, and the Jacob dentty s eqvalent to (0.23 { f λ {g μ h}} {g μ { f λ h}} = {{ f λ g} λ +μ h}. A commtatve assocatve ntal dfferental algebra V, endowed wth a λ- bracket V V C[λ] V, denoted by a b {a λ b}, s called a Posson vertex algebra (PVA f t satsfes all the denttes (0.20 (0.23. If the λ-bracket (0.17 defnes a PVA strctre on V, we say that H( =(H (,,...;, I n (0.12 s a Hamltonan operator, and that eqaton (0.11 (or, eqvalently, (0.19 s a system of Hamltonan eqatons assocated to the Hamltonan operator H(. It follows from (0.17 that { } h, = {h λ } λ =0 = X H( δh/ for h V / V. It s easy to check that, provded that Jacob dentty (0.23 holds, the map f XH( δ f / defnes a homorphsm of the Le algebra V / V of local fnctonals (wth the bracket (0.18 to the Le algebra of evoltonary vector felds. Recall that, by defnton, a local fnctonal f s an ntegral of moton of the Hamltonan eqaton (0.16 f (0.24 df dt = { h, } f = 0. By the above observaton ths mples that the correspondng evoltonary vector felds X H( δh/ and X H( δ f / commte: (0.25 [X H( δh/,x H( δ f / ]=0. In fact, we wll see n Secton 1.6, that n many cases ths s a necessary and sffcent condton for the commtatvty relaton (0.24. The basc problem n the theory of Hamltonan eqatons s to establsh ntegrablty. A system of Hamltonan eqatons d/dt={ h,}=h( δh/,

8 148 A. Barakat, A. De Sole and V.G. Kac s sad to be ntegrable f, frst, h les n an nfnte-dmensonal abelan sbalgebra of the Le algebra (V / V,{, }, n other words, f there exsts an nfnte seqence of lnearly ndependent local fnctonals h 0 = h, h 1, h 2,... V / V, commtng wth respect to the Le bracket (0.18 on V / V, and, second, f the evoltonary vector felds X H( δhn span an nfnte-dmensonal vector space. By the above observatons, f sch a seqence h n exsts, the correspondng Hamltonan eqatons d { = dt n } h n, = H( δh n, n Z +, form a herarchy of compatble Hamltonan eqatons. Ths, f the center of the Le algebra V / V wth bracket (0.18 s fnte-dmensonal, then we get a herarchy of ntegrable evolton eqatons (Note that the herarchy (0.10 of all lnear evolton eqatons s not Hamltonan, bt ts part wth n odd s. The problem of classfcaton of ntegrable Hamltonan eqatons conssts of two parts. The frst one s to classfy all λ-brackets on V, makng t a PVA. The second one s to classfy all maxmal nfnte-dmensonal abelan sbalgebras n the Le algebra V / V wth bracket (0.18. What apparently complcates the problem s the possblty of havng the same evolton eqaton (0.19 n two dfferent Hamltonan forms: (0.26 d { dt = } h 1λ = λ =0 0 { } h 0λ, λ =0 where { λ }, = 0,1, are two lnearly ndependent λ-brackets on V and h, = 0,1, are some local fnctonals. However, ths dsadvantage happens to be the man sorce of ntegrablty n the Hamltonan approach, as can be seen by tlzng the so called Lenard scheme [M], [O], [D]. Under some mld condtons, ths scheme works, provded that the two λ-brackets form a b- Hamltonan par, meanng that any ther lnear combnaton makes V apva. Namely, the scheme generates a seqence of local fnctonals h n, n Z +, extendng the gven frst two terms, sch that for each n Z + we have (0.27 { } h n+1λ = λ =0 0 { } h nλ 1 1. λ =0 In ths case all h n V / V, n Z +, parwse commte wth respect to both brackets {, }, = 0,1, on V / V, and hence they are ntegrals of moton n nvolton for the evolton eqaton (0.26. In Secton 2 we provde some condtons, whch garantee that the Lenard scheme works. As applcatons, we dscss n detal the example of the KdV and the dspersonless KdV herarches, based on the Gardner Faddeev Zakharov and the Vrasoro Magr PVA. In partclar, we prove that the polynomals n

9 Posson vertex algebras n the theory of Hamltonan eqatons 149 form a maxmal abelan sbspace of ntegrals of moton for the dspersonless KdV, and derve from ths that the nfnte-dmensonal space of ntegrals of moton for the KdV herarchy, obtaned va the Lenard scheme, s maxmal abelan as well. In fact, de to the presence of a parameter n the Vrasoro Magr λ-bracket (the central charge, we have a trple of compatble λ-brackets. Ths allows one to choose another b-hamltonan par, whch leads to the so called HD ntegrable herarchy, whch we dscss as well. We also dscss the example of the copled non-lnear wave (CNW system of Ito, based on the PVA correspondng to the Vrasoro Le algebra actng on fnctons on the crcle. Here agan, de to the presence of a parameter n the λ-brackets, we can make another choce of a b-hamltonan par, leadng s to a new herarchy, whch we call the CNW herarchy of HD type. Another mportant class of Hamltonan eqatons s provded by symplectc operators. Let S( =(S (,,...;, I be a matrx dfferental operator wth fntely many non-zero entres. The operator S( s called symplectc f t s skew-adont and t satsfes the followng analoge of the Jacob dentty (,,k I: (0.28 { λ S k (μ} B { μ S k (λ} B + {S (λ λ +μ k } B = 0. Here { λ } B s called the Beltram λ-bracket, whch we defne by { λ } = δ and extended to V V C[λ] V sng (0.20 and (0.21. In the symplectc case there s agan a Le algebra bracket, smlar to (0.18, bt t s defned only on the followng sbspace of V / V of Hamltonan fnctonals: { FS = f δ f S( V l}. It s gven by the followng formla (cf. (0.15: { } δg (0.29 f, g = S P, where δ f = S( P. Consder an evolton eqaton of the form (0.30 d dt = P, where P V l s sch that S( P = δh for some h FS. A local fnctonal f F S s called an ntegral of moton of the evolton eqaton (0.30 f { h, f } S = 0, and eqaton (0.30 s called ntegrable f t admts nfntely many lnearly ndependent commtng ntegrals of moton h n,n Z +,commtng wth h = h 0, and the correspondng evoltonary vector felds X P n commte and span an nfnte-dmensonal vector space. Ths, classfcaton of ntegrable Hamltonan eqatons assocated to the symplectc operator S( redces to the classfcaton of nfnte-dmensonal maxmal abelan sbspaces

