S D "V V " CUOLA DI OTTORATO ITO OLTERRA D R M XXIV OTTORATO DI ICERCA IN ATEMATICA (cid:21) CICLO W Classical -algebras Dottorando Relatore Daniele Valeri Prof. Alberto De Sole A A 2011(cid:21)2012 NNO CCADEMICO Contents Introduction 1 Chapter 1. Poisson vertex algebras and Hamiltonian equations 5 1.1. Algebras of di(cid:27)erential functions and Poisson vertex algebras 5 1.2. Hamiltonian operators and Hamiltonian equations 9 1.3. Compatible Poisson vertex algebra structures and integrability of Hamiltonian equations 11 Chapter 2. Classical W-algebras via Drinfeld-Sokolov Hamiltonian reduction 13 2.1. Review of classical Drinfeld-Sokolov Hamiltonian reduction 13 2.2. Classical W-algebras in the Poisson vertex algebra theory 15 2.3. Explicit construction of W (g,f,s), where g is of type B ,C 20 z n n Chapter 3. Gelfand-Dickey algebras 31 3.1. The algebra of formal pseudodi(cid:27)erential operators 31 3.2. Poisson vertex algebra structures attached to a general pseudodi(cid:27)erential operator 32 3.3. Reduction to the case u =0 36 −N 3.4. Poisson vertex algebra structure attached to a general di(cid:27)erential operator: Gelfand-Dickey algebras 42 3.5. Some examples 44 3.6. The Kupershmidt-Wilson theorem and the Miura map 44 Chapter 4. Isomorphisms between classical W-algebras and Gelfand-Dickey algebras 49 4.1. From (cid:28)rst order matrix di(cid:27)erential operators to n-th order pseudodi(cid:27)erential operators 49 4.2. W (gl ,f,s)∼=W for f principal nilpotent 51 z n n 4.3. Wz(sln,f,s)∼=W(cid:98)n for f principal nilpotent 60 Chapter 5. Integrable hierarchies for classical W-algebras 63 5.1. The homogeneuos case: integrable hierarchies for a(cid:30)ne Poisson vertex algebras 63 5.2. Integrable hierarchies arising from the classical Drinfeld-Sokolov Hamiltonian reduction 69 Appendix A. Remarks on formal calculus 77 Appendix B. Identities involving binomial coe(cid:30)cients 81 Appendix C. Virasoro (cid:28)eld in classical W-algebras 83 Bibliography 85 -1 Introduction The (cid:28)rst appearence of (quantum) W-algebras as mathematical objects is related to the conformal (cid:28)eld theory. The main problem of the conformal (cid:28)eld theory is a description of (cid:28)elds having conformal symmetry. Only in dimension D =2, the group of conformal di(cid:27)eomorphisms is rich enough to give rise to a meaningful theory. AfterthefundamentalpaperbyBelavin,PolyakovandZamolodchikov[4]itwasrealizedbyZamolod- chikov [24] that extended symmetries in two dimensional conformal (cid:28)eld theory in general do not give rise to (super)algebras with linear de(cid:28)ning relations. He constructed the so-called W -algebra, which 3 is an extension of the Virasoro algebra obtained adding one primary (cid:28)eld of weight 3. Later, this con- struction was generalized by Fateev and Lukyanov [14] to construct which are known as W -algebras. n Roughly speaking, these algebras are non-linear extensions of the Virasoro algebra obtained by adding primary (cid:28)elds. An exhaustive reference about extended symmetries can be found in [5]. The key point in the construction of the algebras W by Fateev and Lukyanov was the relation n between W-algebras and integrable systems. By a work of Gervais [18], the Virasoro algebra was found hidden in the second Poisson structure of the Korteweg-de Vries (KdV) equation, which is the so-called Virasoro-Magri structure. Magri [21] (cid:28)rst revealed the bi-Hamiltonian nature of the KdV equation. FateevandLukyanovidenti(cid:28)edZamolodchikov’sW algebrawiththeso-calledsecondPoissonstructure 3 of the equations of n-th KdV type, for n=3 (the KdV equation corresponds to n=2). These Poisson structuresareknownasGelfand-Dickeyalgebras,afterpioneristicworksonthesubjectbythem(onecan havealookto[10]forareviewoftheseworksandalotofmaterialontheargument)andareobtainedas Poisson algebras of local functionals on the algebra of di(cid:27)erential operators. After quantizing the n=3 structure, Fateev and Lukyanov found the same commutator formulas of Zamolodchikov W algebra. 