aa r X i v : . [ m a t h . QA ] M a y IPMU-20-0061
ON MIURA MAPS FOR W -SUPERALGEBRAS SHIGENORI NAKATSUKA
Abstract.
We prove the injectivity of the Miura maps for W -superalgberasand the isomorphisms between the Poisson vertex superalgebras obtained asthe associated graded of the W -superalgebras in terms of the Li’s filtrationand the level 0 Poisson vertex superalgebras associated with the arc spaces ofthe corresponding Slodowy slices in full generality. Introduction
Let g be a basic classical (simple) Lie superalgebra over the field of complex num-bers C , k ∈ C a complex number, a non-zero even nilpotent element f of g , andΓ : g = L j ∈ Z g j a good grading for f . Then a vertex superalgebra W k ( g , f ; Γ),called the (affine) W -superalgebra, is defined as the quantum Drinfeld-Sokolov re-duction of the universal affine vertex superalgebra V k ( g ) of g at level k [KRW].The principal W -algebras W k ( g ) associated with simply-laced simple Lie algebrasand principal nilpotent elements are known to have the triality (T):(T1) (Feigin-Frenkel duality) W k ( g ) ≃ W L k ( L g ),(T2) (GKO construction) W k ( g ) ≃ Com (cid:16) V k ′ +1 ( g ) , V k ′ ( g ) ⊗ L ( g ) (cid:17) . Here L g is the Langlands dual Lie algebra of g , L ( g ) is the simple quotient of V ( g ), Com( W, V ) denotes the coset vertex superalgebra of V by W , and the levels( k, L k, k ′ ) satisfy certain relations, see [FF, ACL] for details.Recently, Gaiotto-Rapˇc´ak [GR] and Proch´azka-Rapˇc´ak [PR] proposed a gener-alization of the triality for W -superalgebras and their cosets by affine vertex sub-algebras in terms of 4 dimensional N = 4 topologically twisted super Yang-Millstheories. Since the Kazama-Suzuki coset construction of the N = 2 superacon-formal algebra appears as a very special case, these isomorphisms are expected tobe efficient for the study of the representation theory of vertex superalgebras, see[CGN, CL] for a mathematical approach.In the proofs of the triality (T), a vertex superalgebra homomorphism, calledthe Miura map [A1, FBZ, G1] µ : W k ( g , f ; Γ) → V τ k ( g ) ⊗ Φ( g ) (0.1)plays a fundamental role and its injectivity is very important. (See Section 2 fordetails.) The situation is expected to be the same for the generalized triality in thesense of [GR, PR]. The injectivity of µ is proved in [F, A2] when g is non-super and f is principal, whose proof also applies for an arbitrary f and is proved in [G1] when g and f are arbitrary but the level k is generic. Unfortunately, it is not enough forthe application to the representation theory of vertex superalgebras based on thetriality since the representation theory is rich only for special levels out of genericlevels. The main theme of the paper is to improve the situation and, especially, toprove the injectivity in full generality.In Section 1, we extend some results on unipotent algebraic groups to the supersettings for the application to the study of W -superalgebras. We explain the cor-respondence between formal supergroups and finite dimensional Lie superalgebras [Ma1, Se] by using the Campbell-Hausdorff formal supergroups. This correspon-dence gives the one between unipotent algebraic supergroups and finite dimensionalnilpotent Lie superalgebras [MO]. This implies that every unipotent algebraic su-pergroup is an affine superspace (i.e., C p | q for some p, q ∈ Z ≥ ) as is well-knownin the non-super cases, cf [Mil]. Let G be an unipotent algebraic supergroup withLie superalgebra g . Then we have a left g -module structure on the coordinate ring C [ G ]. We prove that under a certain condition, the Lie superalgebra cohomology H n ( g ; C [ G ]) is isomorphic to δ n, C . The proof is very standard: we use a filtrationon the complex, which gives a spectral sequence E ≃ H • dR ( G ) ⇒ H • ( g ; C [ G ]) andthen use Poincar´e lemma H n dR ( G ) ≃ δ n, since G is an affine superspace.In Section 2, after we review the definition of W -superalgebras and the construc-tion of the Miura maps (0.1), we extend some results of principal W -algebras to thesuper cases, following [A1]. We first consider the Li’s filtrations [L] on W k ( g , f ; Γ)and V τ k ( g ) ⊗ Φ( g ), respectively. The associated graded vector superspaces admita natural structure of Poisson vertex superalgebras (PVAs) and µ induces a PVAhomomorphism ¯ µ : gr F W k ( g , f ; Γ) → gr F (cid:16) V τ k ( g ) ⊗ Φ( g ) (cid:17) . The injectivity of µ reduces to that of ¯ µ (Lemma 2.2). Let { e, h, f } ⊂ g be an sl -triple in the even subsalgebra of g containing the nilpotent element f . We write g ≥− / = ⊕ j ≥− / g j , g + = ⊕ j> g j , and denote by G + the unipotent algebraicsupergroup whose Lie superalgebra is g + . Then the affine subspace f + g ≥− / is stable under the Adjoint action of G + and, moreover, admits a G + -equivalentisomorphism of affine supervarieties S f × G + ≃ f + g ≥− / , ( X, g ) g − Xg, where S f = f + g e ⊂ g , called the Slodowy slice of f (Proposition 2.5). This isa generalization to the super setting of [A2, eq.(4)] and [GG, Lemma 2.1]. Wehave a Poisson structure on C [ f + g ≥− / ] (see Section 2.4), which restricts to C [ S f ] ≃ C [ f + g ≥− / ] G + . If Γ is a Z -grading, then the Poisson supervariety S f isalso obtained as the Hamiltonian reduction of g equipped with the Kostant-KirillovPoisson structure with respect to G + and character χ = ( f | -). It turns out that wehave an isomorphism gr F W k ( g , f ; Γ) ≃ C [ J S f ] as PVAs (Proposition 2.8) where J S f is the arc space of S f (see Section 1.6) and C [ J S f ] is inherited with the level 0PVA structure induced by the Poisson structure on C [ S f ]. On the other hand, weidentify gr F (cid:16) V τ k ( g ) ⊗ Φ( g ) (cid:17) ≃ C [ J ( f + ( g − / ⊕ g )] and thus¯ µ : C [ J S f ] → C [ J ( f + ( g − / ⊕ g )] . This map is induced by its finite dimensional analogue¯ µ fin : C [ S f ] ≃ C [ f + g ≥− / ] G + → C [ f + ( g − / ⊕ g )] , which is just the restriction map. We prove the injectivity of ¯ µ fin (Proposition 2.9),which immediately implies the injectivity of ¯ µ and thus of the original Miura map µ (Theorem 2.1). Acknowledgments
The author would like to express his gratitude to ThomasCreutzig and Naoki Genra for suggesting the problem and useful discussions, tohis supervisor Masahito Yamazaki for reading the draft and to Yuto Moriwaki foruseful discussions on algebraic supergroups. This work was supported by WorldPremier International Research Center Initiative (WPI Initiative), MEXT, Japan.The author is also supported by the Program for Leading Graduate Schools, MEXT,Japan and by JSPS KAKENHI Grant Number 20J10147.
