A geometric construction of integrable Hamiltonian hierarchies associated with the classical affine W-algebras
aa r X i v : . [ m a t h . QA ] M a y A GEOMETRIC CONSTRUCTION OF INTEGRABLEHAMILTONIAN HIERARCHIES ASSOCIATED WITH THECLASSICAL AFFINE W -ALGEBRAS SHIGENORI NAKATSUKA
Abstract.
A class of classical affine W -algebras are shown to be isomorphicas differential algebras to the coordinate rings of double coset spaces of certainprounipotent proalgebraic groups. As an application, integrable Hamiltonianhierarchies associated with them are constructed geometrically, generalizing thecorresponding result of Feigin-Frenkel and Enriquez-Frenkel for the principalcases. Introduction
Since the discovery of the Drinfel’d-Sokolov hierarchy [14], a number of integrablesystems have been constructed in the same spirits cf. [4, 6, 18, 21, 22]. Thoseintegrable systems are not of finite dimension, which are formulated by Poissonalgebras, but of infinite dimension, reflecting classical field theory as their origin.In [2], Barakat, De Sole and Kac used
Poisson vertex algebras , which play a roleof Poisson algebras for integrable systems as above, and introduced the notion of integrable Hamiltonian hierarchies as a framework of integrability for Poisson vertexalgebras. Since then De Sole, Kac, and Valeri e.g. [9, 10, 11, 12, 13] have studiedsystematically integrable Hamiltonian hierarchies associated with Poisson vertexalgebras, called the classical (affine) W -algebras , which are obtained as classicallimit of vertex algebras, called the (affine) W -algebras [7, 26], see also [8, 30].The W -algebras are parametrized by a triple ( g , f, k ) consisting of a finite di-mensional simple Lie algebra g over C , a nonzero nilpotent element f ∈ g , and acomplex number k ∈ C , called the level and are denoted by W k ( g , f ). The classical W -algebras are also parametrized by the same data and thus we denote them alsoby W k ( g , f ). De Sole, Kac and Valeri recovered the earlier results on the integrableHamiltonian hierarchies mentioned above in their study and obtained a culminatingresult in [13] on the existence of integrable Hamiltonian hierarchies associated with W k ( g , f ), which states: the classical affine W -algebra W k ( g , f ) admits an integrableHamiltonian hierarchy for any finite dimensional simple Lie algebra g of classicaltype, nonzero nilpotent element f , and k = 0 . Besides the algebraic theory of integrable Hamiltonian hierarchies mentionedabove, there is a result of Feigin-Frenkel [19, 20] and Enriquez-Frenkel [16] whichconstructs the Drinfel’d-Sokolov hierarchies by using a geometric realization of theclassical affine W -algebra W ( g , f prin ), where f prin denotes the principal nilpotentelement in g . More precisely, they proved that W ( g , f prin ) are isomorphic as differ-ential algebras to the coordinate rings of certain double coset spaces of prounipotentproalgebraic groups and that the Drinfel’d-Sokolov hierarchies are induced fromnatural group actions on these spaces.The aim of this paper is to generalize the those results to the classical W -algebras W k ( g , f ) when f satisfies certain properties (see the condition (F) in the below)and the level k ∈ C is generic. We note that the choice of f is a special case of theso-called Type I in [4, 6, 18]. The integrable Hamiltonian hierarchies obtained in the paper for the case k = 1 coincide with special cases of those considered in [3]by construction, and are special cases of the result [9].Let g be a finite dimensional Lie algebra over C , and f ∈ g be a non-zero nilpotentelement. We fix an sl -triple { e, h = 2 x, f } containing f . We denote by Γ : g = ⊕ dj = − d g j the ad x -grading and by g = h ⊕ ( ⊕ α ∈ ∆ g α ) a root space decomposition of g which is homogeneous with respect to Γ. Then the root system ∆ admits an inducedgrading ∆ = ⊔ dj = − d ∆ j . Let Π denote the subset of ∆ > = ⊔ j> ∆ j consisting of theelements indecomposable in ∆ > . Let V k ( g ) denote the universal affine Poissonvertex algebra associated with the reductive Lie algebra g at level k and F ( g / )the βγ -system Poisson vertex algebra associated with the symplectic vector space g / . Then the W -algebra W k ( g , f ) is defined as a 0-th cohomology of the BRSTcomplex C k ( g , f ), see Section 3.1 for details.For generic level k , the W -algebras are realized as the joint kernel of certainscreening operators [25]. Our first result (Theorem 3.6) is a Poisson vertex algebraanalogue of this result. Namely, for generic k ∈ C , we realize the classical affine W -algebra W k ( g , f ) as the Poisson vertex subalgebra of V k ( g ) ⊗ F ( g / ) invariantunder the derivations Q Wα , ( α ∈ Π):(1.1) W k ( g , f ) ∼ = \ α ∈ Π Ker (cid:16) Q Wα : V k ( g ) ⊗ F ( g / ) → V k ( g ) ⊗ F ( g / ) (cid:17) . See ( 3.16) and ( 3.17) for the definition of Q Wα .We now suppose that the pair ( g , f ) satisfies the following condition (F):(F1) The grading Γ is a Z -grading.(F2) There exists an element y ∈ g d such that s = f + yt − ∈ g [ t ± ] is semisimple.(F3) The Lie subalgebra Ker(ad s ) ⊂ g [ t ± ] is abelian and Im(ad s ) ∩ g [ t ± ] = g .Here g [ t ± ] = g ⊗ C [ t, t − ] and g [ t ± ] = ⊕ j ∈ Z g [ t ± ] j is the Z -grading given bydeg( Xt n ) = j + ( d + 1) n , ( X ∈ g j ). The nilpotent elements f satisfying (F1)-(F2) are called Type I in the literature e.g. [18, 4, 6]. We consider the completion L g = g ⊗ C (( t )). It has subalgebras L g + , which is the completion of g [ t ± ] > , and L g − = g [ t ± ] ≤ . We have the corresponding closed subgroups LG + , LG − of theloop group LG of G .Let a denote the completion of Ker(ad s ) in L g , and a ± = a ∩ L g ± . Let A , (resp. A ± ) be the closed subgroup of LG corresponding to a , (resp. a ± ).The right A -action LG − \ LG/A + × A → LG − \ LG/A + , induces a Lie algebrahomomorphism a → Der( C [ LG − \ LG/A + ]), and so that a → Der( C [ LG + /A + ])since LG + /A + ⊂ LG − \ LG/A + is an open subset. We denote this action by a a R .In particular, we obtain a differential algebra ( C [ LG + /A + ] , s R ). On the other hand,the natural left LG + -action LG + × LG + /A + → LG + /A + induces a Lie algebrahomomorphism L g + → Der( C [ LG + /A + ]), which we denote by X X L .Our second result (Theorem 4.3) is the following geometric interpretation of( 1.1) under the condition (F): there exists an isomorphism of differential algebrasΨ k : V k ( g ) → C [ LG + /A + ] , such that the derivation Q Wα is identified with X Lα for some element X α ∈ g α ,( α ∈ Π). Since such root vectors X α generate a Lie algebra g + = ⊕ j> g j , it impliesthat W k ( g , f ) is isomorphic to C [ G + \ LG + /A + ] as a differential algebra (Corollary4.4). Here G + is the closed subgroup of LG + corresponding to g + .The action a → Der( C [ LG + /A + ]) preserves the subalgebra C [ G + \ LG + /A + ].Thus we obtain a space of mutually commutative derivations of W κ ( g , f ) as the im-age of a − . We denote this space by H κ ( g , f ). Our third result (Theorem 5.9) statesthat H k ( g , f ) is an integrable Hamiltonian hierarchy associated with W k ( g , f ). -ALGEBRAS AND INTEGRABLE SYSTEMS 3 The double coset spaces G + \ LG + /A + considered here are embedded into theabelianized Grassmanians G [ t − ] \ G (( t )) /A + as a Zariski open subset. The abelian-ized Grassmanians have been used to construct Drinfeld-Sokolov hierarchies geomet-rically [3]. As pointed in loc.cit. , such a construction implies a strong compatibilityof integrable Hamiltonian systems associated with classical affine W -algebras andHitchin systems. The author hope to investigate the relationship between classicalaffine W -algebras and Hitchin systems in future works. Acknowledgements
This paper is the master thesis of the author. He wishes toexpress his gratitude to Professor Atsushi Matsuo for encouragement throughoutthis work and numerous advices.2.
Poisson vertex superalgebras
Poisson vertex superalgebra.