10 150 A. Barakat, A. De Sole and V.G. Kac of the Le algebra FS. Based on the above defntons, the theory proceeds n the same way as n the Hamltonan case, ncldng the Lenard scheme. Followng Dorfman [D], we establs n Secton 4 ntegrablty of the potental KdV eqaton and the Krchever Novkov eqaton, sng ths scheme. In fact, there s a more general setp, n terms of Drac strctres, ntrodced by Dorfman [D], whch embraces both the Hamltonan and the symplectc setp. A Drac strctre nvolves two matrx dfferental operators H( = (H (,,...;, I and S( =(S (,,...;, I, and the correspondng Hamltonan eqatons are of the form d dt = P,whereP V l s sch that (0.31 S( P = H( δh, for some h FH,S, the correspondng Le algebra of Hamltonan fnctonals, defned by { (0.32 FH,S = f V / V wth the Le algebra bracket { } δg (0.33 f, g = S P, H( δ f S( V l}, f where H( δ = S( P. Under a stable ntegrablty condton, expressed n terms of the so-called Corant Dorfman prodct, (0.33 s well-defned and t satsfes the Le algebra axoms. We develop frther the theory of the Lenard scheme for Drac strctres, ntated by Dorfman [D], and, sng ths, complete the proof of ntegrablty of the non-lnear Schrödnger system, sketched n [D]. Applyng the Lenard scheme at the level of generalty of an arbtrary algebra of dfferental fnctons V reqres nderstandng of the exactness of the varatonal complex Ω(V = k Z + Ω k (V at k = 0 and 1. We prove n Secton 3 that, addng fntely many antdervatves to V, one can make any closed k- cocycle exact for any k 1, and, after addng a constant, for k = 0 (whch seems to be a new reslt. The contents of the paper s as follows. In Sectons 1.1 and 1.2 we ntrodce the noton of an algebra V of dfferental fnctons and ts normalty property. The man reslts here are Propostons 1.5 and 1.9. They provde for a normal V (n fact any V after addng a fnte nmber of antdervatves algorthms for comptng, for a gven f V sch that δ f = 0, an element g V sch that f = g + const., and for a gven F V l whch s closed,.e., sch that D F ( =D F (,whered F( s the Fréchet dervatve (defned n Secton 1.2, an element f V sch that δ f = F. These reslts are mportant for the proof of ntegrablty of Hamltonan eqatons, dscssed n Sectons 2 and 4.

11 Posson vertex algebras n the theory of Hamltonan eqatons 151 In Sectons 1.3 and 1.7 we gve several eqvalent defntons of a Posson vertex algebra (PVA and we explan how to ntrodce a strctre of a PVA n an algebra V of dfferental fnctons or ts qotent (Theorems 1.15 and 1.21, Propostons 1.16 and In partclar, we show that an eqvalent noton s that of a Hamltonan operator. In Secton 1.5 we show how to constrct Hamltonan eqatons and ther ntegrals of moton n the PVA langage. In Secton 1.6 we explan how to compte the center of a PVA. In Secton 1.4 we ntrodce the Beltram λ-bracket (whch, nlke the Posson λ-bracket, s commtatve, rather than skew-commtatve, and nterpret the basc operators of the varatonal calcls, lke the varatonal dervatve and the Fréchet dervatve, n terms of ths λ-bracket. Other applcatons of the Beltram λ-bracket appear n Secton 3. In Secton 2.1 we ntrodce the noton of compatble PVA strctres on an algebra V of dfferental fnctons and develop n ths langage the well-known Lenard scheme of ntegrablty of Hamltonan evolton eqatons [M], [O], [D]. The new reslts here are Propostons 2.9, 2.10 and 2.13, and Corollary 2.12, whch provde sffcent condtons for the Lenard scheme to work. In Secton 2.2, sng the Lenard scheme, we dscss n detal the ntegrablty of the KdV herarchy, whch ncldes the classcal KdV eqaton d dt = 3 + c, c C. The new reslt here s Theorem 2.15 on the maxmalty of the seqence of ntegrals of moton of the KdV herarchy. We dscss the ntegrablty of the HD herarchy, whch ncldes the Harry Dm eqaton d dt = 3 ( 1/2 +α ( 1/2, α C, n Secton 2.3, and that of the CNW herarchy, whch ncldes the CNW system of Ito (c C: d dt = c vv, dv dt = (v n Secton 2.4. In Secton 2.5 we prove ntegrablty of the CNW herarchy of HD type, whch seems to be new. The smplest non-trval eqaton of ths herarchy s (c C: d ( 1 dt = v ( + c 3 d 1 v dv (. dt = v 2 In Sectons 3.1 and 3.2 we ntrodce, followng Gelfand and Dorfman [GD2], the varatonal complex Ω(V as the redcton of the de Rhamg-complex Ω(V

