Rationality of the exceptional W-algebras W k ( sp 4 , f subreg ) associated with subregular nilpotent elements of sp 4
aa r X i v : . [ m a t h . R T ] S e p RATIONALITY OF THE EXCEPTIONAL W -ALGEBRAS W k ( sp , f subreg ) ASSOCIATED WITH SUBREGULAR NILPOTENT ELEMENTS OF sp JUSTINE FASQUEL
Abstract.
We prove the rationality of the exceptional W -algebras W k ( g , f )associated with the simple Lie algebra g = sp and a subregular nilpotentelement f = f subreg of sp , proving a new particular case of a conjecture ofKac-Wakimoto. Introduction
Let g be a finite dimensional simple Lie algebra, f a nilpotent element of g and k ∈ C a complex number. The universal affine W -algebra W k ( g , f ) associ-ated with ( g , f ) is a certain vertex algebra obtained from the quantized Drinfeld-Sokolov reduction of the universal affine vertex algebra V k ( g ). The W -algebrascan be regarded as affinizations of finite W -algebras (introduced by Premet [32]),and can also be considered as generalizations of affine vertex algebras and Virasorovertex algebras. The construction of W -algebras was firstly introduced by Feiginand Frenkel [19] for f a principal nilpotent element, and was extended for generalnilpotent elements by Kac, Roan and Wakimoto [26]. The theory of W -algebrasis related with integrable systems [24], the two-dimensional conformal field theory,the geometric Langlands program [23, 20, 10], and the 4d/2d duality [8, 12, 13, 33]in physics.The nicest (conformal) vertex algebras are those which are both rational andlisse. The rationality means the completely reducibility of Z > -graded modules.The lisse condition is equivalent to the fact that the associated variety has dimen-sion 0. If a vertex algebra V is rational and lisse, then it gives rise to a rationalconformal field theory. The rationality condition implies that V has finitely manysimple Z > -graded modules and that the graded components of each of these Z > -graded modules are finite dimensional [17]. In fact lisse vertex algebras also verifythis property [1, 31, 36]. It is actually conjectured by Zhu [36] that rational vertexalgebras must be lisse (this conjecture is still open).It is known [22] that the simple quotient L k ( g ) of V k ( g ) is rational (and lisse) ifand only if it is integrable as a representation of b g , that is, k ∈ Z > , where b g is theaffine Kac-Moody algebra associated with g . It is also known [35] that the simplequotient Vir c of the Virasoro vertex algebra Vir c is rational (and lisse) if and onlyif c = c ( p, q ), where c ( p, q ) = 1 − p − q ) pq , p, q > , ( p, q ) = 1 . More difficult is the study of the rationality of the simple quotient W k ( g , f ) of W k ( g , f ). It was conjectured by Frenkel, Kac and Wakimoto [21], and proved by Arakawa[7], that if k = − h ∨ + p/q ∈ Q is non-degenerate admissible for g and if f = f princ is a principal nilpotent element, then W k ( g , f princ ) is rational. Here, h ∨ is thedual Coxeter number of g . More generally, Kac and Wakimoto [29] conjecturedthat W k ( g , f ) is rational whenever k = − h ∨ + p/q is admissible and ( f, q ) is an exceptional pair [29].It was shown by Arakawa [6] that, for k = − h ∨ + p/q an admissible level, theassociated variety of the simple affine vertex algebra L k ( g ) is precisely the closure ofsome nilpotent orbit O q which only depends on the denominator q . It is also provedin [6] that if f ∈ O q then W k ( g , f ) is lisse. Following [9], we extend the notion ofexceptional pair and say that the pair ( f, q ) is exceptional if f ∈ O q , with q ∈ Z , q >
1. Thus, for k = − h ∨ + p/q an admissible level, the pair ( f, q ) is exceptionalonly if W k ( g , f ) is lisse. It was conjectured in [6] that W k ( g , f ) is rational whenever k is admissible and ( f, q ) is an exceptional pair, now in the broader sense.Arakawa and van Ekeren recently gave strong evidences of this conjecture byshowing that the exceptional W -algebra W k ( g , f ) is rational for a large class ofexceptional pairs ( f, q ). In particular, they proved the conjecture for exceptional W -algebras in type A (where all notions of exceptional pairs coincide) and forexceptional W -algebras associated with a subregular nilpotent element f when g issimply-laced. A particular case of this result was previously established by Arakawa[5] using different methods for the Bershadsky-Polyakov vertex algebra, that is, thesimple W -algebra associated with sl (type A ) and f subregular (in this case, f isalso a minimal nilpotent element of sl ).In this article, we prove the rationality of the exceptional W -algebra W k ( g , f )associated with g = sp ∼ = so (type C = B ) and f = f subreg a subregularnilpotent element of g . Note that the subregular nilpotent orbit O subreg of sp isassociated with the partition (2 ) of 4 and that f subreg is of Levi type which meansthat f is a principal nilpotent element in a Levi subalgebra of sp containing it(see Remark 5.1). It follows from [6] that O q = O subreg if and only if q = 3 or q = 4. The level k is admissible for sp with denominator q = 3 or q = 4 if either k = − p/ p,
3) = 1 and p >
3, or k = − p/ p,
2) = 1 and p > g = sp and f = f subreg . Main Theorem 1.