N MIURA MAPS FOR W -SUPERALGEBRAS 3 Unipotent algebraic supergroup
Formal supergroups.
We review the contravariant equivalence between for-mal supergroups and Lie superalgebras, following [Se, Ma1]. The results in Section1.1-1.4 are proved in much more generality in [Ma1]. Given a commutative C -superalgebra A , we denote by A = A ¯0 ⊕ A ¯1 the parity decomposition and by ¯ a the parity of a ∈ A . In particular, given a basis { x α } α of A , the parity ¯ x α is alsodenoted by ¯ α . The multiplication m : A × A → A is also denoted by ab = m ( a, b ).Let { x α } α ∈ S be a set equipped with parity, i.e., S = S ¯0 ⊔ S ¯1 and ¯ x α = ¯ α . The ringof (commutative) superpolynomials in the variables { x α } α ∈ S is the commutative C -superalgebra C [ x α | α ∈ S ] := C [ x α | α ∈ S ¯0 ] ⊗ ^ x β , ( β ∈ S ¯1 ) , which is the tensor product of the polynomial ring generated by the variables { x α } α ∈ S ¯0 and the exterior algebra generated by the variables { x α } α ∈ S ¯1 .We denote by ˆ R p | q a formal power series ringˆ R p | q = C [[ x , · · · , x p ]] ⊗ ^ φ , ··· ,φ q with p even variables and q odd variables, and by ˆ m = ( x , · · · , x p , φ , · · · , φ q ) ⊂ ˆ R p | q its unique maximal ideal. It is complete in the linear topology whose basisof open neighborhoods around 0 is { ˆ m n } ∞ n =0 . Let m : ˆ R p | q × ˆ R p | q → ˆ R p | q denotethe product and u : C → ˆ R p | q the unit morphism. We denote by ˆ R p | q b ⊗ ˆ R p | q thecompleted tensor product ˆ R p | q b ⊗ ˆ R p | q := lim ←− n,m ˆ R p | q / ˆ m n ⊗ ˆ R p | q / ˆ m m . Then we havea natural isomorphismˆ R p | q b ⊗ ˆ R p | q ≃ C hh x (1)1 , · · · , x (1) p , x (2)1 , · · · , x (2) p ii ⊗ ^ φ (1)1 , ··· ,φ (1) q ,φ (2)1 , ··· ,φ (2) q . A formal supergroup is a super bialgebra ( ˆ R p | q , ∆ , ǫ ) with continuous homomor-phisms • coproduct ∆ : ˆ R p | q → ˆ R p | q b ⊗ ˆ R p | q , • counit ǫ : ˆ R p | q → C .It has an unique continuous homomorphism S : ˆ R p | q → ˆ R p | q such that m ◦ ( S ⊗ ◦ ∆ = ǫ ◦ u = m ◦ (1 ⊗ S ) ◦ ∆ , called the antipode. Thus formal supergroups are Hopf superalgebras, see [CCF]. Remark 1.1.
Let G be an algebraic supergroup with the identity e . Then the germof the structure sheaf O G at e , which we denote by O G,e , is a local ring. Let m G,e denote its unique maximal ideal. Then the completion b O G,e := lim ←− O
G,e / m nG,e has a natural structure of formal supergroup. A morphism of formal supegroups ( ˆ R p | q , ∆ , ǫ ) and ( ˆ R p ′ | q ′ , ∆ ′ , ǫ ′ ) is a morphismof Hopf superalgebras F : ˆ R p | q → ˆ R p ′ | q ′ which is continuous in the linear topology. We denote by FG the category of formalsupergroups. SHIGENORI NAKATSUKA
Point distributions of formal supergroups.
Let F = ( ˆ R p | q , ∆ , ǫ ) be a for-mal supergroup and ˆ m ⊂ ˆ R p | q its maximal ideal. A point distribution is a continuouslinear map ξ : ˆ R p | q → C where C is inherited with the discrete topology. Let U F := Hom c. C ( ˆ R p | q , C ) denotethe vector superspace of point distributions. Since any point distribution factorsˆ R p | q ։ ˆ R p | q / ˆ m ℓ → C for some ℓ ∈ Z ≥ , we have U F = lim −→ ℓ Hom C ( ˆ R p | q / ˆ m ℓ , C ). It has an associativealgebra structure by the dual maps∆ ∗ : U F → U F ⊗ U F , ǫ ∗ : C → U F of ∆ and ǫ , respectively. Note that these maps are well-defined since ∆ and ǫ arecontinuous. Moreover, U F has a Hopf superalgebra structure F ∨ = ( U F , m ∗ , u ∗ , S ∗ ).Let us describe F ∨ more explicitly. The elements X α = x i · · · x i p p φ j . . . φ j q q ∈ ˆ R p | q , ( α = ( i , · · · , i p , j , · · · , j q ) ∈ S = Z p ≥ × { , } q ), form a topological basis ofˆ R p | q . We denote by ξ α , ( α ∈ S ), its dual basis of U F , i.e., ξ α satisfies ξ α ( X β ) = δ α,β .Then ξ α ⊗ ξ β , ( α, β ∈ S ), form a basis of U F ⊗ U F . Here ξ α ⊗ ξ β satisfies ξ α ⊗ ξ β ( X α ′ ⊗ X β ′ ) = ( − ¯ α ′ ¯ β ξ α ( X α ′ ) ξ β ( X β ′ ) . In the following, we also use the symbols X α ∈ ˆ R p | q , ( α ∈ e S = Z p ≥ × Z q ≥ ),standing for 0 if α / ∈ S and the notation α = ( α , α ), | α | = P pr =1 i r where α = ( i , · · · , i p ) ∈ Z p ≥ and | α | = P qs =1 i s where α = ( j , · · · , j q ) ∈ Z q ≥ . Wedefine a degree on U F by deg( ξ α ) = | α | := | α | + | α | .Then we have ǫ ∗ : C → U F , u ∗ : U F → C ξ , ξ α δ α, . (1.1)The product F ∗ : U F × U F → U F satisfies ξ α · ξ β = ( − ¯ α ¯ β (cid:18) α + β α (cid:19) ξ α + β + lower degree terms , (1.2)where (cid:18) pq (cid:19) := ℓ Y i =1 (cid:18) p i q i (cid:19) , p = ( p , · · · , p ℓ ) , q = ( q , · · · , q ℓ )and that the lower degree terms do not the contain constant term C ξ . In particular,it follows that the elements ξ α with | α | = 1 generate U F as a superalgebra. Thecoproduct m ∗ is the unique superalgebra homomorphism such that m ∗ ( ξ α ) = ξ α ⊗ ⊗ ξ α (1.3)for α ∈ S such that | α | = 1. Lemma 1.2 ([Ma1, Lemma 4.1]) . The bilinear map [ - , - ] : U F × U F → U F , ( a, b ) ab − ( − ¯ a ¯ b ba defines a Lie superalgebra structure on the subspace U F = ⊕ | α | =1 C ξ α .Proof. Since U F is an associative superalgebra, it suffices to prove that [- , -] pre-serves the subspace U F . But it follows from (1.2). (cid:3) N MIURA MAPS FOR W -SUPERALGEBRAS 5 Let U ( U F ) denote the universal enveloping superalgebra of the Lie superalgebra U F . Recall that it admits a Hopf superalgebra structure with coproduct U ( U F ) → U ( U F ) ⊗ U ( U F ) , X X ⊗ ⊗ X, ( X ∈ U F ) , counit U ( U F ) ։ U ( U F ) / ( U F ) ≃ C , and antipode U ( U F ) → U ( U F ) , X
7→ − X, ( X ∈ U F ) . Here ( U F ) ⊂ U ( U F ) denotes the left ideal generated by the subspace U F . Proposition 1.3 ([Ma1, Proposition 4.8]) . The Lie superalgebra homomorphism U F ֒ → U F induces a C -superalgebra homomorphism U ( U F ) → U F . It is an isomor-phism of Hopf superalgebras.Proof. It is obvious that we have a C -superalgebra homomorphism U ( U F ) → U F .Let ι denote this map. Then it follows from (1.1) and (1.3) that ι is a superbialgebra homomorphism. Since an antipode on a super bialgebra is unique if itexists, ι intertwines the antipodes of U ( U F ) and U F . Thus ι is a homomorphism ofHopf superalgebras. Since the elements ξ α with α ∈ S such that | α | = 1 generate U F as a C -superalgebra, ι is surjective. Now the PBW type theorem for U ( U F )implies that ι is an isomorphism. (cid:3) Equivalence between Formal supergroups and Lie superalgebras.