We recall here some basics about Poissonvertex superalgebras and their relation to the theory of integrable systems, following[2, 30]. We remove the prefix “super” whenever we consider the non-super cases.A differential C -superalgebra is a pair ( V, ∂ ) consisting of a supercommutative C -superalgebra V and an even derivation ∂ on it. We denote by ¯ a the parity of For a ∈ V and by Der( V ) the set of super derivations of V . A differential C -superalgeraof the form V = C [ u ( n ) i | i ∈ I ¯0 , n ≥ ⊗ ^ ⊕ i ∈ I ¯1 C u ( n ) i as C -superalgebras for some index set I = I ¯0 ⊔ I ¯1 where u ( n ) i = ∂ n u i , is called the superalgebra of differential polynomials in the variables u i , ( i ∈ I ).A Poisson vertex superalgebra is a triple (
V, ∂, {− λ −} ) consisting of a differential C -superalgebra ( V, ∂ ) and an even C -bilinear map(2.2) {− λ −} : V × V → V [ λ ] , ( f, g )
7→ { f λ g } = X n ≥ λ n n ! f ( n ) g, called the λ -bracket, satisfying { ∂f λ g } = − λ { f λ g } , { f λ ∂g } = ( λ + ∂ ) { f λ g } , (2.3) { g λ f } = − ( − ¯ f ¯ g ← { f − ∂ − λ g } , (2.4) { f λ { g µ h }} − ( − ¯ f ¯ g { g µ { f λ h }} = {{ f λ g } λ + µ h } , (2.5) { f λ gh } = { f λ g } h + ( − ¯ g ¯ h { f λ h } g, (2.6) { f g λ h } = ( − ¯ g ¯ h { f λ + ∂ h } → g + ( − ¯ f (¯ g +¯ h ) { g λ + ∂ h } → f, (2.7)for f, g, h ∈ V . Here we denote { f λ g } → = X n ! f ( n ) gλ n , ← { f λ g } = X λ n n ! f ( n ) g. Remark 2.1.
Given an algebra of differential polynomials V in the variables u i , ( i ∈ I ) , a linear map F : span C { u i } i ∈ I → V [ λ ] uniquely extends to F : V → V [ λ ] by ( 2.3) , ( 2.7) . Let V be a superalgebra of differential polynomials in the variables u i , ( i ∈ I ).We denote by ∂∂u ( n ) i , ∂ R ∂ R u ( n ) i ∈ Der( V )denote the derivation with respect to u ( n ) i from the left and the right respectively,(which coincide if V is non-super). Then the λ -bracket on V is determined by thevalues { u iλ u j } in the following sense. SHIGENORI NAKATSUKA
Theorem 2.2 ([2, 30]) . Let ( V, ∂ ) be a superalgebra of differential polynomials inthe variables u i , ( i ∈ I ) , and H ij ( λ ) , ( i, j ∈ I ) , an element of V [ λ ] with the sameparity as u i u j . Then there is a unique Poisson vertex superalgebra structure on V satisfying { u iλ u j } = H ji ( λ ) if and only if the λ -bracket satisfies ( 2.4) and ( 2.5) for u i , ( i ∈ I ) . Moreover, the λ -bracket is given by { f λ g } = X i,j ∈ I,n,m ≥ ( − ¯ f ¯ g +¯ i ¯ j ∂ R g∂ R u ( n ) j ( λ + ∂ ) n H ji ( λ + ∂ ) → ( − λ − ∂ ) m ∂f∂u ( m ) i , f, g ∈ V. Example 2.3 (Universal affine Poisson vertex algebra) . Given a finite dimensionalLie algebra L over C and a nondegenerate symmetric invariant bilinear form κ on L , let V κ ( L ) denote the algebra of differential polynomials in the variablesgiven by a basis of L . Then a λ -bracket {− λ −} : L × L → V κ ( L )[ λ ] given by { u λ v } = [ u, v ] + κ ( u, v ) λ , ( u, v ∈ L ), defines a Poisson vertex algebra structure on V κ ( L ). This is called the universal affine Poisson vertex algebra associated with L at level κ .Given a Poisson vertex superalgebra V , a Poisson vertex module over V (cf. [1])is a vector superspace M which is a V -module as a supercommutative algebra V and endowed with an even C -bilinear map {− λ −} : V × M → M [ λ ] satisfying(2.8) { ∂f λ m } = − λ { f λ m } , (2.9) { f λ { g µ m }} − ( − ¯ f ¯ g { g µ { f λ m }} = {{ f λ g } λ + µ m } , (2.10) { f λ g · m } = { f λ g } · m + ( − ¯ f ¯ g g · { f λ m } , (2.11) { f · g λ m } = ( − ¯ m ¯ g { f λ + ∂ m } → g + ( − ¯ f (¯ g + ¯ m ) { g λ + ∂ m } → f for f, g ∈ V , m ∈ M . Here the right V -action M × V → M is defined by the actionof V as a supercommutative algebra. In this case, we define the λ -bracket {− λ −} : M × V → M [ λ ] , ( m, f )
7→ { m λ f } = − ( − ¯ m ¯ f ← { f − ∂ − λ m } . For m ∈ M , the linear map { m λ −} : V → M [ λ ] is called an intertwining operator and m is called the Hamiltonian.For a Poisson vertex superalgebra V , the vector superspace Lie( V ) = V /∂V iscalled the space of local functionals . We denote by R : V → Lie( V ) , f R f the canonical projection. Proposition 2.4 ([2, 30]) . (1) The bilinear map
Lie( V ) × Lie( V ) → Lie( V ) , ( R f, R g ) R { f λ g } λ =0 is well-defined and defines a Lie superalgebra structure. Moreover, if V iseven and an algebra of differential polynomials in the variables { u i } i ∈ I , then (cid:2)R f, R g (cid:3) = P i.j ∈ I R δ R gδ R u j { u i∂ u j } → δfδu i , where δfδu i = P n ≥ ( − ∂ ) n ∂f∂u ( n ) i and δ R fδ R u i = P n ≥ ( − ∂ ) n ∂ R f∂ R u ( n ) i denote theleft and right variational derivative of f with respect to u i . (2) The Lie superalgebra
Lie( V ) acts on V by η : Lie( V ) → Der( V ) , R f
7→ { f λ −}| λ =0 . We use the following lemma in the below. -ALGEBRAS AND INTEGRABLE SYSTEMS 5
Lemma 2.5 (cf. [2, Proposition 1.33]) . Let L be a finite dimensional Lie algebraover C equipped with a nondegenerate symmetric invariant bilinear form κ . Forthe universal affine Poisson vertex algebra V κ ( L ) , the kernel of η : Lie( V κ ( L )) → Der( V κ ( L )) is Ker( η ) = span C (cid:8)R , R u | u ∈ Z ( L ) (cid:9) , where Z ( L ) is the center of L .Proof. Let { u i } i ∈ I be a basis of L . Suppose R F ∈ Ker( η ) ⊂ Lie( V κ ( L )). ByTheorem 2.2, η (cid:0)R F (cid:1) = P i,j,n ∂ n (cid:16) ([ u i , u j ] + κ ( u i , u j ) ∂ ) δFδu i (cid:17) ∂∂u ( n ) j .It follows M j = P i ([ u i , u j ] + κ ( u i , u j ) ∂ ) δFδu i = 0 for j ∈ I . Define a degree on V κ ( L )by deg( u ( n ) i ) = n and deg( AB ) = deg( A )+deg( B ). Then δ/δu preserves the degree.Let G i be the top degree component of δF/δu . Then the top component M top j of M j is M top j = P i κ ( u i , u j ) ∂G i . Since κ is nondegenerate, we obtain ∂G i = 0, whichimplies G i ∈ C . Since V κ ( L ) is an algebra of differential polynomials, we conclude F ∈ C ⊕ L . (See the proof of [2, Proposition 1.5].) Set F = a + P i b i u i , ( a, b i ∈ C ).Then M j = [ P i b i u i , u j ] = 0, ( j ∈ I ), which implies P i b i u i ∈ Z ( L ). (cid:3) Finally, given a Poisson vertex superalgebra V , an element R f ∈ Lie( V ) is called integrable if there exists an infinite dimensional abelian Lie subsuperalgebra H ofLie( V ) which contains R f . In this case, H is called an integrable Hamiltonianhierarchy associated with V .2.2. Differential graded Poisson vertex superalgebra. A differential gradedPoisson vertex superalgebra (d.g. Poisson vertex superalgebra) is a pair ( V, d ) con-sisting of a Poisson vertex superalgebra V = ⊕ n ∈ Z V n and a linear map d : V → V ,called the differential, satisfying • V is a Z -graded Poisson vertex superalgebra, i.e., V = ⊕ n ∈ Z V n as a vectorsuperspace satisfying V n · V m ⊂ V n + m , { V nλ V m } ⊂ V n + m [ λ ] , • the linear map d is of homogeneous parity and satisfies d = 0, d : V n → V n +1 ,d ( ab ) = d ( a ) · b + ( − ¯ d ¯ a a · d ( b ) , d ( { a λ b } ) = { d ( a ) λ b } + ( − ¯ d ¯ a { a λ d ( b ) } . The cohomology H ∗ ( V ; d ) = ⊕ n ∈ Z H n ( V ; d ) inherits a Z -graded Poisson vertexalgebra structure. Moreover, H ( V ; d ) is a Poisson vertex subsuperalgebra and H n ( V ; d ), ( n ∈ Z ), is a Poisson vertex module over H ( V ; d ). In the sequel, we alsouse the notion of a differential graded vertex superalgebra . The definition is similarand therefore we omit the details.2.3. Classical limit.