12 152 A. Barakat, A. De Sole and V.G. Kac by the acton of,.e., Ω(V = Ω(V / Ω(V. Or man new reslt on exactness of the complexes Ω(V and Ω(V provded that V s normal, s Theorem 3.2, proved n Secton 3.3. Another verson of the exactness theorem, whch s Theorem 3.5, s well-known (see e.g. [D], [D]. However t s not always applcable, whereas Theorem 3.2 works for any V, provded that we add to V a fnte nmber of antdervatves. In Sectons 3.4 and 3.5, followng [DSK], we lnk the varatonal complex Ω(V to the Le conformal algebra cohomology, developed n [BKV], va the Beltram λ-bracket, whch leads to an explct constrcton of Ω(V. Asan applcaton, we gve a classfcaton of symplectc dfferental operators and fnd smple formlas for the well-known Sokolov and Dorfman symplectc operators [S], [D], [W]. In Secton 3.6 we explore an alternatve way of nderstandng the varatonal complex, va the proecton operators Pk of Ω k onto a sbspace complementary to Ω k. In Sectons we defne the Drac strctres and the correspondng evolton eqatons. In Sectons 4.4 and 4.5 we ntrodce the noton of compatble Drac strctres and dscss the correspondng Lenard scheme of ntegrablty. The exposton of these sectons follows closely the book of Dorfman [D], except for Proposton 4.16 and Corollary 4.17, garanteeng that the Lenard scheme works, whch seem to be new reslts. In Secton 4.6 we prove ntegrablty of the non-lnear Schrödnger (NLS system d dt = v + 2v( 2 + v 2, dv dt = 2( 2 + v 2 sng a compatble par of Drac strctres and the reslts of Secton 4.5. In Secton 4.7 we prove that the graph of a Hamltonan dfferental operator s a Drac strctre (see Theorem 4.21, and a smlar reslt on the b- Hamltonan strctre vs. a par of compatble Drac strctres (see Proposton 4.24, whch allows s to derve the key property of a b-hamltonan strctre, stated n Theorem 2.7 n Secton 2.1. Lkewse, n Secton 4.8 we relate the symplectc dfferental operators to Drac strctres and prove Dorfman s crtera of compatblty of a par of symplectc operators [D]. In Sectons 4.9 and 4.10 we derve the ntegrablty of the potental KdV eqaton (c C d dt = 3( 2 + c, and of the Krchver Nvkov eqaton d dt = 3 ( 2 2.

13 Posson vertex algebras n the theory of Hamltonan eqatons 153 The latter ses the compatblty of the Sokolov and Dorfman symplectc operators. In ths paper we are consderng only the translaton nvarant case, when fnctons f V do not depend explctly on x. The general case amonts to addng x n the defnton (0.3 of. Many mportant ntegrable herarches, lke the KP, the Toda lattce, and the Drnfeld Sokolov, are not treated n ths paper. We are plannng to fll n these gaps n ftre pblcatons. 1. Posson vertex algebras and Hamltonan operators 1.1. Algebras of dfferental fnctons By a dfferental algebra we shall mean a ntal commtatve assocatve algebra over C wth a dervaton. Recall that 1 = 0. One of the most mportant examples s the algebra of dfferental polynomals n l varables R = C[ (n {1,...,l} = I, n Z + ], where s the dervaton of the algebra R, defnedby ( (n = (n+1. Defnton 1.1. An algebra of dfferental fnctonsv n a set of varables { } I s an extenson of the algebra of polynomals R = C[ (n I, n Z + ], endowed wth lnear maps : (n V V, for all I and n Z +, whch are commtng dervatons of the prodct n V, extendng the sal partal dervatves n R, and sch that, gven f V, f = 0 for all bt fntely many I and n Z (n +. Unless otherwse specfed, we shall assme that V s an ntegral doman. Typcal examples of algebras of dfferental fnctons that we wll consder are: the algebra of dfferental polynomals tself, R = C[ (n I, n Z + ],anylocalzaton of t by some element f R, or, more generally, by some mltplcatve sbset S R, sch as the whole feld of fractons Q = C( (n I, n Z +, or any algebrac extenson of the algebra R or of the feld Q obtaned by addng a solton of certan polynomal eqaton. An example of the latter type, whch we wll consder n Secton 2.3, s V = C[ ±1,,,...], obtaned by startng from the algebra of dfferental polynomals R = C[ (n n Z + ], addng the sqare root of the element, and localzng by. Onany algebra ofdfferental fnctons V we have a well-defned dervaton : V V, extendng the sal dervaton of R (defned above, gven by (1.1 = (n+1 I, n Z + (n.

14 154 A. Barakat, A. De Sole and V.G. Kac Moreover we have the sal commtaton rle [ ] (1.2 (n, = (n 1, where the rght hand sde s consdered to be zero f n = 0. These commtaton relatons mply the followng lemma, whch wll be sed throghot the paper. Lemma 1.2. Let D (z = n Z+ z n. Then for every h(λ = N (n m=0 h mλ m C[λ] V and f V the followng dentty holds: (1.3 D (z(h( f =(D (z(h( f + h(z + (D (z f, where D (z(h( s the dfferental operator obtaned by applyng D (z to the coeffcents of h(. Proof. In the case h(λ=λ, eqaton (1.3 means (1.2. It follows that D (z n =(z + n D (z for every n Z +. The general case follows. As before, we denote by V r V r the sbspace of all F =(F I wth fntely many non-zero entres (r may be nfnte, and ntrodce a parng V r V r V / V (1.4 (P, F P F, where, as before, denotes the canoncal map V V / V. Recall the defnton (0.7 of the varatonal dervatve δ follows mmedately from (1.2 that : V V l.it (1.5 δ = 0, I,.e., V Ker δ. In fact, we are gong to prove that, nder some assmptons on V, apart from constant fnctons, there s nothng else n Ker δ Frst, we make some prelmnary observatons. Gven f V, we say that t has dfferental order n, and we wrte ord( f =n, f 0forsome I and f (m f (n = 0forall I and m > n. In other words, the space of fnctons of dfferental order at most n s { (1.6 Vn := f V f (m } = 0 for all I, m > n..