LetLA denote the category of finite dimensional Lie superalgebras. Taking the univer-sal enveloping superalgebra g
7→ U ( g ) for a Lie superalgebra g defines a functor U : LA → BAfrom LA to the category of super bialgebras BA. The functor U is fully faithfuland its quasi-inverse is given by taking the primitive elements: P : BA → LA , ( R, ∆ , ǫ )
7→ P ( R ) := { a ∈ R | ∆( a ) = a ⊗ ⊗ a } . The Lie superalgebra structure is given by ( a, b ) ab − ( − ¯ a ¯ b ba for a, b ∈ P ( R )Thus LA is regarded as a full subcategory of BA. By Proposition 1.3, the association F F ∨ for a formal supergroup F gives a contravariant functor (-) ∨ : FG → LA.
Theorem 1.4 ([Ma1, Theorem 4.4]) . The functor ( - ) ∨ : FG → LA gives a con-travariant equivalence of categories.
A quasi-inverse is given again by taking the dual vector superspace(-) ∨ : LA → FG , U ( g ) Hom C ( U ( g ) , C ) , (1.4)where g is an arbitrary finite dimensional Lie superalgebra. The super coalgebra(algebra) structure on Hom C ( U ( g ) , C ) is induced by the super algebra (coalgebra)structure on Hom C ( U ( g ) , C ). In the purely even setting, the equivalence betweenthe categories of formal groups and of finite dimensional Lie algebras is a specialcase of Cartier duality between the category of linearly compact commutative Hopfalgebras and the category of cocommutative Hopf algebras, see [D] for details.The quasi-inverse (1.4) is known naturally isomorphic to the functor of takingthe Campbell-Hausdorff formal supergroup. Let g denote a finite dimensional Liesuperalgebra over C . By viewing g as an affine superscheme, we denote by C [ g ]its coordinate ring. We identify C [ g ] with the symmetric superalgebra S ( g ∗ ) ofthe dual vector superspace g ∗ of g . Then the Lie superbracket [- , -] on g induces asuperalgebra homomorphism D : S ( g ∗ ) → S ( g ∗ ) ⊗ S ( g ∗ ) , (1.5) SHIGENORI NAKATSUKA called the co(super)bracket of g . Consider the completion ˆ S ( g ∗ ) = lim ←− n S ( g ∗ ) / I n where I is the argumentation ideal and the completed tensor productˆ S ( g ∗ ) b ⊗ ˆ S ( g ∗ ) = lim ←− n,m S ( g ∗ ) / I n ⊗ S ( g ∗ ) / I m . Then the vector superspace Hom C ( g ∗ , ˆ S ( g ∗ ) b ⊗ ˆ S ( g ∗ )) admits a Lie superalgebrastructure by[ f, g ] := m ◦ ( f ⊗ g ) ◦ D, f, g ∈ Hom C (cid:16) g ∗ , ˆ S ( g ∗ ) b ⊗ ˆ S ( g ∗ )) (cid:17) (1.6)where m denotes the multiplication of ˆ S ( g ∗ ) ⊗ ˆ S ( g ∗ ). Define a superalgebra homo-morphism ∆ : ˆ S ( g ∗ ) → ˆ S ( g ∗ ) b ⊗ ˆ S ( g ∗ ) by∆( X ) = ∞ X n =1 ( − n +1 n X ( p i ,q i ) > i =1 , ··· ,n [ ι p , ι q , · · · , ι p n , ι q n ]( X ) P ( p i + q i ) Q p i ! q i ! , ( X ∈ g ∗ ) (1.7)where the summation ( p i , q i ) is over Z ≥ \{ (0 , } , ι i denotes the natural inclusion ι i : g ∗ → g ∗ ⊗ C ⊕ C ⊗ g ∗ ⊂ ˆ S ( g ∗ ) b ⊗ ˆ S ( g ∗ ) , for i = 1 , ι p , ι q , · · · , ι p n , ι q n ]= [ ι , · · · , [ ι | {z } p , [ ι , · · · , [ ι | {z } q · · · [ ι , · · · , [ ι | {z } p n , [ ι , · · · , [ ι | {z } q n − , ι ] , ( q n = 0) , [ ι , · · · , [ ι | {z } p , [ ι , · · · , [ ι | {z } q · · · [ ι , · · · , [ ι | {z } p n − , ι ] , ( q n = 0) . Then by [Ma1, Proposition 4.11], ( ˆ S ( g ∗ ) , ∆ , ǫ ) defines a formal supergroup where ǫ : ˆ S ( g ∗ ) → C is the obvious counit. Moreover, by the proof of [Ma1, Theorem 4.4],the functor CH : LA → FG , g ( ˆ S ( g ∗ ) , ∆ , ǫ )is a quasi-inverse of (-) ∨ : FG → LA.1.4.
Unipotent algebraic supergroup and Nilpotent Lie superalgebra.