3 This observation enabled them to construct all the W , since the second Poisson structure of the n-th n KdV type equation were known for any n. At the classical level, Gelfand-Dickey algebras are the (cid:28)rst examples of classical W-algebras. After, the second Poisson structure of Gelfand-Dickey type was considered in a more general setup, namely,inasimilarfashion,itcanbede(cid:28)nedaPoissonstructureonthespaceoflocalfunctionalsonthe larger algebra of pseudodi(cid:27)erential operators. The corresponding Poisson structures are related to the Kadomtsev-Petviashvili (KP) equation and the n-th KdV equations can be obtained with a reduction procedure from the KP hierarchy of equations. A quick reference is given by the lecture notes [9]. ThefactthatW-algebrashaveingeneralnon-linearde(cid:28)ningrelationsputsthemoutsideofthescope of the Lie algebra theory. However, they are intimately related to Lie algebras via the Drinfeld-Sokolov Hamiltonianreduction[12]. GivenaLiealgebraganditsprincipalnilpotent,thisreductionallowsusto construct a classical W-algebra. Furthermore, this procedure also emphasizes again the fact that these structures are related to certain hierarchies of partial di(cid:27)erential equations. Moreover, they proved that Gelfand-Dickey algebras correspond to the Drinfeld-Sokolov Hamiltonian reduction performed for the Lie algebra of n by n traceless matrices sl . n In this work we will be interested in the classical aspect of the theory, rather than the quantum one. So, from now on, we can skip the adjective classical. We have seen that W-algebras appear in at least three interrelated contexts. In particular, for which concerns integrable systems, there is also another Poisson structure for the equations of n-th KdV type (or for the KP equation in general). In this case we say that we have a bi-Hamiltonian structure and as pointed out in [21] this is one of the main tool to prove integrability for such equations. Recently, Barakat, De Sole and Kac [3] established a deep relation between Poisson vertex algebras andHamiltonianequationsandprovedthatPoissonvertexalgebrasprovideaveryconvenientframework to study (both classical and quantum) Hamiltonian systems. The aim of this thesis is to develop the theory of classical W-algebras in the Poisson vertex algebras language. This leads to a better understanding of the Hamiltonian structure underlying W-algebras and to a generalization of results, both in Gelfand-Dickey and Drinfeld-Sokolov approach to integrability of Hamiltonian eqautions. 1 In the (cid:28)rst chapter, we review the basic notions and foundations of the theory of Poisson vertex algebras aimed to the study of Hamiltonian equations as laid down in [3]. ThesecondchapterisaboutDrinfeld-SokolovHamiltonianreduction. Theorginalconstructiongiven in [12] involved a semisimple Lie algebra g and its principal nilpotent element f. After reviewing the Drinfeld-Sokolov Hamiltonian reduction, we will de(cid:28)ne it in a purely Poisosn vertex algebra language. This will enable us to perform such reduction for any nilpotent element of g. After showing that our construction is equivalent to the original one, we will construct, in the case of the classical Lie algebra B ,C and D , the corresponding W-algebras as quotients of particular Poisson vertex subalgebras of n n n the W-algebra corresponding to gl . n ThethirdchapterisdevotedtotheanalysisoftheGelfand-DickeyalgebrasapproachtoW-algebras, although we radically change point of view. Instead of considering a pseudodi(cid:27)erential operator and thende(cid:28)ningthePoissonstructureonthealgebraoflocalfunctionalsonit, weattachtoanydi(cid:27)erential algebra a particular pseudodi(cid:27)erential operator, which we call "general" and prove that in this case the Adler map [1] gives rise to a Hamiltonian operator, using the generating series of its matrix entries. The use of the λ-bracket language surprisingly simpli(cid:28)es the proof if compared to the usual one [10]. This allows us to think about the Adler map as a map from pseudodi(cid:27)erential operator to λ-bracket structures. When the λ-bracket corresponding under this map to a pseudodi(cid:27)erential operator de(cid:28)nes a Poisson vertex algebra structure then the Adler map gives rise to a Hamiltonian structure. We will give an example of an operator in which this does not happen and one in which it happens. In the (cid:28)rst case we can still modify the Ader map and get a Hamiltonian structure. In the Drinfeld-Sokolov Hamiltonian reduction, this structure corresponds to the Hamiltonian reduction of sl and its principal n nilpotentelement. Intheothercasewewillbeabletorecoverandtogiveaverysimpleproofofafamous theorem of Kupershmidt and Wilspon [20]. The fourth section is devoted to establish the well known fact that Gelfand-Dickey algebras are W-algebras corresponding to some special cases of Drinfeld-Sokolov Hamiltonian reduction. Namely, we will prove that the Poisson vertex algebras we construct using Drinfeld-Sokolov approach, in the case of the Lie algebra gl (respectively sl ) and its principal nilpotent element, is isomorphic to the Poisson n n vertex algebra we got, using Gelfand-Dickey approach, in the case of a general di(cid:27)erential operator of order n (respectively the same di(cid:27)erential operator with missing ∂n−1 term). In the last chapter we will be interested in (cid:28)nding integrable systems attached to W-algebras and proving their integrability. First, we will consider the homogeneous case, namely we will (cid:28)nd integrable hierarchiesfora(cid:30)nePoissonvertexalgebras(seeExample1.6forthede(cid:28)nition). Thenwewillgeneralize the results of [12], about integrable systems attached to W-algebras via Drinfeld-Sokolov Hamiltonian reduction, to a larger class of nilpotent elements. Ringraziamenti Poche parole non bastano per esprimere la mia profonda gratitudine e la stima che ho nei confronti del mio relatore, Alberto De Sole. I suoi consigli, rimproveri e incoraggiamenti sono stati fondamentali durante questi miei anni di dottorato. In confronto a quando ho iniziato, adesso, grazie alla sua guida, mi sento matematicamente moltomigliorato,epermeŁgi(cid:224)unprimotraguardo. Etuttoquestononsarebbestatopossibilesenzala sua assidua professionalit(cid:224) e il tempo che mi ha dedicato. Per questo mi scuso con Alessandra e i piccoli Federico, Daniele e Stefano per tutto il tempo a cui l’ho tolto loro. RingrazioinparticolareVictorKacperavermipermessoditrascorrerevariperiodidistudio,dissemi- natiquael(cid:224)duranteladuratadelmiodottorato,pressoildipartimentodimatematicadelMassachusetts Institute of Technology e per i suoi utili e preziosi consigli e discussioni matematiche. Semisonosempresentitoacasa,ancheseerodall’altrepartedell’oceano,lodevoadAndrea,Giorgia, Salvatore e Tonino. Li ringrazio moltissimo per avermi messo a disposizione i loro divani durante le mie lunghericercheperuntettosottocuitrovareriparoepernonavermimaifattosentiresmarritoinquesto nuovo "mondo" che avevo davanti. Studiare oltreoceano Ł stato faticoso, ma anche piacevole grazie a tutte le persone, del dipartimento di matematica e non, che ho conosciuto. In particolare, devo ringraziare per questo il team dei Perverse Sheaves, nonchŁ Andrei, Bhairav, Nikola, Roberto, gli "zii accademici" Jethro e Uhi Rinn, per le lunghe chiacchierate matematiche e non, e il grande amico Sasha, compagno di avventure/sventure! Perquelcheriguardalapartedell’oceanochepiømiappartiene,ilprimopensierovaallaprofessoressa de Resmini. Lei mi ha sempre spinto a dare il massimo quando ero uno studente e incoraggiato a continuare con il dottorato di ricerca. A livello personale, credo che ne sia valsa la pena. Grazie Professoressa! 