Let f = f subreg be a subregular nilpotent element of g = sp .Then the exceptional W -algebra W − p/ ( g , f ) , with ( p,
3) = 1 , p > , and theexceptional W -algebra W − p/ ( g , f ) , with ( p,
2) = 1 , p > , are rational (andlisse). Moreover, we have a complete classification of their simple modules. The W -algebras of the main theorem can be viewed as the “easiest” exceptional W -algebras which are not covered by Arakawa and van Ekeren works. It is alsoa natural analog to the Bershadsky-Polyakov vertex algebra for the type C . Infact, the W -algebra W − p/ ( sp , f min ), with ( p,
2) = 1 and p >
4, associatedwith g = sp and f = f min a minimal nilpotent element of sp is also a naturalanalog. The latter is a bit more difficult to study than the one of our main theorembecause of the number of its generators, but we also plan to study its rationality. The level k is admissible if L k ( g ) is admissible as a representation of b g , cf. Sect. 3 for a precisedefinition. It is non-degenerate if the associated variety of L k ( g ) is the whole nilpotent cone of g . The notion of exceptional pair coincides with [18] if f is of Levi type , and with [29] if moreover( q, r ∨ ) = 1, where r ∨ is the lacity of g . ATIONALITY OF THE EXCEPTIONAL W -ALGEBRAS W k ( sp , f subreg ) 3 More generally, we plan to study the rationality of the exceptional minimal W -algebra W − h ∨ + p/ ( sp n , f min ), n >
4, with h ∨ = n/ sp n , for any n >
4, appears as associated varietyof simple admissible affine vertex algebras L k ( sp n ), while it does not for so n for n > W -algebras,operator product expansions (OPE) are not known for an arbitrary W -algebra. Ourfirst step is to compute explicit generators of W k ( sp , f subreg ), k ∈ C , and OPEsbetween them. We refer to Sect. 4 for general facts about W k ( g , f ), for f even , andto Sect. 5 for the computation of OPEs in the case where g = sp and f = f subreg .Next, following ideas of [5], we use the twist action introduced in [30] to obtaina finite set of possible simple W k ( sp , f subreg )-modules, with k as in the theorem(see Sect. 6). Unfortunately, contrary to the case where f = f min is minimal, thefunctor H f (?) so that W k ( g , f ) = H f ( V k ( g )) is not exact for an arbitrary nilpotentelement f . So we cannot use directly the methods of [5] to conclude that our setis exactly the set of simple W k ( sp , f subreg )-modules, with k as in the theorem. Toencounter the difficulty, we exploit technics of [9] to show that all elements of ourset are simple W k ( sp , f subreg )-modules, and that there is no nontrivial extensionsbetween them (see Sect. 7).Some standard facts on vertex algebras are summarized in Sect. 2, and the mainnotation on affine Kac-Moody algebras and corresponding affine vertex algebrascan be found in Sect. 3.As a by-product of our proof, we obtain an explicit description of the irreducible W k ( sp , f subreg )-modules. One can use this to compute the characters of the mod-ules and thus the fusion rules corresponding to the simple W -algebra. We hope togo back to this topic in a future work. Acknowledgments.