Anaffine algebraic supergroup G is called unipotent if the following equivalent condi-tions hold: • The coordinate ring C [ G ] is irreducible as a coalgebra, i.e., C [ G ] has aunique simple subcoalgebra, which is C , • The isomorphism classes of simple rational G -supermodules are trivial mod-ules { C | , C | } ,see [Ma2, Definition 2.9]. Let Unip-AG denote the full subcategory of the categoryof affine algebraic supergroup consisting of unipotent affine algebraic supergroups.Then we have a functor ˆ O - ,e : Unip-AG → FG , G ˆ O G,e , by Remark 1.1 and T e - : Unip-AG → LA , G T e G. The latter one is just taking the Lie superalgebra on the tangent space at theidentity e , see [CCF] for details. By construction, we have a natural isomorphism(-) ∨ ◦ ˆ O - ,e ≃ T - e : Unip-AG → LA . (1.8) N MIURA MAPS FOR W -SUPERALGEBRAS 7 A finite dimensional Lie superalgebra g is called nilpotent if the descending cen-tral series g = g , g n = [ g , g n − ] , ( n > , (1.9)vanishes, i.e., g n = 0 for some n . Let Nil-LA ⊂ LA denote the full subcategoryconsisting of (finite dimensional) nilpotent Lie superalgebras.
Proposition 1.5 ([MO]) . The functor T e - : Unip-AG → Nil-LA gives an equiva-lence of categories.
Let g be a finite dimensional Lie superalgebra and D : g ∗ → g ∗ ⊗ g ∗ denote thecobracket. Then g is nilpotent if and only if the map D n defined by D n = ( D ⊗ ⊗ · · · ⊗ | {z } n − ) ◦ D n − , n ≥ , vanishes, i.e., D n = 0 fore some n ≥
0. Thus the coproduct (1.7) of the correspond-ing formal group CH ( g ) stabilize the supersymmetric algebra S ( g ∗ ):∆ : S ( g ∗ ) → S ( g ∗ ) ⊗ S ( g ∗ ) . (1.10)Equivalently, Spec( S ( g ∗ )) has a structure of affine algebraic supergroup. By Propo-sition 1.5 and (1.8), Spec( S ( g ∗ )) is an unipotent affine algebraic supergroup and,conversely, any unipotent affine algebraic supergroup is obtained in this way. SinceSpec( S ( g ∗ )) is an affine superspace, we obtain the following assertion. Corollary 1.6.
Any unipotent affine algebraic supergroup is an affine superspace C p | q for some p, q ∈ Z ≥ . Lie superalgebra cohomology.
Let G be an affine algebraic supergroupwith Lie superalgebra g = T e G . For any commutative superalgebra A and an A -point g ∈ G ( A ), we have a right multiplication R g : G ( A ) → G ( A ) , h hg. This induces a left g -module structure on the coordinate ring R : g → End C ( C [ G ]) . (1.11)We call G positively graded if g is Z > -graded: g = M n> g n , [ g n , g m ] ⊂ g n + m . In this case, g ∗ is naturally Z < -graded wt : g ∗ = ⊕ n< g ∗ n where g ∗ n = Hom C ( g − n , C )for n <
0. We extend the grading on g ∗ to S ( g ∗ ) multiplicatively: wt( ab ) =wt( a ) + wt( b ). Then the cobracket D in (1.5) preserves the grading and so does thecoproduct ∆ : C [ G ] → C [ G ] ⊗ C [ G ] by (1.7). Proposition 1.7.
For a positively graded unipotent affine algebraic supergroup G , H n ( g ; C [ G ]) ≃ δ n, C . Proof.
Although the assertion is well-known, we include a proof for the complete-ness of the paper. Since g is nilpotent, we have g ⊃ [ g , g ] ⊃ [ g , [ g , g ]] ⊃ · · · ⊃ g m = 0for some m ∈ Z ≥ . We write N = dim g and take a basis { v α } Nα =1 of g such that thefirst dim g / [ g , g ] elements gives a basis of g / [ g , g ], and the next dim [ g , g ] / [ g . [ g , g ]]elements gives a basis of [ g , g ] / [ g , [ g , g ]] etc. Let c γαβ denote the structure constantsof g , i.e., [ v α , v β ] = P γ c γαβ v γ . Note that c γαβ is non-zero only if γ ≥ α, β . TheChevalley-Eilenberg complex of g with coefficients in C [ G ] is C • ( g ; C [ G ]) = C [ G ] ⊗ S (Π g ∗ ) , SHIGENORI NAKATSUKA d = X α ( − ¯ α R ( v α ) ⊗ ϕ α − X α,β,γ ( − ¯ α ¯ γ c γαβ ϕ γ ϕ α ϕ β . (1.12)Here Π g ∗ denotes the parity-reversed superspace of g ∗ , spanned by { ϕ α } Nα =1 withparity ¯ ϕ α = ¯ α + 1 ∈ Z . The symbols ϕ α in the differential d is the multiplicationby ϕ α and ϕ α is the contracting operator for ϕ α , i.e., ϕ α ( ϕ β ) = δ α,β , which has thesame parity as ϕ α , (see e.g. [DK]). By (1.10), we may take C [ g ] as the coordinatering C [ G ] of G . Let { x α } Nα =1 denote the linear coordinates of g corresponding tothe basis { v α } Nα =1 . Then C [ g ] = C [ x α | α = 1 , · · · , N ]. By (1.7), (1.11) is expressedas R ( v α ) = X β (cid:18) ǫ (2) ∂∂x (2) α ∆( x β ) (cid:19) ∂∂x β , where ǫ (2) ∂∂x (2) α ∆( x β ) := X x β (1) ǫ (cid:18) ∂∂x α x β (2) (cid:19) and ∆( x β ) = P x β (1) ⊗ x β (2) . Thus the map R preserves the grading. Extend thegrading on C [ G ] to C • ( g ; C [ G ]) by wt( ϕ α ) = wt( x α ). Then we have the gradingdecomposition C • ( g ; C [ G ]) = M n ≤ C • n ( g ; C [ G ]) (1.13)and d preserves the grading. Define a degree for each monomial bydeg( x α ) = 1 = deg( ϕ α ) , deg( ab ) = deg( a ) + deg( b )and F p C • ( g ; C [ G ]) by the subspace spanned by all the monomials of degree greaterthan or equal to p . Then (1.12) implies d : F p C • ( g ; C [ G ]) → F p +1 C • ( g ; C [ G ]).Hence, there is a spectral sequence E r ⇒ H • ( g ; C [ G ]) such that E = H (gr F C • ( g ; C [ G ]); gr d ) . The filtration gives a filtration on each C • n ( g ; C [ G ]) by F p C • n ( g ; C [ G ]) = C • n ( g ; C [ G ]) ∩ F p C • n ( g ; C [ G ]) . It is of finite length and satisfies C • n ( g ; C [ G ]) = ∪ p F p C • n ( g ; C [ G ]). Therefore, thespectral sequence E r converges. Since the differential on gr F C • ( g ; C [ G ]) isgr d = N X α =1 ( − ¯ α ∂∂x α ⊗ ϕ α , the complex (gr F C • ( g ; C [ G ]) , gr d ) is the (algebraic) de Rham complex of G . ByCorollary 1.6, H n (gr F C • ( g ; C [ G ]) , gr d ) ≃ H n dR ( C p | q )for some p, q ∈ Z ≥ . Then by Poincar´e lemma below, we have H n dR ( C p | q ) ≃ δ n, C .It follows that the spectral sequence collapses at r = 1:gr F H n ( g ; C [ G ]) ≃ E n ≃ δ n, C . This completes the proof. (cid:3)
The following assertion is proved by Kostant in the analytic setting [Ko, Theorem4.6] and is also well-known in the algebraic setting.