Let V be a vertex superalgebra over a polynomial ring C [ ǫ ].Suppose that V is free as a C [ ǫ ]-module and the λ -bracket satisfies(2.12) [ V λ V ] ⊂ ǫV. Define the vector superspace V cl = V /ǫV and let V → V cl , ( f ¯ f ) denote thecanonical projection. Then V cl × V cl → V cl , ( ¯ f , ¯ g ) f ( − g is well-defined and defines an associative supercommutative algebra structure on V cl . Since the translation operator ∂ of V preserves ǫV , it induces a linear map ∂ : V cl → V cl , ∂ ¯ f ∂ ( f ) , SHIGENORI NAKATSUKA which is a derivation of V cl . Since the ǫV is an ideal of V by ( 2.12), the λ -bracketof V induces a bilinear map {− λ −} : V cl × V cl → V cl [ λ ] , ( ¯ f ¯ g ) [ f λ g ] . The triple ( V cl , ∂, {− λ −} ) defines a Poisson vertex superalgebra, called the classicallimit of V (cf. [7, 23]).3. Screening operators for classical affine W -algebras In this section, we describe the classical affine W -algebras by using screeningoperators. They will be obtained as a classical limit of the screening operators forthe affine W -algebras obtained in [25]. We will use the same notation for Poissonvertex algebras as vertex algebras since there will be no confusion.3.1. Affine W -algebras. Let g be a finite dimensional simple Lie algebra over C with the normalized symmetric invariant bilinear form κ = ( −|− ). Let f ∈ g be anonzero nilpotent element, fix an sl -triple { e, h = 2 x, f } containing f and denoteby Γ : g = ⊕ − d ≤ j ≤ d g j the Z -grading given by ad x , with d the largest number suchthat g d = 0. We fix a triangular decomposition g = n + ⊕ h ⊕ n − so that x ∈ h , g > = ⊕ j> g j ⊂ n + , and g < = ⊕ j< g j ⊂ n − . Let g = h L ⊕ α ∈ ∆ g α be a rootspace decomposition, ∆ j = { α ∈ ∆ | g α ⊂ g j } , and ∆ > = ⊔ j> ∆ j . Fix a nonzeroroot vector e α in g α and a basis e i , ( i ∈ I ), of h . Then e α , ( α ∈ I ⊔ ∆), form abasis of g . We denote by { e ¯ α } α ∈ I ⊔ ∆ its dual basis of g with respect to κ . Let c γα,β denote the structure constants of g , i.e., [ e α , e β ] = P γ ∈ I ⊔ ∆ c γα,β e γ .Let V k ( g ) be the universal affine vertex algebra of g at level k , generated bythe even elements e α , ( α ∈ I ⊔ ∆), with λ -bracket [ e αλ e β ] = [ e α , e β ] + k ( e α | e β ) λ .Let F ch ( g > ) be the charged free fermion vertex superalgebra associated with thesymplectic odd vector superspace g > ⊕ g ∗ > , generated by the odd elements ϕ α , ϕ α , ( α ∈ ∆ > ), with λ -bracket [ ϕ αλ ϕ β ] = δ α,β , [ ϕ αλ ϕ β ] = [ ϕ αλ ϕ β ] = 0. Let F ( g / ) be the βγ -system vertex algebra associated with the symplectic vectorspace g / , generated by Φ α , ( α ∈ ∆ / ), with λ -bracket [Φ αλ Φ β ] = χ ([ e α , β ]),where χ ( − ) = ( f | − ).The affine W -algebra W k ( g , f ) associated with the triple ( g , f, k ), ( k ∈ C ), isthe vertex algebra defined as the 0-th cohomology of the differential graded vertexalgebra C k ( g , f ) = V k ( g ) ⊗ F ch ( g > ) ⊗ F ( g / ) , with differential d (0) = h X α ∈ ∆ > (cid:0) ( e α + Φ α + χ ( e α ) (cid:1) ϕ α − X α,β,γ ∈ ∆ > c γα,β ϕ γ ϕ α ϕ βλ − i | λ =0 , called the BRST complex . The grading C k ( g , f ) = ⊕ n ∈ Z C kn ( g , f ) is given bygr( e α ) = gr(Φ β ) = 0, gr( ϕ α ) = − gr( ϕ α ) = 1 with gr( AB ) = gr( A ) + gr( B )and gr( ∂A ) = gr( A ). Then we have d (0) : C kn ( g , f ) → C kn +1 ( g , f ). We have H n ( C k ( g , f )) = 0 for n = 0 (see [27, 28]).3.2. Classical limit.
By [27, 28], we have vertex subsuperalgebras C k ± ( g , f ), whichgives a decomposition of a complex C k ( g , f ) = C k − ( g , f ) ⊗ C k + ( g , f ) and satisfies H n ( C k + ( g , f )) ∼ = δ n, C . Thus H ( C k − ( g , f )) ∼ = W k ( g , f ). As a vertex superalgebra, C k − ( g , f ) is generated by J u = u + X β,γ ∈ ∆ > c γu,β ϕ γ ϕ β , ( u ∈ g ≤ ) , Φ α , ( α ∈ ∆ / ) , ϕ α , ( α ∈ ∆ > ) . Following [25], we introduce the classical affine W -algebra as the cohomology ofthe differential graded Poisson vertex algebra in the classical limit of C k − ( g , f ). -ALGEBRAS AND INTEGRABLE SYSTEMS 7 Suppose k + h ∨ = 0. Set ǫ = k ′ k + h ∨ , ( k ′ ∈ C \{ } ), ¯ J u = ǫJ u , ( u ∈ g ≤ ), and¯Φ α = ǫ Φ α , ( α ∈ ∆ / ). Then we have(3.13) [ ¯ J uλ ¯ J v ] = ǫ (cid:0) ¯ J [ u,v ] + (cid:0) k ′ ( u | v ) + o ( ǫ )) λ (cid:1) , [ ϕ αλ ¯ J u ] = ǫ X β ∈ ∆ > c αu,β ϕ β , (3.14) [ ¯Φ αλ ¯Φ β ] = ǫ · χ ([ e α , e β ]) , [ ¯ J uλ ¯Φ α ] = [ ϕ αλ ϕ β ] = [ ϕ αλ ¯Φ β ] = 0 , (cf. [25]). Viewing ǫ as an indeterminate in ( 3.13), ( 3.14), we obtain a vertexsuperalgebra ˜ C k − over the polynomial ring C [ ǫ ]. By Section 3.2, we obtain a Poissonvertex superalgebra ˜ C k − /ǫ ˜ C k − , which we denote by C cl k ′ = C cl k ′ ( g , f ). We have anisomorphism C cl k ′ ∼ = V k ′ ( g ≤ ) ⊗ F ( g ) ⊗ Sym( C [ ∂ ] g ∗ > )of Poisson vertex superalgebra where V k ′ ( g ≤ ) is the universal affine Poisson ver-tex algebra generated by g ≤ with λ -bracket { u λ v } = [ u, v ] + k ′ ( u | v ) λ , F ( g / ) the βγ -system Poisson vertex algebra generated by Φ α , ( α ∈ ∆ / ), with λ -bracket { Φ αλ Φ β } = ( f | [ e α , e β ]), and Sym( C [ ∂ ] g ∗ > ) the differential C -superalgebra gener-ated by odd elements ϕ α , ( α ∈ ∆ > ), which satisfy { ϕ αλ u } = X β ∈ ∆ > c αu,β ϕ β , { ϕ αλ ϕ β } = { ϕ βλ Φ α } = 0 . Decompose the differential d (0) as d (0) = d st(0) + d ne(0) + d χ (0) where d st = X α ∈ ∆ > e α ϕ α − X α,β ∈ ∆ > γ ∈ ∆ > c γα,β ϕ γ ϕ α ϕ β , d ne = X α ∈ ∆ Φ α ϕ α , d χ = X α ∈ ∆ > χ ( e α ) ϕ α . Then we have[ d st λ ¯ J u ] = − X α ∈ I ⊔ ∆ ≤ β ∈ ∆ > c αu,β ¯ J e α ϕ β + X β ∈ ∆ > ( k ′ ( u | v )( ∂ + λ ) + o ( ǫ )) ϕ β , [ d ne λ ¯ J u ] = X α ∈ ∆ / β ∈ ∆ > c αβ ¯Φ α ϕ β , [ 1 ǫ d χλ ¯ J u ] = X β ∈ ∆ > χ ([ u, e β ]) ϕ β , [ d st λ ϕ α ] = − X β,γ ∈ ∆ > c αβ,γ ϕ β ϕ γ , [ d ne λ ¯Φ α ] = X β ∈ ∆ / χ ([ e β , e α ]) ϕ β , [ d ne λ ϕ α ] = [ d χλ ϕ α ] = [ d st λ ¯Φ α ] = [ 1 ǫ d χλ ¯Φ α ] = 0 . The differential d (0) = { ( d st + d ne + d χ ) λ −}| λ =0 is given by { d st λ −} , { d ne λ −} , { d χλ −} : C cl k ( g , f ) → C cl k ( g , f )[ λ ] , which satisfy { d st λ u } = − X α ∈ I ⊔ ∆ ≤ β ∈ ∆ > c αu,β e α ϕ β + X β ∈ ∆ > k ( u | e β )( ∂ + λ ) ϕ β , { d st λ e α } = 0 , { d st λ ϕ α } = − X β,γ ∈ ∆ > c αβ,γ ϕ β ϕ γ , { d ne λ u } = X α ∈ ∆ / β ∈ ∆ > c αu,β e α ϕ β , { d ne λ e α } = X β ∈ ∆ / ( f | [ e β , e α ]) ϕ β , { d ne λ ϕ α } = 0 , and { d χλ u } = X β ∈ ∆ > ( f | [ u, e β ]) ϕ β , { d χλ ϕ α } = { d χλ e α } = 0 , SHIGENORI NAKATSUKA and are extended to C cl k by ( 2.3), ( 2.6) (see Remark 2.1). The 0-th cohomologyof C cl k ( g , f ), which we denote by W k ( g , f ) = H ( C cl k ( g , f )), is a Poisson vertexalgebra called the classical affine W -algebra associated with ( g , f, k ) ([25]). Notethat H n ( C cl k ( g , f )) = 0 holds for all n = 0.3.3. Screening operators.