15 Posson vertex algebras n the theory of Hamltonan eqatons 155 Clearly ord( f =n f and only f f Vn\Vn 1. We also denote by C V f the space of constant fnctons f, namely sch that = 0forall I and (m m Z +, or, eqvalently, sch that f = 0 (ths eqvalence s mmedate by (1.1. We wll also let C ( V0 the sbspace of fnctons f dependng only f on, namely sch that = 0 nless = and n = 0. (n We can refne fltraton (1.6 as follows. For I and n Z + we let { f } (1.7 Vn, := f Vn (n = 0 forall > Vn. In partclar, V n,l = Vn. WealsoletVn,0 = Vn 1 for n 1, and V0,0 = C. It s clear by the defnton (1.1 of that f f = g 0andg Vn,\Vn, 1, then f Vn+1,\Vn+1, 1, and n fact f has the form (1.8 f = h (n+1 + > h (n + r, where h Vn, for all I, r Vn,, andh 0. In partclar, t mmedately follows that (1.9 C V = 0. Proposton 1.3. (a The parng (1.4 s non-degenerate, namely P F = 0 for every F V r f and only f P = 0. (b Moreover, let I be an deal of V r contanng a non-zero element F whch s not a zero dvsor. If P F = 0 for every F n the deal I, then P = 0. Proof. We can assme, wthot loss of generalty, that r = 1. Sppose that P 0 s sch that P F = 0foreveryF V. In ths case, lettng F = 1, we have that P V has the form (1.8. Bt then t s easy to see that (n+1 P does not have ths form, so that (n+1 P 0. Ths proves (a. The argment for (b s the same, by replacng P by P F. We let g be the Le algebra of all dervatons of V of the form (1.10 X = I, n Z + h,n (n, h,n V. By (1.1, s an element of g, and we denote by g the centralzer of n g. ElementsX g are called evoltonary vector felds. ForX g we have X( (n =X( n = n X(,sothatX = I, n Z+ X( (n s completely (n

16 156 A. Barakat, A. De Sole and V.G. Kac determned by ts vales X( =P, I. We ths have a vector space somorphsm V l g,gvenby (1.11 V l P =(P I X P = ( n P I, n Z + (n g. The l-tple P s called the characterstc of the evoltonary vector feld X P Normal algebras of dfferental fnctons Defnton 1.4. We call an algebra of dfferental fnctons normal f (Vn,=Vn, for all I, n Z +. Gven f Vn,, wedenoteby d (n f (n Vn, a premage of f nder the map addng elements from Vn, 1. (n. Ths antdervatve s defned p to Clearly any algebra of dfferental fnctons can be embedded n a normal one. We wll dscss n Secton 3 the exactness of the varatonal complex Ω. In ths langage, Propostons 1.5 and 1.9 below provde algorthms to fnd, for a closed element n V, a premage of (p to addng a constant, and, for a closed element n Ω 1, a premage of δ. Proposton 1.5. Let V be a normal algebra of dfferental fnctons. Then (1.12 Ker δ = C V. In fact, gven f Vn, sch that δ f g n 1, = = 0, we have f (n d (n 1 f (n Vn 1,, Vn 1,, and lettng we have f g n 1, Vn, 1. Ths allows one to compte an element g V sch that f = g + const. by ndcton on the par (n, n the lexcographc order. Proof. Eqalty (1.12 s a specal case of exactness of the varatonal complex, whch wll be establshed n Theorem 3.2. Hence, f f Ker δ V n,, wehave f (n f = g + c for some g Vn 1, and c C. By eqaton (1.8, Vn 1, for every I. The rest mmedately follows by the defnton (1.1 of. Example 1.6. In general (1.12 does not hold. For example, n the non-normal algebra V = C[ ±1,,,...] the element δ s n Ker, bt not n V.Btn a normal extenson of V ths element can be wrtten as log.

17 Posson vertex algebras n the theory of Hamltonan eqatons 157 In order to state the next proposton we need to ntrodce the Fréchet dervatve. Defnton 1.7. The Fréchet dervatve D f of f V s defned as the followng dfferental operator from V l to V : D f ( P = X P ( f = I, n Z + f (n n P. One can thnk of D f ( P as the dfferental of the fncton f ( n the sal sense, that s, p to hgher order n P, f ( f ( + P f ( I, n Z + (n P (n = D f ( P, hence the name Fréchet dervatve. More generally, for any collecton F =(f α α A of elements of V (where A s an ndex set, the correspondng Fréchet dervatve s the map D F : V l V A gven by (1.13 (D F (P α = D fα ( P(= X P ( f α, α A. Its adont wth respect to the parng (1.4 s the lnear map D A F ( : V V l,gvenby ( (1.14 (D F( G = ( n Fα α A, n Z + (n G α, I. On the other hand, f F has only fntely many non-zero entres, namely F V A V A, t s clear from the defnton (1.14 that D F can be extended to a map V A V l. In partclar, for F V l, both D F ( and D F ( are maps from V l to V l. The followng lemma wll be sefl n comptatons wth Fréchet dervatves. Lemma 1.8. (a Both the Fréchet dervatve D F ( and the adont operator D F ( are lnear n F V A. (b Let M be an A B matrx wth entres n V. We have, for F V B, G V A and P V l, and D MF( G = D MF ( P = F β (D Mαβ ( P+ M αβ (D Fβ ( P β B β B =(D M ( P F + M (D F ( P, α A, β B D M αβ ( (F β G α + α A, β B = D M( (G T F+D F( (G T M. D F β ( (M αβ G α

18 158 A. Barakat, A. De Sole and V.G. Kac (c Let F V A, G V A D F ( P = (D F ( P, and P V l. We have D F ( G = D F( ( G. Proof. Part (a s obvos. For part (b the frst eqaton follows mmedately from the defnton (1.13 of the Fréchet dervatve, whle the second eqaton can be derved takng the adont operator. Part (c can be easly proved sng eqaton (1.2. We have the followng formla for the commtator of evoltonary vector felds n terms of Fréchet dervatves: (1.15 [X P,X Q ]=X DQ ( P D P ( Q. Elements of the form δ f V l are called exact. AnelementF V l s called closed f ts Fréchet dervatve s a self-adont dfferental operator: (1.16 D F( =D F (. It s well-known and not hard to check, applyng Lemma 1.2 twce, that any exact element n V l s closed: (1.17 D δ f ( =Dδf( for every f V / V. If V s normal, the converse holds. In fact, the next proposton provdes an algorthm for constrctng f V sch that δ f = F, provded that F s closed, by ndcton on the followng fltraton of V l. Gven a trple (n,, wth n Z + and, I, the sbspace (V l n,, V l conssts of elements F V l sch that F k Vn, for every k I and s the maxmal ndex sch that F Vn,\Vn, 1.Wealsolet(V l n,0, =(V l n 1,l, and (V l n,,0 =(V l n, 1,l. Proposton 1.9. Let V be a normal algebra of dfferental fnctons. Then F V l s closed f and only f t s exact,.e., F = δ f for some f V. In fact, f F (V l n,, s closed, then there are two cases: (a f n s even, we have and F (n (1.18 f n,, =( 1 n/2 d (n/2 V n/2,, and lettng d (n/2 F (n V n/2,, we have F δ f n,, (V l n,, 1 ; (b f n s odd, we have < and F (n V (n 1/2,, and lettng (1.19 f n,, =( 1 (n+1/2 we have F δ f n,, (V l n,, 1. d ((n+1/2 d ((n 1/2 F (n V (n+1/2,,