Lemma 1.8 (Poincar´e lemma) . H n dR ( C p | q ) ≃ δ n, C . N MIURA MAPS FOR W -SUPERALGEBRAS 9 Proof.
The algebraic de Rham complex of C p | q is C • dR ( C p | q ) = C [ C p | q ] ⊗ S (Π C p | q ) . We write C [ C p | q ] = C [ x , · · · , x p , θ , · · · , θ q ] where x i , (resp. θ j ), are even, (resp.odd), variables, and S (Π C p | q ) = C [ dx , · · · dx p , dθ , · · · , dθ q ] where dx i , (resp. dθ j ),are odd, (resp. even), variables. Then the differential is d = p X i =1 ∂∂x i ⊗ dx i − q X j =1 ∂∂θ j ⊗ dθ j . Thus by K¨unneth formula, H • dR ( C p | q ) ≃ (cid:0) H • dR ( C | ) (cid:1) ⊗ p ⊗ (cid:0) H • dR ( C | ) (cid:1) ⊗ q . There-fore, it suffices to show the assertion in the cases C | and C | respectively. Thefirst one is the usual Poincar´e lemma. For the second one, note that the d -closedforms are linear combinations of 1 ⊗ dθ n , ( n ≥ n > d ( − θ ⊗ dθ p − ) = 1 ⊗ dθ p . This completes the proof. (cid:3)
Superscheme of formal arcs.
Let SSch denote the category of superschemesover C . An object D := Spec( C [[ t ]]) , of SSch is called the formal disc. Proposition 1.9 ([KV, Proposition 4.2.1]) . Let X be a superscheme over C . Thecontravariant functor SSch → Set , Y Hom
SSch ( Y b × D, X ) is represented by a superscheme JX , that is, Hom
SSch ( Y b × D, X ) ≃ Hom
SSch ( Y, JX ) for any object Y of SSch. Here Y b × D is the completion of Y × D with respect tothe subsuperscheme Y b ×{ } . The superscheme JX in the above proposition is called the superscheme of formalarcs in X or the arc space of X . The association X JX gives a functor: J - : SSch → SSch . (1.14)The arc space JX of X has a canonical projection π X : JX → X satisfying the fol-lowing functoriality: for any morphism f : X → Y of supersshemes, the morphism Jf : JX → JY makes the following diagram commutative: JX Jf / / π X (cid:15) (cid:15) JY π Y (cid:15) (cid:15) X f / / Y. If X is an affine superscheme with C [ X ] = C [ x , · · · , x N ] / ( f , · · · , f M )where C [ x , · · · , x N ] is the ring of superpolynomials in the variables x , · · · , x N and f , · · · , f M ∈ C [ x , · · · , x N ], then JX is again an affine superscheme with C [ X ] = C [ x , ( n ) , · · · , x N, ( n ) | n < / ( f , ( n ) , · · · , f M, ( n ) | n < . X n< f j ( n ) z − n − = f j ( x ( z ) , · · · , x N ( z )) with x i ( z ) = P n< x i ( n ) z − n − . In this case, the canonical projection π X is givenby C [ X ] → C [ JX ] , x i x i ( − . Later, we use the following properties of the functor (1.14).
Lemma 1.10 (cf. [A1]) . For superschemes X and Y , the following holds: (1) X ≃ Y implies JX ≃ JY . (2) J ( X × Y ) ≃ JX × JY . Let G be an algebraic supergroup with Lie superalgebra g = T e G . The arc space JG of G has a structure of group superscheme. The Lie superalgebra on the tangentspace T e JG is J g := g [[ t ]]. The following proposition can be proved in the sameway as Proposition 1.7. Proposition 1.11.
For a positively graded unipotent affine algebraic supergroup G , H n ( J g ; C [ JG ]) ≃ δ n, C . W -superalgebras and Miura maps W -superalgebras. We review here the definition of (affine) W -superalgebrasintroduced by Kac, Roan and Wakimoto [KRW]. Given a vertex superalgebra V ,(see [Ka2] for details) we denote by Y ( A, z ) = A ( z ) = P n ∈ Z A ( n ) z − n − the fieldcorresponding to an element A ∈ V and set R A ( z ) dz = A (0) . Given A, B ∈ V ,we denote by : A ( z ) B ( z ) : the normally ordered product and by A ( z ) B ( w ) ∼ P n ≥ C n ( w )( z − w ) n +1 the operator product expansion (OPE) of A ( z ) and B ( z ).Let g be a basic classical simple Lie superalgebra (over C ) and g = g ¯0 ⊕ g ¯1 the parity decomposition. The Lie superalgebra g admits a non-degenerate eveninvariant bilinear form κ . Since such forms are all proportional, we may write κ = kκ for some k ∈ C where κ = ( · , · ) is the one satisfying κ ( θ, θ ) = 2 for thehighest root θ of g ¯0 , see [Ka1, Mu]. Let f ∈ g ¯0 be a non-zero nilpotent element. Agood grading of g with respect to f is a Z -grading on g Γ : g = M j ∈ Z g j such that f ∈ g − and the adjoint action ad f of f is injective g j ֒ → g j − for j ≥ / g j ։ g j − for j ≤ /