Introduce another grading wt on C cl k bywt( u ) = − j, ( u ∈ g j ) , wt(Φ α ) = 0 , ( α ∈ ∆ / ) , wt( ϕ α ) = 2 j ( α ∈ ∆ j ) , wt( ∂A ) = wt( A ) , and wt( AB ) = wt( A ) + wt( B ) . and a decreasing filtration { F p C cl k } p ≥ on C cl k by F p C cl k = span { A ∈ C cl k | wt( A ) ≥ p } , This filtration is exhaustive, separated, and compatible with the grading of C cl k asa complex. The associated spectral sequence { E r , d r } r ≥ has the differentials d = d st(0) , d = d ne(0) , d = d χ (0) , d r = 0 , ( r ≥ , and thus converges at r = 3. We will describe W k ( g , f ) = H ( C cl k ( g , f )) by usingit. Since the calculation is straightforward, we omit the details. (The analogousargument for vertex algebras can be found in [25].)To calculate E = H ∗ ( C cl k ; d ), notice that d acts by 0 on V k ( g ) ⊗ F ( g / ) andthat ( V ( ⊕ α ∈ ∆ > C ϕ α ) , d ) is a subcomplex isomorphic to the Chevalley-Eilenbergcomplex of the Lie algebra g > with coefficients in the trivial representation C .Thus ( V k ( g ) ⊗ F ( g / ) ⊗ V ( ⊕ α ∈ ∆ > C ϕ α ) , d ) is a subcomplex, whose cohomologyis V k ( g ) ⊗ F ( g / ) ⊗ H ∗ ( g > ; C ). Lemma 3.1. (1)
The natural map V k ( g ) ⊗ F ( g / ) ⊗ H ∗ ( g > ; C ) → E is an isomorphismof graded vector spaces for generic k ∈ C . (2) The isomorphism E (0)1 ∼ = V k ( g ) ⊗ F ( g / ) is an isomorphism of Poissonvertex algebras. (3) Each cohomology E ( n )1 is a Poisson vertex module over E (0)1 . Moreover, E ( n )1 is isomorphic to V k ( g ) ⊗ F ( g / ) ⊗ H n ( g > ; C ) as vector spaces. Let us describe the Poisson vertex modules E ( n )1 more explicitly. Recall that thecoadjoint representation on g ∗ > of g induces a representation of g on H n ( g > , C )as described as follows. (cf. [29, Chapter 3]) Let W denote the Weyl group of g and set I = { i ∈ I | α i ∈ ∆ } , W ′ = { w ∈ W | w ∆ +0 ⊂ ∆ + } . Let ∗ : W × h ∗ → h ∗ denote the shifted action of W and l : W → Z ≥ the length function. Then thereis an isomorphism of g -modules H n ( g > ; C ) ∼ = M w ∈ W ′ l ( w )= n L ( w − ∗ , where L ( w − ∗
0) is the integrable highest weight g -module with highest weight w − ∗ g -module M , set M k = V k ( g ) ⊗ C M . The space M k has a uniquePoisson vertex module over M k such that V k ( g ) acts as a commutative algebra bymultiplication on the first component and the λ -bracket {− λ −} : V k ( g ) ⊗ M k → M k [ λ ] satisfies { a λ b ⊗ m } = { a λ b }⊗ m + b ⊗ a · m for a ∈ g , b ∈ V k ( g ), and m ∈ M .We denote by L k ( w − ∗
0) the Poisson vertex module obtained from L ( w − ∗ Lemma 3.2.
There is an isomorphism E ( n )1 ∼ = M w ∈ W ′ l ( w )= n L k ( w − ∗ ⊗ F ( g / ) , -ALGEBRAS AND INTEGRABLE SYSTEMS 9 as Poisson vertex modules over V k ( g ) ⊗ F ( g / ) . In particular, we have E (1)1 ∼ = L i ∈ I \ I L k ( − α i ) ⊗ F ( g / ) and the subspace L ( − α i ) is identified as L ( − α i ) ∼ = M β ∈ [ α i ] C ϕ β ⊂ E (1)1 , where [ α i ] = ∆ > ∩ ( α i + Q ) and Q denotes the root lattice of ∆ . For α ∈ [ α i ]with i ∈ I \ I , we have(3.15) ∂ϕ α = 1 k X β ∈ [ α ] ,γ ∈ I ⊔ ∆ c αβ,γ e ¯ γ ϕ β . Let us describe the differentials on E induced from d ne(0) and d χ (0) . Consider theintertwining operators Q Wi : V k ( g ) ⊗ F ( g / ) → L k ( − α i ) ⊗ F ( g / ) given by Q Wi = ( P β ∈ [ α i ] { Φ β ϕ βλ −}| λ =0 , ( i ∈ I / ) , P β ∈ [ α i ] { ( f | e β ) ϕ βλ −}| λ =0 , ( i ∈ I ) . Then we have:
Lemma 3.3.
The differentials on E induced by d ne(0) and d χ (0) are given by d ne(0) = X i ∈ I / Q Wi and d χ (0) = X i ∈ I Q Wi . Recall that the complex C cl k = C cl k ( g , f ) is Z ≥ -graded and that H n ( C cl k ( g , f ) ∼ = δ n, W k ( g , f ). Then we see that W k ( g , f ) is a subalgebra of the 0-th degree C cl k, = V k ( g ≤ ) ⊗ F ( g / ). Since I = V k ( g < ) ⊗ F ( g / ) is a Poisson vertex ideal of C cl k, ,we obtain a homomorphism of Poisson vertex algebras W k ( g , f ) → C cl k, / I ∼ = V k ( g ) ⊗ F ( g / ) . It is injective and, by using the differentials d ne(0) and d χ (0) , the image is describedas in the following theorem. Theorem 3.4.
For generic k ∈ C , there is an isomorphism j : W k ( g , f ) ∼ = \ i ∈ I / ⊔ I Ker (cid:16) Q Wi : V k ( g ) ⊗ F ( g / ) → L k ( − α i ) ⊗ F ( g / ) (cid:17) , of Poisson vertex algebras. The operators { Q Wi } i ∈ I / ⊔ I are called the screening operators for W k ( g , f ) . Wenote that the level k = 1 is generic [25]. The inclusion j : W k ( g , f ) → V k ( g ) ⊗ F ( g / ) in Theorem 3.4 induces a Lie algebra homomorphism between their spacesof local functionals j ∗ : Lie (cid:0) W k ( g , f ) (cid:1) → Lie (cid:0) V k ( g ) ⊗ F ( g / ) (cid:1) . (See Proposition 2.4.) Lemma 3.5.
The Lie algebra homomorphism j ∗ is injective.Proof. It is easy to seeKer (cid:16) ∂ : L k ( − α i ) ⊗ F ( g / ) → L k ( − α i ) ⊗ F ( g / ) (cid:17) = 0 , ( i ∈ I / ⊔ I ) . Take an element f ∈ W k ( g , f ) such that R f ∈ Ker j ∗ . Then there exists an element G ∈ V k ( g ) such that j ( f ) = ∂g . Let H Wi denote the Hamiltonian of Q Wi . Thenwe have 0 = Q Wi j ( f ) = { H Wiλ j ( f ) }| λ =0 = { H Wiλ ∂g }| λ =0 = ∂ { H Wiλ g }| λ =0 , and so that { H Wiλ g }| λ =0 ∈ Ker( ∂ : L k ( α i ) ⊗ F ( g / )) → L k ( α i ) ⊗ F ( g / )) = 0.Therefore, we obtain g ∈ j ( W k ( g , f )) and so that R f = 0. (cid:3) Let Q Wα : V k ( g ) ⊗ F ( g / ) → V k ( g ) ⊗ F ( g / ), ( α ∈ Π), be the derivationdetermined by(3.16) Q Wα e β = P γ ∈ [ α ] c γβα Φ γ , Q Wα Φ β = ( f | [ e α , e β ]) , ( α ∈ Π / ) ,Q Wα e β = ( f | [ e β , e α ]) , Q Wα Φ β = 0 , ( α ∈ Π ) , (3.17) [ Q Wα , ∂ ] = 1 k X β ∈ I ⊔ ∆ γ ∈ [ α ] c γα,β e ¯ β Q Wγ . Theorem 3.6.