2 Un ringraziamento anche ai colleghi "deResminiani" Giuseppe e Riccardo, per la leggerezza che ci ha contraddistinto nell’a(cid:27)rontare l’evento celebrativo della Prof e per tutto il tempo trascorso insieme al Castelnuovo. Per quel che riguarda studiare al Castelnuovo, un grande ringraziamento a tutti gli assidui frequen- tatori e non della stanza dottorandi. Quello che succede l(cid:236) dentro, Ł noto solo agli addetti ai lavori. In particolare, ringrazio quegli scoppiati di Federico, Lorenzo, Marco e Sergio, chi per un motivo chi per un altro e i "napoletani" Giuseppe e Renato, per l’ospitalit(cid:224) durante le mie pratiche in terra partenopea per ottenere il visto e i bei momenti insieme al di fuori della matematica. Anche se forse inconsapevolmente non lo sa, una buona parte di questo lavoro Ł stato merito del continuo supporto psicologico del mio grande amico Piero. Con lui, tra varie altre cose, ho condiviso insalata, mista e polpettone e vissuto momenti all’insegna della "libertŁ du mouvement et (cid:29)ux de co- science". Grazie per esserci in ogni momento! E per fortuna che ora c’Ł Angela che ti tiene a bada. Un grande ringraziamento/in bocca al lupo al mio amico/studente Matteo. Non mollare ora che sei quasi giunto al traguardo. Grazie ad Arianna, che da un p(cid:242) di tempo, oltre ad essermi vicina, sopporta tutte le mie assenze, (cid:28)siche o mentali che siano. In(cid:28)ne, anche se gi(cid:224) pensavano mi fossi dimenticato di loro, un grandissimo grazie ai miei genitori, Elena e Mario, e alla mia "sorellina" Francesca, ai piø noti come Zem. Anche se la solita confusione che regna a casa non Ł sempre l’ambiente ideale per studiare, Ł sicuramente uno degli ambienti piø belli per vivere e se sono riuscito a proseguire gli studi Ł solo merito dei vostri sacri(cid:28)ci, che forse non sono mai riuscito a ripagare abbastanza. State tranquilli per(cid:242), perchŁ anche quando diventer(cid:242) ricco e famoso, non vi dimenticher(cid:242) mai. 3 CHAPTER 1 Poisson vertex algebras and Hamiltonian equations In this chapter we review the connection between Poisson vertex algebras and the theory of Hamil- tonian equations as laid down in [3]. It will be shown that Poisson vertex algebras provide a very convenient framework for systems of Hamiltonian equations associated to a Hamiltonian operator. As themainapplicationweexplainhowtoestablishintegrabilityofsuchpartialdi(cid:27)erentialequationsusing the so called Lenard scheme. 1.1. Algebras of di(cid:27)erential functions and Poisson vertex algebras By a di(cid:27)erential algebra we shall mean a unital commutative associative algebra R over C with a derivation ∂, that is a C-linear map from R to itself such that, for a,b∈R ∂(ab)=∂(a)b+a∂(b). In particular ∂1=0. Oneofthemostimportantexamplesweareinterestedinisthealgebra of di(cid:27)erential polynomials in one independent variable x and l dependent variables u (l may also be in(cid:28)nite) i R [x]=C[x,u(n) |i∈{1,...,l}=I,n∈Z ], l i + where the derivation ∂ is de(cid:28)ned by ∂(u(n))=u(n+1) and ∂x=1. One can also consider the algebra of i i translation invariant di(cid:27)erential polynomials in l variables u i R =C[u(n) |i∈{1,...,l}=I,n∈Z ], l i + where ∂(u(n))=u(n+1). i i De(cid:28)nition 1.1. An algebra of di(cid:27)erential functions V in one independent variable x and a set of dependent variables {u } is a di(cid:27)erential algebra with a derivation ∂ endowed with linear maps i i∈I ∂ : V −→ V, for all i ∈ I and n ∈ Z , which are commuting derivations of the product in V such ∂u(n) + i that, given f ∈ V, ∂f = 0 for all but (cid:28)nitely many i ∈ I and n ∈ Z and the following commutation ∂u(n) + i relations hold (cid:34) (cid:35) ∂ ∂ ,∂ = , (1.1) ∂u(n) ∂u(n−1) i i where the RHS is considered to be