2, (see [EK, H] for the classification). We fix abasis x α , ( α ∈ I = { , · · · , dim g } ) of g such that each x i is of homogeneous parityand Γ-grading. Then we have I = ⊔ j I j where I j = { α | x α ∈ g j } . Let c γα,β denotethe structure constants, i.e., [ x α , x β ] = P γ c γα,β x γ .Let V k ( g ) denote the universal affine vertex superalgebra of g at level k , whichis generated X ( z ), ( X ∈ g ), satisfying the OPEs X ( z ) Y ( z ) ∼ [ X, Y ]( z )( z − w ) + k ( X, Y )( z − w ) , X, Y ∈ g . We define a conformal grading on V k ( g ) by ∆( u ) = 1 − j for u ∈ g j . Let F ch ( g + )be the charged fermion vertex superalgebra associated with g + := ⊕ j> g j . It isgenerated by fields ϕ α ( z ) , ϕ α ( z ), ( α ∈ I + := ⊔ j> I j ) of parity reversed to x α ,satisfying the OPEs ϕ α ( z ) ϕ β ( w ) ∼ δ α,β z − w , ϕ α ( z ) ϕ β ( w ) ∼ ∼ ϕ α ( z ) ϕ β ( w ) , α, β ∈ I + . We define a conformal grading on F ch ( g + ) by ∆( ϕ α ) = 1 − j , ∆( ϕ α ) = j for α ∈ I j ,and a degree on F ch ( g + ) = L n ∈ Z F n ch bydeg( ϕ α ( z )) = 1 = − deg( ϕ α ( z )) , α ∈ I + , N MIURA MAPS FOR W -SUPERALGEBRAS 11 deg( ∂A ( z )) = deg( A ( z )) , deg(: A ( z ) B ( z ) :) = deg( A ( z )) + deg( B ( z )) . Let Φ( g / ) be the neutral Fermion vertex superalgebra associated with g / , whichis generated by fields Φ α ( z ), ( α ∈ I / ) satisfying the OPEsΦ α ( z )Φ β ( w ) ∼ χ ([ x α , x β ]) z − w , α, β ∈ I , where χ ( x ) = ( f, x ) for x ∈ g . We define a conformal grading on Φ( g / ) by∆(Φ α ) = 1 / α ∈ I / . Define a Z -graded vertex superalgebra by C • k ( g , f ; Γ) = V k ( g ) ⊗ Φ( g ) ⊗ F • ch ( g + ) , with cohomological grading given by the degree on F ch ( g + ) and a differential d (0) by d ( z ) = X α ∈ I + : (( − ¯ α x α ( z )+Φ α ( z ) + χ ( x α )) ϕ α ( z ) : − X α,β,γ ∈ I + ( − ¯ α ¯ γ c γα,β : ϕ γ ( z ) ϕ α ( z ) ϕ β ( z ) :where Φ α = 0 for α / ∈ I / . Then ( C • k ( g , f ; Γ) , d (0) ) forms a cochain complex,called the BRST complex . The vertex superalgebra obtained as the cohomology H ( C • k ( g , f ; Γ) , d (0) ) is called the (affine) W -superalgebra associated with ( g , f, k, Γ)and denoted by W k ( g , f ; Γ). By [KW1, Theorem 4.1], we have W k ( g , f ; Γ) = H ( C • k ( g , f ; Γ) , d (0) ) . Since d (0) preserves the conformal grading on C • k ( g , f ; Γ) induced by those on eachcomponent V k ( g ), Φ( g / ) and F • ch ( g + ), W k ( g , f ; Γ) has an induced conformal grad-ing, which is a Z ≥ -grading.2.2. Miura map.
Define a field J u ( z ) = u ( z ) + X α,β ∈ I + ( − ¯ α c αu,β : ϕ α ( z ) ϕ β ( z ) , u ∈ g . Let C + ⊂ C • k ( g , f ; Γ) denote the vertex subalgebra generated by J u ( z ), ( u ∈ g + ),and ϕ α ( z ), ( α ∈ I + ), and C − ⊂ ( C • k ( g , f ; Γ) denote the one generated by J u ( z ),( u ∈ g ≤ := ⊕ j ≤ g j ), Φ α ( z ), ( α ∈ I / ), and ϕ α ( z ), ( α ∈ I + ). By [KW1, KW2], C •± ⊂ C • k ( g , f ; Γ) are subcomplexes and give a decomposition C • k ( g , f ; Γ) ≃ C • + ⊗ C •− . Moreover, we have H n ( C • + , d (0) ) ≃ δ n, C . Thus we have an isomorphism W k ( g , f ; Γ) ≃ H ( C •− , d (0) ) . Since C − is Z ≥ -graded as a complex, it follows that W k ( g , f ; Γ) is a vertex subal-gebra of C − . By [KRW, Theorem 2.4, (c)], we have J u ( z ) J v ( X ) ≃ J [ u,v ] ( w )( z − w ) + τ k ( u, v )( z − w ) , u, v ∈ g ≤ , where τ k ( u, v ) = k ( u, v ) + 12 ( κ g ( u, v ) − κ g ( u, v ))and κ g , (resp. κ g ) denotes the Killing form of g , (resp. g ). It follows that C − ≃ V τ k ( g ≤ ) ⊗ Φ( g ) . Since V τ k ( g < ) ⊗ Φ( g / ) ⊂ V τ k ( g ≤ ) ⊗ Φ( g ) is a (vertex superalgebra) ideal, wehave a natural surjection V τ k ( g ≤ ) ⊗ Φ( g ) ։ (cid:16) V τ k ( g ≤ ) ⊗ Φ( g ) (cid:17) / (cid:0) V τ k ( g < ) ⊗ Φ( g / ) (cid:1) ≃ V τ k ( g ) ⊗ Φ( g ) . The restriction to the subalgebra W k ( g , f ; Γ) is called the Miura map [FBZ, A1, G1] µ : W k ( g , f ; Γ) → V τ k ( g ) ⊗ Φ( g ) . (2.1)Note that V τ k ( g ) ⊗ Φ( g ) has a conformal grading defined by∆( J u ( z )) = 1 , ( u ∈ g ) , ∆(Φ α ( z )) = 12 , ( α ∈ I ) , and that µ preserves the conformal grading ∆. Theorem 2.1.
The Miura map µ is injective. The above theorem is proved in [F, A1] when g is purely even and f is principal.But their proofs apply for an arbitrary nilpotent element. Is is also proved in [G1]when the level k is generic. Our proof is a slight generalization of the one in [A1]together with necessary supergeometry.2.3. Li’s filtration.
By [L], given a vertex superalgebra V , the subspaces F p V ,( p ∈ Z ), spanned by a − n − a − n − · · · a r ( − n r − | i with a , a , · · · , a r ∈ V , n i ≥ n + n + · · · + n r ≥ p , form a descending filtration F • V of V satisfying the following propertires: • F p V ( n ) F q V ⊂ F p + q − n − V , ( p, q ∈ Z ≥ , n ∈ Z ), • F p V ( n ) F q V ⊂ F p + q − n V , ( p, q ∈ Z ≥ , n ≥ F V := ⊕ p ∈ Z gr nF V, gr nF V = F p V /F p +1 V denote the associated graded vector superspace and σ p : F p V ։ gr nF V the canonicalprojection. By [L], gr F V has a Poisson vertex superalgebra structure by ∂σ p ( a ) = σ p +1 ( ∂a ) , σ p ( a ) σ q ( b ) = σ p + q ( a ( − b ) , { σ p ( a ) λ σ q ( b ) } = X n ≥ n ! σ p + q − n ( a ( n ) b ) λ n . (see e.g. [FBZ, L] for the definition of Poisson vertex superalgebras and e.g. [BDK,Su] in terms of λ -brackets). It is obvious that any homomorphism η : V → W ofvertex superalgebras induces a homomorphism of Poisson vertex superalgebras¯ η : gr F V → gr F W, σ p ( a ) σ p ( η ( a )) . Lemma 2.2.