For generic k ∈ C , the classical affine W -algebra W k ( g , f ) is iso-morphic to the Poisson vertex subalgebra (3.18) W k ( g , f ) ∼ = \ α ∈ Π Ker (cid:16) Q Wα : V k ( g ) ⊗ F ( g / ) → V k ( g ) ⊗ F ( g / ) (cid:17) , of V k ( g ) ⊗ F ( g / ) invariant under the derivations Q Wα , ( α ∈ Π) . We call the level k ∈ C generic when ( 3.18) holds. Proof.
Since Q Wi : V k ( g ) ⊗ F ( g / ) → L k ( − α i ) ⊗ F ( g / ) acts by derivation, itdecomposes as Q Wi = P α ∈ [ α i ] ϕ α Q Wα , where Q Wα ∈ Der( V k ( g ) ⊗ F ( g / )) and ϕ α is the multiplication by ϕ α . We check that Q Wα satisfies ( 3.17) and ( 3.16).By direct calculation, ( 3.16) follows from the definition of Q Wi . To show ( 3.17),recall that V k ( g ) has a Virasoro element L = k P α ∈ I ⊔ ∆ e α e ¯ α , i.e., it satisfies { L λ L } = ( λ + ∂ ) L + c k L for some c k ∈ C and ∂ = L (0) . Then we have [ Q Wi , ∂ ] =[ { Q Wi,λ −}| λ =0 , { L µ −}| µ =0 ] = {{ Q Wiλ L } µ −}| λ = µ =0 = {− ∂Q Wiλ −}| λ =0 = 0. Here, wehave used ( 2.5) in the second, ( 2.4)in the third, and ( 2.3) in the last equality.Therefore, 0 = P α ∈ [ α i ] ϕ α [ Q Wα , ∂ ] − ( ∂ϕ α ) Q Wα . Now ( 3.17) follows from this by( 3.15). (cid:3) Geometric Realization of W k ( g , f )4.1. Double coset space.
Consider the Lie algebra g [ t ± ] = g ⊗ C [ t ± ]. By abuseof notation, we denote by κ = ( −|− ) the invariant bilinear form on g [ t ± ] givenby ( at n | bt m ) = ( a | b ) δ n + m, , which extends the one on g . We extend the gradingΓ on g to g [ t ± ] by setting deg( at n ) = deg( a ) + ( d + 1) n (see Section 3.1) and fixa homogeneous basis e α , ( α ∈ I ⊔ ˆ∆), extending the basis e α , ( α ∈ I ⊔ ∆), of g . We denote | α | = deg( e α ) for simplicity and c γα,β the structure constants, i.e.,[ e α , e β ] = P γ c γα,β e γ . Let e ¯ α , ( α ∈ I ⊔ ˆ∆), denote the dual basis of e α , ( α ∈ I ⊔ ˆ∆),with respect to κ .Consider the completion L g = lim ←− n> ( g [ t ± ] / g [ t ± ] >n ), which we call the loopalgebra of g . The grading on g [ t ± ] induces a decomposition L g = L g + ⊕ L g − where L g + = lim ←− n> ( g [ t ± ] > / g [ t ± ] >n ) and L g − = g [ t ± ] ≤ . We decompose X ∈ L g as X = X + + X − ∈ L g + ⊕ L g − . The subspace L g + is an affine scheme of infinite type.By setting z α the coordinates of the basis e α , ( α ∈ ˆ∆ > ⊂ ˆ∆), of L g + , we have C [ L g + ] = C [ z α | α ∈ ˆ∆ > ].Let G be the connected simply-connected algebraic group whose Lie algebra is g and LG the loop group of G , whose Lie algebra is L g . We have closed subgroups LG ± of LG whose Lie algebra is L g ± .Let us consider the quotient space LG − \ LG , which is an ind-scheme. It has anopen subscheme LG + . The C -points of LG + is identified with the LG + -orbits ofthe image of the identity in LG − \ LG . Since LG + is a prounipotent proalgebraic -ALGEBRAS AND INTEGRABLE SYSTEMS 11 group, the exponential map exp : L g + → LG + is an isomorphism of schemes. Thusthe coordinate ring C [ LG + ] is identified with C [ L g + ] = C [ z α | α ∈ ˆ∆ > ].The left multiplication LG + × LG + → LG + , (( g , g ) g − g ) induces a L g + -action on C [ LG + ] as derivations(4.19) ξ L : L g + → Der( C [ LG + ]) , φ φ L . Since LG + is embedded into LG − \ LG as an open subscheme, the right multiplica-tion LG − \ LG × LG − → LG − \ LG , (([ g ] , g ) [ g g ]), induces a L g − -action on C [ LG + ] as derivations(4.20) ξ R : L g → Der( C [ LG + ]) , φ φ R . The commutator of these actions ξ L , ξ R can be expressed by using the distinguishedelement K = exp( P α ∈ ˆ∆ > z α e α ) ∈ LG + ( C [ LG + ]). Note that this element is co-ordinate independent since it is identified with the identity morphism under thecorrespondence LG + ( C [ LG + ]) = Hom( C [ LG + ] , C [ LG + ]) . Lemma 4.1.
For u ∈ L g + and v ∈ L g , [ u L , v R ] = [ u, ( KvK − ) − ] L + holds.Proof. We use the notation e X = exp( X ) for brevity. Let ǫ i be dual numbers, i.e.,satisfies ǫ i = 0. For F ∈ C [ LG + ] and g ∈ LG + , we have( v R F )( g ) = the ǫ -linear term of F ( g e ǫ v )= the ǫ -linear term of F ((e ( gvg − ) + ) g ) , and ( u L v R F )( g ) = the ǫ -linear term of ( v R F )(e − ǫ u g )= the ǫ ǫ -linear term of F ((e (e − ǫ u gvg − e ǫ u ) + ) e − ǫ u g )= the ǫ ǫ -linear term of F ( g ( ǫ , ǫ )) , where g ( ǫ , ǫ ) = e − uǫ +( gvg − ) + ǫ − ([ u,gvg − ] + +( gvg − ) + u ) ǫ ǫ g. Similarly, we obtain( v R u L )( F )( g ) = the ǫ ǫ -linear term of F ( g ( ǫ , ǫ ))where g ( ǫ , ǫ ) = e − uǫ +( gvg − ) + ǫ − u ( gvg − ) + ǫ ǫ g. Since g ( ǫ , ǫ ) = e − [ u, ( gvg − ) − ] + ǫ ǫ g ( ǫ , ǫ ) , we obtain [ u L , v R ] F ( g ) = [ u, ( gvg − ) − ] L + F ( g ). (cid:3) In the sequel, we assume that ( g , f ) satisfies the condition (F) introduced inSection 1:(F1) The grading Γ is a Z -grading.(F2) There exists an element y ∈ g d such that s = f + yt − ∈ g [ t ± ] is semisimple.(F3) The Lie subalgebra Ker(ad s ) ⊂ g [ t ± ] is abelian and Im(ad s ) ∩ g [ t ± ] = g . Example 4.2.
The following pairs ( g , f ) satisfy (F).(1) ( g , f ) with f principal nilpotent element.(2) ( g ( C n ) , f (2 n ) ) and ( g ( C n ) , f (4 n ) ), ( n ≥ g ( X n ) , f ) with X = E, F, G listed in Table 4.1.
Table 4.1Dynkin diagram weights of vertices Dynkin diagram weights of vertices G E ❝ ❝ ❍✟ E F ❝ ❝ ❝ ❝ ❝ ❝❝ ❝ ❝ ❝ ❝ ❍✟ E ❝ ❝ ❝ ❝ ❝ ❝ ❝❝ E ❝ ❝ ❝ ❝ ❝❝ g ( C n ) by symplecticpartitions and that of g ( X n ), ( X = E, F, G ), by the weighted Dynkin diagram, (see[5].)Let a denote the completion of Ker(ad s ) in L g . It is abelian by (F2). Let A ⊂ LG , (resp. A ± ⊂ LG ) be the closed subgroup of LG whose Lie algebra is a ,(resp. a ± = a ∩ L g ± ). We consider the quotient space LG + /A + . It admits a left LG + -action LG + × LG + /A + → LG + /A + , ( g, hA + ) g − hA + , and a right A − -action LG + /A + × A − → LG − \ LG/A + , ( hA + , g ) hgA + . The right A − -action is well-defined since A is abelian. These actions induce infini-tesimal actions ξ L : L g + → Der C [ LG + /A + ], ( φ φ L ) and ξ R : a − → C [ LG + /A + ],( φ φ R ) respectively. In particular, C [ LG + /A L +] becomes a differential algebraby letting s R be the differential.Set(4.21) E α = k ( e α | KsK − ) ∈ C [ LG + ] , ( α ∈ I ⊔ ∆ ) . We have E α ∈ C [ LG + /A + ] since a R ( e α | KsK − ) = the ǫ -linear term of ( e α | K e ǫa s ( K e ǫa ) − )= ( e α | K [ a, s ] K − ) = 0 , for a ∈ a + . Since the Poisson vertex algebra V k ( g ) is an algebra of differentialpolynomials, there exists a unique homomorphism of differential algebrasΨ k : V k ( g ) → C [ LG + /A + ] , e α E α . Theorem 4.3.