19 Posson vertex algebras n the theory of Hamltonan eqatons 159 (Note that f n,, n (1.18 s defned p to addng an arbtrary element g V n/2, sch that V n/2, 1, and smlarly for f n,, n (1.19. Eqatons (1.18 g (n/2 and (1.19 allow one to compte an element f V sch that F = δ f by ndcton on the trple (n,, n the lexcographc order. Proof. The frst statement of the proposton s a specal case of the exactness of the complex of varatonal calcls, whch wll be establshed n Theorem 3.2. Hence there exsts f V, defned p to addng elements from V, sch that F = δ f. Lemma (a Up to addng elements from V, f can be chosen sch that f Vm,k\Vm,k 1 and Vm,k\Vm 1,k for some m Z + and k I. f (m k (b If f s chosen as n part (a, let (n,, be the maxmal trple (wth respect to the lexcographc order sch that ( f (n p (p 0 for some p Z +. Then, there are two possbltes: A. n = 2m, k; B. n = 2m 1, > = k. In both cases the only p Z + for whch (1.20 holds s p = m. (c δ f (V l n,, \(V l n,, 1,where(n,, s as n (b. Proof. For (a t sffces to notce that, f f V m,k \V m,k 1 s sch that Vm 1,k, then f d (m 1 k f (m k Vm,k 1, f (m k and after fntely many sch steps we arrve at the f that we want. Let (n,, be as n (b. Notce that, snce condton (1.20 s symmetrc n and, we necessarly have. Snce f V m,k, t s clear that (n,, (2m,k,k. Moreover, snce, by (a, f (m k Vm,k\Vm 1,k, t easly follows that (n,, (2m 1,k + 1,k n the lexcographc order. These two neqaltes mmedately mply that ether A. or B. occrs. In order to prove (c we consder the two cases A. and B. separately. In case A. condton (1.20 exactly means that: f (m h f (m Vm, 1 for every h >, sothat δ f h V2m, 1; Vm,\Vm, 1,sothat δ f V2m,\V2m, 1;

20 160 A. Barakat, A. De Sole and V.G. Kac 3. f (m h Vm, for every h <, sothat δ f h V2m,. By defnton, the above condtons mply that δ f (V l 2m,,. Case B. s treated n a smlar way. Retrnng to the proof of Proposton 1.9, let F = δ f,where f V s chosen as n Lemma 1.10 (a. We then have F (n = (n = (n δ f ( ( m f (m +( m 1 f (m 1 + =( 1 m 2 f (n m (m where, for the last eqalty, we sed the commtaton relaton (1.2 and Lemma 1.10 (b (c. The rest follows easly. In Secton 3 we shall prove exactness of the whole complex Ω n two dfferent contexts. Frst, nder the assmpton that V s normal,and second, nder an assmpton nvolvng the degree evoltonary vector feld: (n (1.21 Δ = 1 l (n, n Z + for whch we are gong to need the followng commtaton rles: (1.22 Δ = Δ, (n Δ =(Δ + 1 (n, δ Δ =(Δ + 1 δ. Usng ths, one fnds a smpler formla (f applcable for the premages of the varatonal dervatve, gven by the followng well-known reslt. Proposton Let V be an algebra of dfferental fnctons and sppose that F V l s closed, namely (1.16 holds. Then, f f Δ 1 ( F, we have δ f (1.23 F Ker(Δ + 1, for all I, where F = I F. Proof. We have, for I, δ ( F = I, n Z + ( n = F + I, n Z + (n F (n ( ( F =F + ( n F I, n Z + n =(Δ + 1F. (n,

21 Posson vertex algebras n the theory of Hamltonan eqatons 161 In the thrd eqalty we sed the assmpton that F s closed,.e., t satsfes (1.16, whle n the last eqalty we sed defnton (1.21 of the degree evoltonary vector feld. If we then sbsttte F = Δ f n the left hand sde and se the last eqaton n (1.22, we mmedately get (1.23. Example Consder the closed element F =( 3, 2, 1 R3 2,2,2,where R s the algebra of dfferental polynomals n 1, 2, 3.SnceKer(Δ + 1=0 n R, we can apply Proposton 1.11 to fnd that f = Δ 1 ( = 1 2 ( satsfes F = δ f. On the other hand, consder the localzaton V by 4 of the algebra of dfferental polynomals n 1, 2, 3, 4 and the followng closed element F (V 4 2,2,2 : F 1 = 2 4 3, F 2 = , F 3 = , F 4 = ( 2 2. In ths case Δ( F=0, hence formla (1.23 s not applcable, bt we stll can se the algorthm provded by Proposton 1.9, to fnd f = f 2,2,2 + f 1,4,3,where f 2,2,2 = ( 2 2 and f 1,4,3 = Posson vertex algebras Defnton Let V be a C[ ]-modle. A λ-bracket on V s a C-lnear map V V C[λ] V, denoted by f g {f λ g}, whchssesqlnear, namely one has ( f,g,h V (1.24 { f λ g} = λ{ f λ g}, { f λ g} =( + λ{ f λ g}. If, moreover, V s a commtatve assocatve ntal dfferental algebra wth a dervaton,aλ-bracket on V s defned to obey, n addton, the left Lebnz rle (1.25 { f λ gh} = { f λ g}h + { f λ h}g, and the rght Lebnz rle (1.26 { fg λ h} = { f λ + h} g + {g λ + h} f. λ One wrtes { f λ g} = n n Z+ n! ( f (ng, where the C-blnear prodcts f (n g are called the n-th prodcts on V (n Z +, and f (n g = 0fornsffcently large. In (1.26 and frther on, the arrow on the rght means that λ + shold be moved (λ + n to the rght (for example, { f λ + h} g = n Z+ ( f (n h n! g. An mportant property of a λ-bracket { λ } s commtatvty (resp. skewcommtatvty: (1.27 {g λ f } = { f λ g} (resp. = { f λ g}.