Let V and W be vertex superalgebras equipped with conformal Z ≥ -gradings and η : V → W be a homomorphism of vertex superalgebras preserving theconformal gradings. Then η is injective if ¯ η is injective.Proof. Let V = ⊕ ∆ ∈ Z ≥ V ∆ , (resp. W = ⊕ ∆ ∈ Z ≥ W ∆ ) denote the conformal grad-ing on V , (resp W ). It is immediate from the definition that the Li’s filtration F • V defines a filtration on each V ∆ by setting F p V ∆ = V ∆ ∩ F p V. Moreover, it is of finite length, i.e., V ∆ ∩ F p V = 0 for p >>
0. Since η preservesthe conformal gradings, gr F η is restricted togr F η : gr F V ∆ → gr F W ∆ , ∆ ≥ . N MIURA MAPS FOR W -SUPERALGEBRAS 13 It is straightforward to show the injectivity of F on V ∆ inductively from that of gr F η on gr F V ∆ starting with the subspace gr NF V ∆ such that gr nF V ∆ = 0 for n > N . (cid:3) By [A1, Section 3], the Poisson vertex superalgebra gr F V τ k ( g ) ⊗ Φ( g / ) isisomorphic to the algebra of differential superpolynomials S ∂ (cid:16) g ⊕ g (cid:17) = C h ∂ n x α | α ∈ I ⊔ I , n ≥ i with λ -brackets { x αλ x β } = [ x α , x β ] , α, β ∈ I , { x αλ x β } = χ ([ x α , x β ]) , α, β ∈ I , { x αλ x β } = 0 , α ∈ I , β ∈ I . Thus the Miura map µ in (2.1) induces a homomorphism of Poisson vertex super-algebras ¯ µ : gr F W k ( g , f, Γ) → S ∂ (cid:16) g ⊕ g (cid:17) . (2.2)Lemma 2.2 implies the following. Corollary 2.3.
The injectivity of ¯ µ implies that of µ . Arc space of Slodowy slice.
Consider the Li’s filtration F • C •− on the com-plex C •− . The associated graded vector superspace ¯ C •− = gr F C •− is again a complexwith differential ¯ d (0) . Note that gr F C •− is algebra of differential superpolynomialsgr F C •− ≃ S ∂ ( g ≤ ⊕ g ) ⊗ S ∂ (Π g ∗ + ) , equipped with the λ -brackets { u λ v } = [ u, v ] , ( u, v ∈ g ≤ ) , { u λ v } = χ ([ u, v ]) , ( u, v ∈ g ) , { ϕ αλ u } = X β ∈ I + c αu,β ϕ α , ( α ∈ I + , u ∈ g ≤ ) , { u λ v } = 0 = { ϕ αλ ϕ β } , ( u ∈ g ≤ , v ∈ g , α, β ∈ I + )where { ϕ α } α ∈ I + ⊂ g ∗ + is the dual basis of { x α } α ∈ I + of g + with reversed parity.The differential Q = ¯ d (0) is given by Q ( u ) = X α ∈ I + ϕ α (cid:0) ( − ¯ α [ x β , u ] ≤ − ( − ¯ u (cid:0) [ x α , u ] / + χ ([ x α , u ] ≥ ) (cid:1)(cid:1) , ( u ∈ g ≤ ) , X α ∈ I + χ ([ x α , u ]) ϕ α , ( u ∈ g / ) , (2.3) Q ( ϕ α ) = − X β,γ ∈ I + ( − ¯ α ¯ γ c γα,β ϕ β ϕ γ , α ∈ I + , (2.4) Q ( ∂a ) = ∂Q ( a ) , Q ( ab ) = Q ( a ) b + ( − ¯ a aQ ( b ) . (2.5)Here we have used the projections g → g ≤ ⊕ g / ⊕ g ≥ , u ( u ≤ , u / , u ≥ ) . We will interpret the complex (gr F C •− , Q ) geometically. To this end, let us firstconsider the finite analogue: the the quotient graded superspacegr fin F C •− := gr F C •− / ( a∂ ( b ) | a, b ∈ gr F C •− ) . It is easy to check that for a vector superspace A , S ∂ ( A ) / ( a∂ ( b ) | a, b ∈ A ) ≃ S ( A ) , where S ( A ) is the symmetric superalgebra of A . Thus we havegr fin F C − ≃ S ( g ≤ ⊕ g / ) ⊗ S (Π g ∗ + ) . Let gr F C − → gr fin F C − , ( a [ a ]), denote the canonical projection. It is naturallya Poisson superalgebra by[ a ][ b ] = [ ab ] , { [ a ] , [ b ] } = [ { a λ b }| λ =0 ] . The Poisson superalgebra gr fin F C •− is called the Zhu’s C -algebra of gr F C •− . By(2.5), the differential Q induces a differential Q fin on gr fin F C •− , which is determinedby the formulas (2.3)-(2.5) with u replaced by [ u ], e.g., Q fin ([ ϕ α ]) = − X β,γ ∈ I + ( − ¯ α ¯ γ c γα,β [ ϕ β ][ ϕ γ ] , α ∈ I + . Then the complex (gr fin F C •− , Q fin ) is identified with the Chevalley-Eilenberg complexof g + with coefficients in S ( g ≤ ⊕ g / ) since differential Q fin defines a left g + -modulestructure on S ( g ≥ ⊕ g / ) by x α · u = ( ( − ¯ α [ x β , u ] ≤ − ( − ¯ u (cid:0) [ x α , u ] / + χ ([ x α , u ] ≥ ) (cid:1) , ( u ∈ g ≤ ) ,χ ([ x α , u ]) , ( u ∈ g / ) , for α ∈ I + . Let G + be the unipotent algebraic supergroup whose Lie superalgebrais g + . The right G + -action( f + g ≥− / ) × G + → ( f + g ≥− / ) , ( f + X, g ) g − ( f + X ) g, induces a left g + -module structure on the coordinate ring C [ f + g ≥− / ]. Then theisomorphism of C -superalgebras S ( g ≤ ⊕ g / ) ≃ C [ f + g ≥− / ] ,u ( − ( − ¯ u ( u | ?) , u ∈ g ≤ , ( u | ?) , u ∈ g / , is a g + -homomorphism. Thus we have proved the following. Lemma 2.4.
The complex (gr fin F C •− , Q fin ) is quasi-isomorphic to the Chevalley-Eilenberg complex of g + with coefficients in C [ f + g ≥− / ] . By Jaconson-Morozov theorem, there exits an sl -triple { e, h, f } ⊂ g ¯0 containing f . Set g e = { X ∈ g | [ e, X ] = 0 } . The subvariety S f = f + g e ⊂ g is called the Slodowy slice of g associated with f . Proposition 2.5.