Suppose that ( g , f ) satisfies (F) and k ∈ C is generic. Then Ψ k isan isomorphism of differential algebras and satisfies (4.22) Ψ k Q Wα Ψ − k = − k e Lα , α ∈ Π . Proof.
Since V k ( g ) and C [ LG + /A + ] are polynomial rings, it suffices to show thatthe linear map d Ψ k : T ∗ Spec V k ( g ) → T ∗ [ e ] LG + /A + between the cotangent spacesis an isomorphism. By the identification L g ∗ + ∼ = L g < and a ∗ + ∼ = a − induced by κ ,we obtain T ∗ [ e ] LG + /A + ∼ = ( L g + / a + ) ∗ ∼ = L g < / a − . This isomorphism is given by -ALGEBRAS AND INTEGRABLE SYSTEMS 13 dz α e ¯ α . Under this isomorphism, we have d Ψ k ( ∂ n e α ) = k ad n +1 s e α , as provedby inductive use of the n = 0 case: d Ψ k ( e α ) = kd ( e α | KsK − ) = kd ( e α | [ X | β | =1 z β e β , s ])= X | β | =1 k ([ s, e α ] | e β ) dz β = k X | β | =1 ([ s, e α ] | e β ) e ¯ β = k [ s, e α ] . It follows from (F2) and (F3) that Ψ k is an isomorphism.To show ( 4.22), it suffices to show u Lα E β = ( f | [ e β , e α ]) , [ u Lα , s R ] = 1 k X ρ ∈ I ⊔ ∆ γ ∈ [ α ] c γα,ρ E ¯ ρ u Lγ , α ∈ Π , β ∈ I ⊔ ∆ , where u α = − k e α by ( 3.17), ( 3.16). For the first one, we have u Lα E β = the ǫ -linear term of k ( e β | e − ǫe α /k KsK − e ǫe α /k )= k ( e β | [ − k e α , KsK − ]) = ([ e β , e α ] | KsK − )= ([ e β , e α ] | s ) = ([ e β , e α ] | f ) . The second one follows from Lemma 4.1 since[ u Lα , s R ] = [ u α , ( KsK − ) ] L + = − k X ρ ∈ I ⊔ ∆ γ ∈ [ α ] c γα,ρ E ¯ ρ e Lγ = 1 k X ρ ∈ I ⊔ ∆ γ ∈ [ α ] c γα,ρ E ¯ ρ u Lγ . (cid:3) Note that the Lie subalgebra g + = ⊕ j> g j is generated by the subspace ⊕ α ∈ Π g α .Let G + ⊂ LG + denote the closed subgroup whose Lie algebra is g + . Corollary 4.4.
The isomorphism Ψ k restricts to an isomorphism of differentialalgebras W k ( g , f ) ∼ = C [ G + \ LG + /A + ] .Proof. The claim follows from Theorem 4.3 since W k ( g , f ) ∼ = \ α ∈ Π Ker (cid:0) Q Wα : V k ( g ) → V k ( g ) (cid:1) , by Theorem 3.6 and (F1). (cid:3) Mutually commutative derivations on W k ( g , f ) . By (F3), we have a Liealgebra homomorphism ξ R : a − → Der( C [ LG + /A + ]) , ( a a R ). Proposition 4.5.
The action ξ R preserves C [ G + \ LG + /A + ] ⊂ C [ LG + /A + ] . Proof.
By Lemma 4.1, we have, for e α ∈ g , ( α ∈ Π ), and a ∈ a − ,[ e Lα , a R ] = [ e α , ( KaK − ) − ] L + = [ e α , ( KaK − ) ] L = X β ∈ I ⊔ ∆ F β [ e α , e β ] L , for some polynomials F β ∈ C [ LG + /A + ]. Hence, for G ∈ C [ G + \ LG + /A + ], we have e Lα ( a R G ) = [ e Lα , a R ] G = X β ∈ I ⊔ ∆ F β [ e α , e β ] L G = 0. (cid:3) Since a − is abelian, its image ξ R ( a − ) ⊂ Der C [ G + \ LG + /A + ] is also abelian. ByTheorem 4.3, ξ R ( a − ) is identified with a set of commutative derivations of W k ( g , f ),which we denote by H k ( g , f ). In the next section, we will prove that H k ( g , f ) isan integrable Hamiltonian hierarchy associated with the classical affine W -algebra W k ( g , f ). H k ( g , f ) as an integrable Hamiltonian hierarchy In this section, we always assume that the condition (F) holds and k ∈ C isgeneric, and identify V k ( g ) with C [ LG + /A + ] by Ψ k , (Theorem 4.3).5.1. Construction of Hamiltonians.
Let Ω dR ( LG + ) = C [ LG + ] ⊗ V L g ∗ + denotethe algebraic de Rham complex of LG + . Here L g ∗ + is the vector space dual to thespace L g + of the right invariant vector fields L g L + on LG + . We take a basis ϕ α , ( α ∈ ˆ∆ > ), of L g ∗ + so that ϕ α ( e β ) = δ α,β , ( α, β ∈ ˆ∆ > ), holds. Note that the complexΩ dR ( LG + ) coincides with the Chevalley-Eilenberg complex of the Lie algebra L g + with coefficients in ( C [ LG + ] , ξ L ). Similarly, let Ω dR ( A + ) = C [ A + ] ⊗ V a ∗ + denotethe algebraic de Rham complex of A + . Then the inclusion A + ֒ → LG + induces theprojection Ω dR ( LG + ) → Ω dR ( A + ). It restricts to a R + -invariant subcomplexes:(5.23) π : C [ LG + /A + ] ⊗ ^ L g ∗ + → C ⊗ ^ a ∗ + . As a L g + -module, C [ LG + /A + ] is isomorphic to the L g + -module Coind L g + a + C coin-duced from the trivial a + -module C . Then π induces an isomorphism H ∗ ( L g + ; C [ LG + /A + ]) ∼ = H ∗ (cid:0) L g + ; Coind L g + a + C (cid:1) ∼ = H ∗ ( a + ; C )by Shapiro’s lemma, (cf. [24]). The right hand side is isomorphic to V a ∗ + since a + is abelian. The action ξ R of a − induces a − -actions on the complexes Ω dR ( LG + ),Ω dR ( A + ), which commute with π . Since a is abelian, a − acts on Ω dR ( A + ) trivially.Thus a − acts on the cohomology H ∗ ( L g + ; C [ LG + /A + ]) trivially. In particular, s acts on H ∗ ( L g + ; C [ LG + /A + ]) trivially. By abuse of notation, we write the above a − -actions by ξ R .Consider the double complex C C ι −−−−→ Ω dR ( LG + ) ± s R −−−−→ Ω dR ( LG + ) ǫ −−−−→ C . Here ι is the unit morphism and ǫ the counit morphism. It has the following shape: C (0 , ǫ x ( − , −−−−→ C [ LG + /A + ] (0 , d −−−−→ C [ LG + /A + ] ⊗ L g ∗ > , d −−−−→ · · · ∂ = s R x − s R x ( − , −−−−→ C [ LG + /A + ] (0 , d −−−−→ C [ LG + /A + ] ⊗ L g > , d −−−−→ · · · , ι x C (0 , − where the subscript ( i, j ) denotes the bidegree of C . Then the calculation of thecohomology H ∗ ( C ) via spectral sequences gives the isomorphisms(5.24) a ∗ + ∼ = H ( C ) ∼ = Ker (cid:16) d : C [ LG + /A + ] C ⊕ Im( s R ) → C [ LG + /A + ] ⊗ L g ∗ + Im( s R ) (cid:17) . By Theorem 4.3, we have an isomorphism C [ LG + /A + ] / C ⊕ Im( s R ) ∼ = Lie( V k ( g )) / C as vector spaces. Since V k ( g ) is an algebra of differential polynomials, R : C → Lie (cid:0) V k ( g ) (cid:1) is injective. Thus we may lift an element R f ∈ Lie( V k ( g )) / C to R ˜ f ∈ Lie( V k ( g )) / C which is without the constant term ˜ f (0) = 0. Identifying a − with a ∗ + by κ , we obtain an isomorphism a − ∼ = a ∗ + ∼ = (cid:26)Z f ∈ Lie( V k ( g )) | f (0) = 0 , d Z f = 0 (cid:27) , a Z H ( a ) . -ALGEBRAS AND INTEGRABLE SYSTEMS 15 The statement of the following proposition makes sense by Lemma 3.5.
Proposition 5.1.
The image R H ( a − ) lies in Lie( W k ( g , f )) . Proof.