22 162 A. Barakat, A. De Sole and V.G. Kac Here and frther the arrow on the left means that λ s moved to the left, ( λ.e., { f λ g} = n n Z+ n! ( f (n g=e d dλ { f λ g}. In case there s no arrow, t s assmed to be to the left. Another mportant property of a λ-bracket { λ } s the Jacob dentty: (1.28 { f λ {g μ h}} {g μ { f λ h}} = {{ f λ g} λ +μ h}. Defnton A Posson vertex algebra (PVA s a dfferental algebra V wth a λ-bracket { λ } : V V C[λ] V (cf. Defnton 1.13 satsfyng skewcommtatvty (1.27 and Jacob dentty (1.28. ALe conformal algebra s a C[ ]-modle,endowedwth aλ-bracket, satsfyng the same two propertes [K]. We next want to explan how to extend an arbtrary non-lnear λ-bracket on a set of varables { } I wth vale n some algebra V of dfferental fnctons, to a Posson vertex algebra strctre on V. Theorem Let V be an algebra of dfferental fnctons, whch s an extenson of the algebra of dfferental polynomals R = C[ (n I, n Z + ].For each par, I, choose { λ } C[λ] V. (a Formla (1.29 { f λ g} =, I m,n Z + g (n (λ + n { λ + } ( λ m f (m defnes a λ-bracket on V (cf. Defnton 1.13, whch extends the gven λ- brackets on the generators, I. (b The λ-bracket (1.29 on V satsfes the commtatvty (resp. skew-commtatvty condton (1.27, provded that the same holds on generators: (1.30 { λ } = ± { λ }, for all, I. (c Assmng that the skew-commtatvty condton (1.30 holds, the λ-bracket (1.29 satsfes the Jacob dentty (1.28 (ths makng V a PVA, provded that the Jacob dentty holds on any trple of generators: (1.31 { λ { μ k}} { μ { λ k }} = {{ λ } λ +μ k }, for all,,k I. Proof. Frst, notce that the sm n eqaton (1.29 s fnte (cf. Defnton 1.1, so that (1.29 gves a well-defned C-lnear map { λ } : V V C[λ] V. Moreover, for f =, g =,, I, sch map clearly redces to the gven polynomals { λ } C[λ] V. More generally, for f = (m, g = (n, eqaton (1.29 redces to (1.32 { (m λ (n } =( λ m (λ + n { λ }..

23 Posson vertex algebras n the theory of Hamltonan eqatons 163 It s also sefl to rewrte eqaton (1.29 n the followng eqvalent forms, whch can be checked drectly: (1.33 { f λ g}= I, n Z + g (n or, sng (1.32, n the followng frther form: (1.34 { f λ g} = ( e d dλ, I m,n Z + (λ + n { f λ }= I, m Z + { λ + g} ( λ m f (m { (m λ (n } g (n where the parentheses ndcate that n the exponent s actng only on f d dλ, (m f (m.,bt acts on λ. Here and frther we se the followng dentty for f (λ V [λ], comng from Taylor s formla: f ( + λ =(e d dλ f (λ. We start by provng the sesqlnearty relatons (1.24. For the frst one we have, by the second dentty n (1.33 and the commtatvty relaton (1.2: ( { f λ g} = { λ + g} ( λ m I, m Z + = I, m Z + { λ + g} ( λ m (m ( f = λ I, m Z + { λ + g} ( λ m f (m 1 (m 1 f (m f + f (m = λ{ f λ g}, where, as n (1.2, we replace by zero for m = 0. Smlarly, for the frst sesqlnearty condton, we can se the frst dentty n (1.33 and (1.2 to get: ( { f λ g} = I, n Z + (n ( g = I, n Z + =(λ + (n 1 I, n Z + g (λ + n { f λ } + g (n (λ + n { f λ } g (n (λ + n { f λ } =(λ + { f λ g}. In order to complete the proof of part (a we are left to prove the left and rght Lebnz rles (1.25 and (1.26. For both we se that the partal dervatves (m

24 164 A. Barakat, A. De Sole and V.G. Kac are dervatons of the prodct n V. For the left Lebnz rle, we se the frst dentty n (1.33 to get: { f λ gh} = ( I,n Z + h g (n + g h (n (λ + n { f λ } = h{ f λ g} + g{ f λ h}, and smlarly, for the rght Lebnz rle, we can se the second dentty n (1.33 to get: ( f { fg λ h} = { λ + h} ( λ m I,m Z + (m g + = { f λ + h} g + {g λ + h} f. g (m We next prove part (b. Wth the notaton ntrodced n eqaton (1.27 we have, by (1.29: (1.35 {g λ f } = e d dλ {g λ f } = e d dλ, I m,n Z + f (m ( λ + m { λ + } (λ n f g (n It mmedately follows from (1.30 that { λ + } F = ±e d dλ ({ λ }F, for arbtrary F V. Eqaton (1.35 then gves {g λ f }=±, I m,n Z + g (n (λ + n { λ+ } ( λ m f (m. =±{ f λ g}, ths provng (skew-commtatvty. We are left to prove part (c. Frst, observe that, snce, by (1.31, Jacob dentty holds on any trple of generators,, k, applyng ( λ m ( μ n (λ + μ + p to both sdes of (1.31 and sng sesqlnearty, Jacob dentty holds on any trple of elements of type (m for I and m Z + : (1.36 { (m λ { (n μ (p k }} { (n μ{ (m λ (p k }} = {{ (m λ (n } λ +μ (p k }. We have to prove that for every f,g,h V Jacob dentty (1.28 holds as well. We wll stdy separately each term n eqaton (1.28. For the frst term n the left hand sde of (1.28 we have, by the frst dentty n (1.33, combned wth sesqlnearty (1.24 and the left Lebnz rle (1.25: { f λ {g μ h}} = { k I, p Z + f λ h (p k } {g μ (p k }