We have an isomorphism ξ : S f × G + ≃ f + g ≥− , ( X, g ) g − Xg (2.6) of affine supervarieties.Proof. It suffices to show that ξ gives an isomorphism of all the A -valued points foran arbitrary commutative C -superalgebra A . Let f + X ∈ f + g e ( A ) and g ∈ G + ( A ).Let Y ∈ g + ( A ) denote the element corresponding to g by the isomorphism g + ≃ G + .It satisfies Z := g − ( f + X ) g = X n ≥ n ! [ · · · [[ f + X, Y ] , Y ] · · · Y ] | {z } n . N MIURA MAPS FOR W -SUPERALGEBRAS 15 For u ∈ g , decompose it as u = P p u p by the grading Γ. Then we have Z p = , ( p < − ,f, ( p = − ,X p + [ f, Y p +1 ] + R p , ( p ≥ − ) , where R p is the term determined by X i , ( i < p ), and Y j , ( j < p + 1). Now, theassertion follows from the decomposition g p = g ep ⊕ [ f, g p +1 ] , p ≥ − / , which is an immediate consequence of the definition of the good grading Γ. (cid:3) Note that the Poisson superalgeba structure on S ( g ≤ ⊕ g / ) ≃ C [ f + g ≥− / ]restricts to C [ S f ] ≃ C [ f + g ≥− / ] G + . Thus S f is a Poisson supervariety. We remarkthat if Γ is a Z -grading, then S f is the Poisson supervariety obtained as the Hamil-tonian reduction of g with respect to the Adjoint G + -action and an infinitesimalcharacter χ = ( f | -) : g + → C . Corollary 2.6. (1) H n (gr fin F C •− , Q fin ) = 0 for n = 0 .(2) H (gr fin F C •− , Q fin ) ≃ C [ S f ] as Poisson superalgebras.Proof. By Lemma 2.4 and Proposition 2.5, we have H n (gr fin F C •− , Q fin ) ≃ H n ( g + ; C [ f + g ≥− / ]) ≃ H n ( g + ; C [ G + ]) ⊗ C [ S f ] . Since G + is a positively graded unipotent algebraic supergroup, the assertion followsfrom Proposition 1.7. (cid:3) Next, we consider the complex (gr F C •− , Q ). Recall that the arc space JG + of G + is an group superscheme whose Lie superalgebra is J g + = g + [[ t ]]. It followsfrom Lemma 2.4 that (gr F C •− , Q ) is quasi-isomorphic to the Chevalley-Eilenbergcomplex of J g + with coefficients in C [ f + J g ≥− / ] with left J g + -module structureinduced by the right JG + -action J ( f + g ≥− ) × JG + → J ( f + g ≥− ) , ( f + X, g ) g − ( f + X ) g. (2.7)The following is a generalization of [A1, Theorem 5.7]. Lemma 2.7. (1) H n (gr F C •− , Q ) = 0 for n = 0 .(2) H (gr F C •− , Q ) ≃ C [ J S f ] as Poisson vertex superalgebras.Proof. By Lemma 1.10 and Proposition 2.5, we have an isomorphism J S f × JG + ≃ J ( f + g ≥− / ) . Therefore, by using Proposition 1.11, we have H n (gr F C •− , A ) ≃ H n ( J g + ; C [ f + J g ≥− / ]) ≃ H n ( J g + ; C [ JG + ]) ⊗ C [ J S f ] ≃ δ n, C [ J S f ] . (cid:3) The following is a generalization of [A1, Theorem 5.8].
Proposition 2.8. gr F W k ( g , f, Γ) ≃ C [ J S f ] as Poisson vertex superalgebras. Proof.
The filtration F • C •− induces a filtration on H (cid:0) C •− , d (cid:1) by F n H (cid:0) C •− , d (cid:1) = Im (cid:0) Ker( d ) ∩ F n C •− → H ( C •− , d ) (cid:1) and thus a spectral sequence E = H (gr F C •− , Q ) ⇒ H ( C •− , Q ). Recall that C •− is Z ≥ -graded C •− = ⊕ ∆ ∈ Z ≥ C •− , ∆ and that F • C •− induces a filtration on each C •− , ∆ by F p C •− , ∆ = F p C •− ∩ C •− , ∆ . Since it is of finite length, the spectral sequence E r converges. By Lemma 2.7, E r collapses at r = 1:gr F H • ( C •− , d ) ≃ E • = H • (gr F C •− , Q ) ≃ δ n, C [ S f ] . Therefore, it suffices to show gr F W k ( g , f ; Γ) ≃ gr F H ( C •− , d ). Since we have amap F n W k ( g , f ; Γ) → F n H ( C •− , d ) by construction, we obtain a homomorphismof Poisson vertex superalgebrasgr F W k ( g , f ; Γ) → gr F H ( C •− , d ) ≃ C [ J S f ] . By [KW1, Theorem 4.1], W k ( g , f ; Γ) is strongly generated by a basis of g e ≃ f + g e = S f , we have a surjection of Poisson vertex superalgebras C [ J S f ] ։ gr F W k ( g , f ; Γ) , by [A1, Theorem 10]. Since the composition of the above two homomorphisms isthe identity of C [ J S f ], we obtain the assertion. (cid:3) Injectivity of Miura map.
It follows from Proposition 2.8 and the proof ofCorollary 2.6 that the map ¯ µ in (2.2) is identified with the composition C [ J S f ] ֒ → C [ J S f ] ⊗ C [ JG + ] ≃ C [ J ( f + g ≥− / )] ։ C [ J ( f + g ini )] , where g ini = g − / ⊕ g . Note that it is the image of the functor J - in (1.14) of itsfinite analogue¯ µ fin : C [ S f ] ֒ → C [ S f ] ⊗ C [ G + ] ≃ C [ f + g ≥− ] ։ C [ f + g ini ] . (2.8)Since the injectivity of ¯ µ fin implies that of ¯ µ , the proof of Theorem 2.1 is reducedto the following by Corollary 2.3. Proposition 2.9.
The map ¯ µ fin is injective.Proof. Recall we have a functor from the category of affine supervarieties to thecategory of affine varieties of taking the reduced variety: X = Spec( C [ X ]) X red := Spec ( C [ X ] / ( C [ X ] ¯1 ))where ( C [ X ] ¯1 ) ⊂ C [ X ] denotes the ideal generated by the odd subspace of C [ X ], see[CCF]. Applying this functor to ¯ µ fin , we obtain the Miura map for the reductiveLie algebra g ¯0 with the same nilpotent element f . By [G2, Lemma 5.12], it isinjective. Note that the map ¯ µ fin preserves the parity. Then it suffices to show thatthe images of the linear coordinates of the odd part S f, ¯1 are linearly independent.Therefore, it suffices to show that the image of the map G + , ¯0 × ( f + g ini , ¯1 ) → f + g ≥− / ։ S f, ¯1 (2.9)is dense in the Zariski topology. Here the last projection is the composition g ≥− / = g ≥− / , ¯0 ⊕ g ≥− / , ¯1 ։ g ≥− / , ¯1 = g e ≥− / , ¯1 ⊕ ( g ≥− / , ¯1 ∩ Im (ad f )) ։ g e ≥− / , ¯1 . 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