We may assume that the basis { e α } α ∈ ˆ∆ > of L g + respects the decomposi-tion L g + = Im(ad s ) ⊕ Ker(ad s ). For a ∈ a − , the element R H ( a ) ∈ Lie( V k ( g ))satisfies, by construction, dH ( a ) = s R A for some element A = P α ∈ ˆ∆ > F α ⊗ ϕ α ∈ C [ LG + /A + ] ⊗ L g ∗ + . Then, dH ( a ) = s R A = s R X α ∈ ˆ∆ > F α ⊗ ϕ α = X α ∈ ˆ∆ > ∂ ( F α ) ⊗ ϕ α + X α ∈ ˆ∆ > F α ⊗ s R ( ϕ α )= X α ∈ Π ∂F α + X β ∈ Π γ ∈⊔ ∆ F β c βα,γ E ¯ γ ⊗ ϕ α + X α ∈ ˆ∆ > \ Π f F nα ⊗ ϕ α , for some element f F nα ∈ C [ LG + /A + ]. In the last equality, we have used Lemma 5.2below. By the identification C [ LG + /A + ] ∼ = V k ( g ) and C [ LG + /A + ] ϕ α ∼ = L k ( − α ),( α ∈ Π ), we obtain X α ∈ Π ∂F α + X β ∈ Π γ ∈ I ⊔ ∆ F β c βα,γ E ¯ γ ⊗ ϕ α = ∂ X α ∈ Π F α ⊗ ϕ α ! = 0 ∈ M α ∈ Π Lie (cid:0) L k ( − α ) (cid:1) . On the other hand, by the definition of d , we have dH ( a ) = P α ∈ ˆ∆ > e Lα H ( a ) ⊗ ϕ α .It follows that e Lα R H ( a ) = 0 for all α ∈ Π , which implies R H ( a ) ∈ Lie( W k ( g , f ))by Theorem 4.3. (cid:3) Lemma 5.2.
The following formula holds (5.25) s R ϕ α = 1 k X β ∈ ˆ∆ > γ ∈ I ⊔ ∆ E ¯ γ c αβ,γ ϕ β + X β ∈ ˆ∆ > c αβ,s ϕ β , α ∈ ˆ∆ > . Proof.
By Lemma 4.1, we have[ s R , e Lα ] = − [ e α , ( KsK − ) − ] L + = − k X β ∈ I ⊔ ∆ E ¯ β [ e α , e β ] L − [ e α , s ] L + . (5.26)Then, from the definition of L g ∗ + , we have (cid:0) s R ϕ α (cid:1) ( e β ) = − ϕ α ( ∂e β ) = 1 k X γ ∈ I ⊔ ∆ E ¯ γ c αβ,γ + c αβ,s . The claim ( 5.25) follows from it immediately. (cid:3)
Poisson vertex superalgebra structure on C [ LG + /A + ] ⊗ V L g ∗ + . We ex-tend the Poisson vertex algebra structure on C [ LG + /A + ] given by Theorem 4.3, tothe whole differential superalgebra ( C [ LG + /A + ] ⊗ V L g ∗ + , s R ). Proposition 5.3.
The differential superalgebra ( C [ LG + /A + ] ⊗ V L g ∗ + , s R ) admitsa unique Poisson vertex superalgebra structure, which satisfies (1) { u λ v } = [ u, v ] + k ( u | v ) λ, ( u, v ∈ g ) , (2) { ϕ αλ ϕ β } = 0 , ( ϕ α , ϕ β ∈ L g ∗ ∪ a ∗ + ) , (3) { u λ ϕ α } = − P c αu,β ϕ β , ( ϕ α ∈ L g ∗ ∪ a ∗ + , u ∈ g ) . It satisfies { u λ ϕ α } = − X c αu,β ϕ β , ( ϕ α ∈ L g ∗ , u ∈ g ) . (5.27) Proof.
It follows from Lemma 5.2 that C [ LG + /A + ] ⊗ V L g ∗ + is an algebra of differ-ential polynomials in the variables given by the union of bases of g and L g ∗ ∪ a ∗ + .By Theorem 2.2, it suffices to show ( 2.4), ( 2.5) for these variables in order to provethat (1)-(3) define a Poisson vertex superalgebra on C [ LG + /A + ] ⊗ V L g ∗ + . The onlynon-trivial ones are ( 2.5) for { e α , e β , ϕ γ } . By ( 2.4), they reduce to the special case(5.28) { e αλ { e βµ ϕ γ }} − { e βµ { e αλ ϕ γ }} = {{ e αλ e β } λ + µ ϕ γ } . Since it coincides with the Jacobi identity (ad * ( e α ) ad * ( e β ) − ad * ( e β ) ad * ( e α )) ϕ γ =ad * ([ e α , e β ]) ϕ γ of the coadjoint g -action on L g ∗ + , ( 5.28) holds.Next, we show ( 5.27). Since ad s is an isomorphism on Im(ad s ) ⊂ L g , we haveits inverse ad − s on Im(ad s ) and on its dual space Im(ad s ) ∗ . Then we have adecomposition L g > = ad − s Im(ad s ) > ⊕ ( L g ∪ a + ). We show ( 5.27) for ϕ α ∈ Im(ad s ) by induction on degree. From Lemma 5.2, we havead − s ϕ a = ∂ϕ a − k X α,b c ab,α e ¯ α ϕ b , ϕ α ∈ Im(ad s ) . Here the sum with respect to Greek letters (resp. Roman letters) is taken over I ⊔ ∆ (resp. ˆ∆ > ). We use the same rule below. Then, for e α ∈ g and ϕ a ∈ Im(ad s ) ,we have { e αλ (ad − s ) n ϕ a } = − X | b | = | a | c aα,b (ad − s ) n ϕ b , which is proved by the inductive use of the n = 1 case: { e αλ ad − s ϕ a } = { e α,λ ∂ϕ a } − k X β,b c ab,β { e αλ e ¯ β ϕ b } = ( ∂ + λ ) { e α,λ ∂ϕ a } − k X β,b c ab,β ( { e αλ e ¯ β } ϕ b + { e αλ ϕ b } e ¯ β )= − ( ∂ + λ ) X b c aαb ϕ b − k X α,b c ab,α ([ e α , e ¯ β ] + k ( e α | e ¯ β ) λ ) ϕ b − X c c bα,c e ¯ β ϕ c ! = − λ X b c aα,b ϕ b + X β,b c ab,β ( e α | e ¯ β ) ϕ b − X b c aα,b ∂ϕ b + 1 k X β,b c ab,β (cid:0) [ e α , e ¯ β ] ϕ b − X c c bα,c e ¯ β ϕ c (cid:1) = − X b c aα,b ad − s ϕ b . In the last equality, we used ( e α | e ¯ β ) = δ α,β for the λ -linear term and use Lemma5.2, c ca,b = c ¯ b ¯ c,a , and the Jacobi identity of the Lie bracket for the constant term.This implies ( 5.27) for ϕ α ∈ Im(ad s ) ∗ . Then ( 5.27) follows from this and (3). (cid:3) The de Rham differential d = P α ∈ ˆ∆ > e Lα ⊗ ϕ α on C [ LG + /A + ] can be describedin term of the Poisson vertex superalgebra structure in Proposition 5.3. Lemma 5.4.
The differential d is determined uniquely by(i) dE β = − k P α ∈ ˆ∆ > ([ e β , s ] | e α ) ⊗ ϕ α , ( β ∈ I ⊔ ∆ ) ,(ii) [ d, ∂ ] = 0 . -ALGEBRAS AND INTEGRABLE SYSTEMS 17 Proof. (1) follows immediately from the formula e Lβ E α = − k ([ e α , s ] | e β ) (see also( 4.21)). (2) follows from ( 5.26) and Lemma 5.2. (cid:3) Let s ∗ ∈ L g ∗ + denote the element corresponding to s ∈ L g − by the identification κ = ( −|− ) : L g − ∼ = L g ∗ + and ¯ s ∈ L g + the element corresponding to s ∗ . Proposition 5.5.
Under the isomorphism Ψ k : V k ( g ) ∼ = C [ LG + /A + ] , d = − k { s ∗ λ −}| λ =0 , holds on C [ LG + /A + ] . Proof.
It suffices to show that − k { s ∗ λ −}| λ =0 satisfies (i), (ii) in Lemma 5.4. (i)follows from the definition of s ∗ and Proposition 5.3, (3). (ii) follows from ( 2.3). (cid:3) The following property of the derivation η a = η ( R H ( a )) ∈ Der( C [ LG + /A + ]),( a ∈ a − ), will be used. Lemma 5.6.
For a ∈ a − , there exist some polynomials F α ( a ) ∈ C [ LG + /A + ] ,( α ∈ I ⊔ ∆ ), which satisfies [ u L , η a ] = X α ∈ I ⊔ ∆ F α ( a )[ u, e α ] L , u ∈ L g . Proof.
By construction of R H ( a ), we have d R H ( a ) = R dH ( a ) = 0 ∈ Lie( C [ LG + /A + ] ⊗ V L g ∗ + ) and thus { dH ( a ) λ −}| λ =0 = 0. By ( 2.5), we have { dH ( a ) λ −}| λ =0 = h { s ∗ λ −}| λ =0 , { H ( a ) µ −}| µ =0 i = h X α ∈ ˆ∆ > e Lα ⊗ ϕ α , η a i = X α ∈ ˆ∆ > (cid:16) [ e Lα , η a ] ⊗ ϕ α − e Lα ⊗ η a ( ϕ α ) (cid:17) . It follows from them that[ e Lα , η a ] = X β ∈ ˆ∆ > h e α | η a ( ϕ β ) i e Lβ , where h− | −i : L g + × L g ∗ + → C denotes the canonical pairing. By Proposition 5.3,we have η a ( ϕ β ) = { H ( a ) λ ϕ β }| λ =0 = X γ ∈ I ⊔ ∆ { e γ∂ ϕ β } → δH ( a ) δe γ = − X γ ∈ I ⊔ ∆ ρ ∈ ˆ∆ > c βγ,ρ ϕ ρ δH ( a ) δe γ . Hence, for α ∈ L g we obtain[ e Lα , η a ] = X β ∈ ˆ∆ > h e α | η a ( ϕ β ) i e Lβ = − X β ∈ ˆ∆ > X γ ∈ I ⊔ ∆ ρ ∈ ˆ∆ > c βγ,ρ δH ( a ) δe γ h e α | ϕ ρ i e Lβ = X γ ∈ I ⊔ ∆ δH ( a ) δe γ [ e α , e γ ] L . This completes the proof. (cid:3)
Integrable Hamiltonian hierarchy.