25 Posson vertex algebras n the theory of Hamltonan eqatons 165 (1.37 = { k I,p Z + f λ h (p k } {g μ (p k } + k I,p Z + h (p k { f λ {g μ (p k }} The fst term n the rght hand sde of (1.37 can be rewrtten, sng agan the frst dentty n (1.33, as (1.38 k,l I p,q Z + (q l 2 h (p k { f λ (q l }{g μ (p k }. Notce that ths expresson s nchanged f we swtch f wth g and λ wth μ, therefore t does not gve any contrbton to the left hand sde of (1.28. For the second term n the rght hand sde of (1.37, we apply twce the second dentty n (1.33, combned wth sesqlnearty (1.24 and the left Lebnz rle (1.25, to get: (1.39 (1.40 k I,p Z + =,,k I m,n,p Z + =,,k I m,n,p Z + +,,k I m,n,p Z + h (p k h (p k h (p k,,k I m,n,p Z + h (p k { f λ {g μ (p k }} ( e d dλ ( e d dλ (p k ( e d dλ f (m f (m f (m { (m λ (e dμ d (e d dμ g (n g (n { (m { (m λ (e dμ d g (n } { (n μ (p k } λ { (n μ (p k }} } { (n μ (p k }. Frthermore, f we se agan both denttes (1.33 and sesqlnearty (1.24, we get the followng dentty for the last term n the rght hand sde of (1.39, h ( e d f { dλ (e dμ d g } (m { = { (n λ +μ+ h} f λ I,n Z + (m λ g (n (n { (n μ (p The second term n the left hand sde of (1.28 s the same as the frst term, after exchangng f wth g and λ wth μ. Therefore, combnng the reslts from eqatons (1.37 (1.40, we get that the left hand sde of (1.28 s }. k }

26 166 A. Barakat, A. De Sole and V.G. Kac (1.41 { f λ {g μ h}} {g μ { f λ h}} h ( =,,k I (p e d f dλ k (m m,n,p Z + { (n μ{ (m λ (p k }}+ I, m Z + { (m λ +μ+ h} { (e d dμ g (n I, n Z + { (n g μ f (m ({ (m λ +μ+ h} { }. λ { (n μ (p k }} f λ g } (n We fnally se (1.36, and then the frst eqaton n (1.33, to rewrte the frst term n the rght hand sde of (1.41 to get: (1.42 { f λ {g μ h}} {g μ { f λ h}} ( = e d f dλ (e d g, I (m dμ (n m,n Z + { + { (n λ +μ+ h} f λ I, n Z + { λ +μ+ h} g μ I, m Z + { (m {{ (m λ (n } λ +μ h} g } (n f (m We next look at the rght hand sde of eqaton (1.28 wth ν n place of λ + μ. By the frst dentty n (1.33, combned wth sesqlnearty (1.24, and sng the rght Lebnz rle (1.26, we get: (1.43 {{ f λ g} ν h} = I,n Z + I, n Z + = I, n Z + + I, n Z + { g (n { g (n ( e dν d }. } { f λ (n } ν h } ν+ h { f λ (n } g (n {{ f λ (n } ν h}. We then expand { f λ (n } sng the second dentty n (1.33 combned wth sesqlnearty (1.24, and then apply the rght Lebnz rle (1.26, to rewrte the last term n the rght hand sde of (1.43 as ( e dν d g {{ f λ (n } ν h} =, I m,n Z + ( e d dν (n g (n {( e d dλ f (m { (m } λ (n } ν h

27 Posson vertex algebras n the theory of Hamltonan eqatons 167 (1.44 = ( e dν d, I m,n Z + +, I m,n Z + ( e d dν g (n g (n ( e d dν e d dλ {( e d dλ f (m f (m {{ (m λ (n } ν h} } ν+ h {(m λ (n }. Frthermore, the last term n the rght hand sde of eqaton (1.44 can be rewrtten, sng eqaton (1.33 combned wth sesqlnearty (1.24, as (1.45, I m,n Z + =, I m,n Z + = I, n Z + = I, n Z + ( e d dν = I,n Z + {( e d dλ g (n {( e d dλ { f (m { f f (m (m {( e d dλ f (m ν+ h f (m } ν+ h g (n } ν+ h { (m λ (n } ν+ h } { (m λ g} } ν+ h { (m λ ν g} } {g ν λ (m }. {(m λ (n } For the last eqalty, we sed that, by part (b, the λ-bracket (1.29 satsfes the skew-commtatvty (1.27. We can then pt together eqatons (1.43 (1.45 wth ν = λ + μ, to get the followng expresson for the rght hand sde of eqaton (1.28, (1.46 {{ f λ g} λ +μ h} =, I m,n Z + (e d dμ g (n { g + I, n Z + (n { f I, n Z + (m ( e d dλ f (m } λ +μ+ h { f λ (n } } λ +μ+ h {g μ (m }. {{ (m λ (n } λ +μ h} To conclde, we notce that the frst term n the rght hand sde of (1.42 concdes wth the frst term n the rght hand sde of (1.46. Moreover, t s not hard to check, sng (1.33 and the fact that and commte, that the second (m (n