zero if n=0. We call C=ker(∂)⊂V the subalgebra of constant functions and denote by F ⊂V the subalgebra of quasiconstant functions, de(cid:28)ned by ∂f F ={f ∈V| =0∀i∈I,n∈Z }. ∂u(n) + i One says that f ∈V has di(cid:27)erential order n in the variable u if ∂f (cid:54)=0 and ∂f =0 for all m>n. i ∂u(n) ∂u(m) It follows by (1.1) that C ⊂ F. Indeed, suppose that f ∈ C has ordier n ∈ Z in siome variable u , then + i (cid:20) (cid:21) 0= ∂ ,∂ f = ∂f which contracdicts our hypothesis. Furthermore, clearly, ∂F ⊂F. ∂u(n+1) ∂u(n) i i The di(cid:27)erential algebras R [x] and R are examples of algebras of di(cid:27)erential functions. Other l l examples can be constructed starting from R [x] or R by taking a localization by some multiplicative l l subset S, or an algebraic extension obtained by adding solutions of some polynomial equations, or a di(cid:27)erential extension obtained by adding solutions of some di(cid:27)erential equations. 5 Inalltheseexamples,butmoregenerallyinanyalgebraofdi(cid:27)erentialfunctionswhichisanextension ofR [x], theactionofthederivation∂ :V−→V, whichextendstheusualderivationinR [x], isgivenby l l ∂ = ∂ + (cid:88) u(n+1) ∂ , (1.2) ∂x i ∂u(n) i∈I,n∈Z+ i which implies that F∩∂V=∂F. Indeed, if f ∈V has di(cid:27)erential order n∈Z in some variable u , then + i ∂f has di(cid:27)erential order n+1, hence, it does not lie in F. The commutation relations (1.1) imply the following lemma ([3, Lemma 1.2]). Lemma 1.2. Let D (z) = (cid:80) zn ∂ . Then for every h(λ) = (cid:80)N h λm ∈ C[λ]⊗V and f ∈ V i n∈Z+ ∂u(n) m=0 m i the following identity holds D (z)(h(∂)f)=D (z)(h(∂))f +h(z+∂)(D (z)f), i i i where D (z)(h(∂)) is the di(cid:27)erential operator obtained by applying D (z) to the coe(cid:30)cients of h(∂). i i Proof. Multiplying by zn and summing over n ∈ Z both sides of (1.1) we get D (z) ◦ ∂ = + i (z+∂)D (z). It follows that i D (z)◦∂n =(z+∂)nD (z), i i for every n∈Z . Thus, if h(λ)=h λn, this implies that + n D (z)◦h(∂)=D (z)◦h ∂n =D (z)(h )◦∂n+h D (z)◦∂n =D (z)(h )◦∂n+h (z+∂)nD (z)= i i n i n n i i n n i =D (z)(h(∂))+h(z+∂)D (z). i i By linearity the general case follows. (cid:3) We denote by V⊕l ⊂ Vl the subspace of all F = (F ) with (cid:28)nitely many non-zero entries (l may i i∈I also be in(cid:28)nite) and introduce a pairing Vl×V⊕l −→V(cid:14)∂V (cid:82) (P,F)−→ PF, (1.3) where (cid:82) denotes the canonical map V −→ V(cid:14)∂V. The pairing (1.3) is non-degenerate [3, Proposition 1.3], namely (cid:82) PF =0 for every F ∈V⊕l if and only if P =0. (cid:16) (cid:17) Let us de(cid:28)ne the operator of variational derivative δ :V−→V⊕l by δf = δf ∈V⊕l, where δu δu δui i∈I δf (cid:88) ∂f = (−∂)n . δui n∈Z+ ∂u(in) By (1.1), it follows immediately that δ ·∂ =0, for each i∈I, then ∂V⊂ker δ . We let Vect(V) be the space of alδluivector (cid:28)elds of V, which is a Lie subalgδuebra of Der(V), the Lie algebra of all derivations of V. An element X ∈Vect(V) is of the form ∂ (cid:88) ∂ X =h + h , h,h ∈V. (1.4) ∂x i,n∂u(n) i,n i∈I,n∈Z+ i By (1.2), ∂ is an element of Vect(V) and we denote by Vect∂(V) the centralizer of ∂ in Vect(V), namely Vect∂(V) = Vect(V)∩Der∂(V). Elements X ∈ Vect∂(V) are called evolutionary vector (cid:28)elds. (cid:104) (cid:105) For X ∈ Vect∂(V) we have X(u(n)) = X(∂nu ) = ∂nX(u ), so that, by (1.4) and ∂ ,∂ = 0, X is i i i ∂x completely determined by its values X(u ) = P , i ∈ I. Thus, we have a vector space isomorphism i i Vl ∼=Vect∂(V) given by (cid:88) ∂ Vl (cid:51)P =(P ) −→X = (∂nP ) ∈Vect∂(V). (1.5) i i∈I P i ∂u(n) i∈I,n∈Z+ i The l-tuple P is called the characteristic of the vector (cid:28)eld X . P The FrØchet derivative D of f ∈V is de(cid:28)ned as the following di(cid:27)erential operator from Vl to V: f (cid:88) ∂f D (∂)P =X (f)= ∂nP . f P ∂u(n) i i∈I,n∈Z+ i We note that D (∂)P is just the (cid:28)rst-order di(cid:27)erential of the function f(u), indeed f(u+εP)=f(u)+ f εD (∂)P +o(ε2). f 6