Let Vect( LG + ) denote the Lie algebraof the algebraic vector fields on LG + . Recall the infinitesimal L g + -actions ( 4.19),( 4.20). Lemma 5.7.
We have
Vect( LG + ) L g L + = L g R + and Vect( LG + ) L g R + = L g L + . Proof.
Although this is well-know, we give a proof for the completeness of the paper.The pairing U ( L g + ) × C [ LG + ] → C , ( X X · · · X n , F ) ( X L X L · · · X Ln F )( e ) , defined by is L g + -invariant and nondegenerate. Here U ( L g + ) denotes the universalenveloping algebra of L g + . Hence we have C [ LG + ] L g L + ∼ = U ( L g + ) /L g + U ( L g + ) ∼ = C .Since L g L + commutes with L g R + and Vect( LG + ) ∼ = C [ LG + ] ⊗ C L g R + , we obtainVect( LG + ) L g L + ∼ = ( C [ LG + ] ⊗ C L g R + ) L g L + = C [ LG + ] L g L + ⊗ C L g R + ∼ = L g R + . The latter claim is proved similarly. (cid:3)
We say that an element X ∈ Vect( LG + ) satisfies (P) if[ u L , X ] ∈ X α ∈ I ⊔ ∆ C [ LG + ][ u, e α ] L , u ∈ L g . Let L denote the set of elements in Vect( LG + ) satisfying (P). It is straightforward toshow that L ⊂
Vect( LG + ) form a Lie subalgebra. The Witt algebra Witt = C (( t )) ∂∂t acts on L g by derivations with respect to t . Since the subalgebra Witt − = C [ t − ] t ∂∂t preserves L g ≤ , it acts on B \ G (( t )) infinitesimally and therefore on the open densesubset LG + . The induced Witt − -action on C [ LG + ] is given by L n exp X α ∈ ˆ∆ > z α e α = the ǫ -linear term of exp X α ∈ ˆ∆ > z α e α e ǫL n , where L n = − t n +1 ∂∂t ∈ Witt − . Lemma 5.8.
We have L = Witt − ⋉ L g R . Proof.
By Lemma 4.1, we have L g R ⊂ L . In particular, this implies that L is a L g -module. The inclusion Witt − ⊂ L is shown in the same way as in the proof ofLemma 4.1. It is obvious that these two actions give an action of their semidirectproduct Witt − ⋉ L g R ⊂ L .To show its equality, consider the quotient L g -module M := L / Witt − ⋉ L g R .Let us show that M belongs to the category O of the affine Lie algebra e g = g [ t ± ] ⊕ C C ⊕ C L at level 0. Let e g = e n + ⊕ e h ⊕ e n − be the standard triangulardecomposition of e g . Indeed, the action of the Cartan subalgebra e h = h ⊕ C C ⊕ L onVect( LG + ) is diagonalizable and, moreover, preserves L and Witt − ⋉ L g R . Henceit is diagonalizable on M = L / Witt − ⋉ L g R . Let us show that the dimensiondim U ( e n + ) w is finite for any w ∈ M . Let ˜ w ∈ L denote a lift of w . The decom-position e n + = g +0 ⋉ e n > where g +0 = L α ∈ ∆ g α and e n > = e n + ∩ L g > inducesa decomposition of the universal enveloping algebra U ( e n + ) ∼ = U ( g +0 ) ⊗ U ( e n > ) asvector spaces. By definition of L , ˜ w satisfies, for any u ∈ L g ,[ u L , w ] = X α ∈ I ⊔ ∆ F α ( w )[ e α , u ] L for some element F α ( w ) ∈ C [ LG + ] . Since e n L> and e n R> commute, for any Z ∈ U ( e n > ),[ u L , Z R ( w )] = Z R ([ u L , w ]) = X α ∈ I ⊔ ∆ Z R (cid:0) F α ( w ) (cid:1) [ e α , u ] L . If Z R ( F α ( w )) = 0 for all α ∈ I ⊔ ∆ and u ∈ L g , then by Lemma 5.7, we have[ Z R , w ] ∈ e n R> , which is zero in M . Considering the e h -weights of Z R ( F α ( w )), weconclude that dim U ( e n > ) R ( F α ( w )) < ∞ for each α and hence dim U ( e n > ) R w < ∞ .Thus it remains to show dim U ( g +0 ) w < ∞ . Take u ∈ e g and v ∈ g +0 , we have -ALGEBRAS AND INTEGRABLE SYSTEMS 19 [ u L , v R ] = [ u, v ] L by Lemma 4.1. Then by setting [ u L , w ] = P α ∈ I ⊔ ∆ G α ( w )[ u, e α ] L for some element G α ∈ C [ LG + ], we have[ u L , [ v R , w ]] = [[ u, v ] L , w ] + [ v R , [ u L , w ]]= X α ∈ I ⊔ ∆ G α ( w )[[ u, v ] , e α ] L + X α ∈ I ⊔ ∆ v R ( G α ( w ))[ u, e α ] L . By induction, we obtain[ u L , [ v R , [ v R , · · · , [ v Rn , w ] · · · ] = X α ∈ I ⊔ ∆ A ⊔ B = { , , ··· ,n } ( v RA G α ( w ))[[ u, v B ] , e α ] L for v , v , · · · , v n ∈ g +0 , where v RA = v Ra v Ra · · · v Ra max for A = { a , a , · · · , a max } ,( a < a < · · · < a max ), and [ u, v B ] = [ · · · [ u, v b ] , v b ] , · · · ] , v b max ] for B = { b , b , · · · ,b max } , ( b < b < · · · < b max ). Since ad N g +0 L g = 0 and (( g +0 ) R ) N G α ( w ) = 0 for N large enough, it follows that [ u L , [ v R , [ v R , · · · , [ v Rn , w ] · · · ] = 0 for n sufficientlylarge and hence [ v R , [ v R , · · · , [ v Rn , w ] , · · · ] ∈ e n R> , which is zero in M . Therefore, weobtain dim U ( g +0 ) R w < ∞ . Thus we conclude that M belongs to the category O .Suppose M = 0. Then it contains a nonzero highest weight vector w . Let ˜ w ∈ L denote a lift of w . Then Witt − ⋉ L g R + C ˜ w is an extension of e n -module Witt − ⋉ L g R by a trivial 1-dimensional e n -module C w , which is nontrivial by construction. Thusthis extension defines a nonzero element in H ( e n ; Witt − ⋉ L g R ). Since the action L g R on C [ LG + ] is faithful, L g R ∼ = L g as e n -modules. On the other hand, thecohomology H n ( e n ; Witt − ⋉ L g R ) is describes as H i ( e n ; Witt − ⋉ L g ) ∼ = H i − ( e n ; C ) ⊗ Witt > , where Witt > = C [[ t ]] t ∂∂t ([19]). This isomorphism is e h -equivariant. In particular,all the degrees of the homogeneous elements with respect to the action of x +( d +1) L in the first cohomology H ( e n ; Witt − ⋉ L g ) ∼ = Witt > are positive. (See Section 1for the definition of x and d .) On the other hand, the cocycle corresponding to C w is negative due to the definition of L , which is a contradiction. Therefore, weconclude M = 0. This completes the proof (cid:3) Theorem 5.9.
Suppose that ( g , f ) satisfies (F) and k ∈ C is generic. Then, H k ( g , f ) is an integrable Hamiltonian hierarchy associated with W k ( g , f ) .Proof. By abuse of notation, let η a ∈ Der( V k ( g )) also denote the correspondingelement in Vect( LG + /A + ) by Ψ k in Theorem 4.3. Since LG + ∼ = ( LG + /A + ) × A + asaffine schemes in an A R + -equivariant way, the elements of Vect( LG + /A + ) naturallylift to elements of Vect( LG + ) which commute with a R + . Again, let η a denote thelift of η a in Vect( LG + ). Then by Lemma 5.6, the lift η a lies in L and, moreover,[ u L , η a ] = P αI ⊔ ∆ F α ( η a )[ u, e α ] L with F α ( η a ) ∈ C [ LG + /A + ] for u ∈ L g . Since¯ s ∈ a + , we have, for u ∈ L g ,[ u L , [¯ s R , η a ]] = [¯ s R , [ u L , η a ]] = (cid:2) ¯ s R , X α F α ( η a )[ u, e α ] L (cid:3) = X α ¯ s R ( F α ( η a ))[ u, e α ] L = 0 . Since L g generates L g + , we obtain [¯ s R , η a ] ∈ Vect( LG + ) L g L + = L g R + . 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