28 168 A. Barakat, A. De Sole and V.G. Kac and thrd terms n the rght hand sde of (1.42 concde, respectvely, wth the second and thrd terms n the rght hand sde of (1.46. Theorem 1.15 (a says that, n order to defne a λ-bracket on an algebra of dfferental fnctons V extendng R = C[ (n I, n Z + ], one only needs to defne for any par, I the λ-bracket (1.47 { λ } = H (λ V [λ]. In partclar λ-brackets on V are n one-to-one correspondence wth l l- matrces H(λ =(H (λ, I wth entres H (λ = N n=0 H ;nλ n n V [λ], or, eqvalently, wth the correspondngmatrx valed dfferental operators H( = (H (, I : V l V l. We shall denote by { λ } H the λ-bracket on V correspondng to the operator H( va eqaton (1.47. Proposton Let H( =(H (, I be an l l matrx valed dfferental operator. (a The λ-bracket { λ } H satsfes the (skew-commtatvty condton (1.27 f and only f the dfferental operator H( s self (skew-adont, meanng that (1.48 H ( := N n=0 ( n H ;n = ±H (. (b If H( s skew-adont, the followng condtons are eqvalent: ( the λ-bracket { λ } H defnes a PVA strctre on V, ( the followng dentty holds for every,,k I: (1.49 ( Hk (μ (n (λ + n H h (λ H k(λ h (n h H kh (λ + μ + ( λ μ n H (λ h I, n Z + h I,n Z + = (n h (μ + n H h (μ ( the followng dentty holds for every F,G V l : (1.50 H( D G ( H( F + H( D H( F ( G H( D F( H( G +H( D F ( H( G = D H( G( H( F D H( F ( H( G. Proof. Eqaton (1.30 s the same as eqaton (1.48, f we se the dentfcaton (1.47. Hence part (a follows from Theorem 1.15 (b. Next, eqaton (1.31 s easly translated, sng the dentfcaton (1.47 and formla (1.29, nto eqaton (1.49. Hence, the eqvalence of ( and ( follows mmedately from Theorem 1.15 (c. We are ths left to prove that, f H( s skew-adont, then condtons ( and ( are eqvalent. Wrtten ot explctly, sng formlas for the Fréchet dervatves n Secton 1.2, eqaton (1.50 becomes (k I:,

29 Posson vertex algebras n the theory of Hamltonan eqatons 169 (1.51 H k(,,h I n Z + ( G (n F (n ( n H h ( G h +( n( F (n =,,h I n Z + ( n H h ( F h +( n( (H h ( F h (n G ( ( n H ( F (H kh( G h (n (1.52,,h I n Z + H k( ( (( n H h ( (n (H h ( G h ( n H ( G (H kh( F h (n We then se Lemma 1.2 to check that the second term n the left hand sde of (1.51 s F h G + F h (H h( G, the frst term n the rght hand sde of (1.51 s (1.53 ( n Hkh ( H ( F (,,h I (n G h + H kh(,,h I n Z + n Z + (n ( Gh (n ( n H ( F, and the second term n the rght hand sde of (1.51 s (1.54 ( n Hkh ( ( Fh H ( G (,,h I (n F h H kh(,,h I (n ( n H ( G. n Z + n Z + We then notce that the second term n (1.53 cancels wth the frst term n the left hand sde of (1.51, the second term n (1.54 cancels wth the thrd term n the left hand sde of (1.51 and, fnally, the second term n (1.52 cancels wth the last term n the left hand sde of (1.51, snce H( s skew-adont. In conclson eqaton (1.51 becomes (1.55 ( ( ( n Hkh ( ( H ( F,,h I (n G h ( n Hkh ( H ( G (n F h n Z + = ( ( n Hh ( H k( ( G,,h I (n F h. n Z + Snce the above eqaton holds for every F,G V l, we can replace actng on F by λ and actng on G by μ and wrte t as an dentty between polynomals n λ and μ. Hence (1.55 s eqvalent to (1.49..

30 170 A. Barakat, A. De Sole and V.G. Kac Defnton A matrx valed dfferental operator H( =(H (, I whch s skew-adont and satsfes one of the three eqvalent condtons ( ( of Proposton 1.16 (b, s called a Hamltonan operator. Example The Gardner Faddeev Zakharov (GFZ PVA strctre on R = C[,,,...] s defned by (1.56 { λ } = λ1 (frther on we shall sally drop 1. In fact, one can replace λ n the rght hand sde of (1.56 by any odd polynomal P(λ C[λ], and stll get a PVA strctre on R. Indeed, the bracket (1.56 s skew-commtatve and t satsfes the Jacob dentty for the trple,,, snce each trple commtator n the Jacob dentty s zero. Example The Vrasoro Magr PVA on R = C[,,,...], wth central charge c C,sdefnedby (1.57 { λ } =( + 2λ + λ 3 c. It s easly seen that the bracket (1.57 s skew-commtatve and t satsfes the Jacob dentty for the trple,,. Example Let g be a Le algebra wth a symmetrc nvarant blnear form (, andletp be an element of g. Let{ } I be a bass for g. TheaffnePVA assocated to the trple (g,(, p s the algebra R = C[ (n I, n Z + ] together wth the followng λ-bracket (1.58 {a λ b} =[a,b]+(p [a,b] + λ(a b, a,b g. Note that, takng n the rght hand sde of (1.58 any of the three smmands, or, more generally, any lnear combnaton of them, endows R wth a PVA strctre. Note that Example 1.18 s a specal case of Example 1.20, when g = C s the 1-dmensonal abelan Le algebra and ( =1. The followng theorem frther generalzes the reslts from Theorem 1.15, as t allows s to consder not only extensons of R, bt also qotents of sch extensons by deals. Theorem Let V be an algebra of dfferental fnctons, whch s an extenson of the algebra of dfferental polynomals R = C[ (n I, n Z + ].For each par, I, let { λ } = H (λ C[λ] V, and consder the ndced λ- bracket on V defned by formla (1.29. Sppose that J V s a sbspace sch that J J, J V J, {J λ V } C[λ] J, {V λ J} C[λ] J, and consder the qotent space V /J wth the ndced acton of, the ndced commtatve assocatve prodct and the ndced λ-bracket.