The author is very grateful to her thesis advisors Anne Moreauand Tomoyuki Arakawa for suggesting the problem and for useful conversations andcomments. She also thanks the members of the W -algebra working group of theUniversity of Lille.The author acknowledges support from the Labex CEMPI (ANR-11-LABX-0007-01). 2. Vertex algebras A vertex algebra is a complex vector space V equipped with a distinguishedvector | i ∈ V and a linear map V ! (End V )[[ z, z − ]] , a a ( z ) = X n ∈ Z a ( n ) z − n − satisfying the following axioms: • a ( z ) b ∈ V (( z )) for all a, b ∈ V , • (vacuum axiom) | i ( z ) = Id V and a ( z ) | i = a + zV [[ z ]] for all a ∈ V , • (locality axiom) ( z − w ) N [ a ( z ) , b ( w )] = 0 for a sufficiently large N for all a, b ∈ V . It means that the Dynkin grading associated with f is even. JUSTINE FASQUEL
The normally ordered product of two fields a ( z ) = P n ∈ Z a ( n ) z − n − and b ( z ) = P n ∈ Z b ( n ) z − n − of a vertex algebra V is defined by: a ( z ) b ( z ) := a ( z ) + b ( z ) + b ( z ) a ( z ) − , where a ( z ) + := P n< a ( n ) z − n − and a ( z ) − := P n > a ( n ) z − n − . The vertex alge-bra V is said to be strongly generated [26] by a family of fields { a i ( z ) } i ∈ I if thespace of fields of V is spanned by normally ordered products of the fields { a i ( z ) } i ∈ I and their derivatives. This means that, as a vector space, V is spanned by a i ( − n ) . . . a i s ( − n s ) | i (1)with s > n r > i r ∈ I . The structure of V is completely determined bythe OPEs among the a i ( z )’s, i ∈ I , or, equivalently, the Lie brackets in End( V )among the a i r ( n ) ’s. If V is strongly generated by the fields { a i ( z ) } i ∈ I , we call the setof monomials (1), where the sequence of pairs ( i , n ) , . . . , ( i r , n r ) is decreasing inthe lexicographical order, a PBW basis of V .A vertex algebra V is called conformal if there exists a vector ω called the conformal vector such that L ( z ) = P n ∈ Z L n z − n − := Y ( ω, z ) satisfies that(a) [ L m , L n ] = ( m − n ) L m + n + m − m δ m + n, c V , where c V is some constantcalled the central charge of V ,(b) L acts semisimply on V ,(c) L − = T on V , where T : V ! V , a a ( − | i is the translation operator.Here, δ i,j stands for the Kronecker symbol. For a conformal vertex algebra V and a V -module M , we set M d = { m ∈ M : L m = dm } . The L -eivenvalue of a nonzero L -eigenvector m is called the conformal weight .If V is conformal and a ∈ V is homogeneous of conformal weight ∆ a , we set a n = a ( n +∆ a − so that a ( z ) = X n ∈ Z a n z − n − ∆ a , (2)which is more standard notation in physics.Let M be a module over a conformal vertex algebra V of central charge c . Themodule M is called a positive energy representation if L acts semisimply withspectrum bounded below, that is, M = M d ∈ χ + Z > M d , with M χ = 0. Let M top be the top degree component M χ of M .By Zhu’s theorem [36], the correspondence M M top gives a bijection betweenthe set of isomorphism classes of irreducible positive energy representations of V and that of simple Zhu( V )-modules, where Zhu( V ) is the Zhu algebra of V (see, forexample, [11] for more detail).With every vertex algebra V one associates a Poisson algebra R V , called the Zhu C -algebra , as follows ([36]). Let C ( V ) be the subspace of V spanned by theelements a ( − b , where a, b ∈ V , and set R V = V /C ( V ). Then R V is naturally aPoisson algebra by1 = | i , ¯ a. ¯ b = a ( − b and { ¯ a, ¯ b } = a (0) b, where ¯ a denotes the image of a ∈ V in the quotient R V . ATIONALITY OF THE EXCEPTIONAL W -ALGEBRAS W k ( sp , f subreg ) 5 The associated variety [4] X V of a vertex algebra V is the affine Poisson varietydefined by X V = Specm R V . A vertex algebra V called lisse if dim X V = 0.3. Affine vertex algebras
Let g be a complex simple Lie algebra as in introduction. Let g = n − ⊕ h ⊕ n + be a triangular decomposition with a Cartan subalgebra h , ∆ the root system of( g , h ), ∆ + a set of positive roots for ∆ and Π = { α , . . . , α ℓ } the corresponding setof simple roots. Let θ be the highest positive root. We also have ∆ ∨ the set ofcoroots. Let P be the weight lattice, Q the root lattice and Q ∨ the coroot lattice.Let e g = g [ t, t − ] ⊕ C K ⊕ C D be the extended affine Kac-Moody algebra, withthe commutation relations:[ xt m , yt n ] = [ x, y ] t m + n + mδ m + n, ( x | y ) K, [ D, xt n ] = − nxt n , [ K, b g ] = 0 , for all x, y ∈ g and all m, n ∈ Z , where ( | ) = 12 h ∨ × κ g is the normalizedinvariant inner product of g , κ g is the Killing form of g and xt n stands for x ⊗ t n ,for x ∈ g , n ∈ Z . Let e g = b n − ⊕ e h ⊕ b n + be the standard triangular decomposition,that is, e h = h ⊕ C K ⊕ C D is a Cartan subalgebra of e g , b n + = n + + t g [ t ] and b n − = n − + t − g [ t − ].Let b g = [ e g , e g ] = g [ t, t − ] ⊕ C K , and let b h = h ⊕ C K ⊂ b g , so that b g = b n − ⊕ b h ⊕ b n + .The Cartan subalgebra e h is equipped with a bilinear form extending that on h by( K | D ) = 1 , ( h | C K ⊕ C D ) = ( K | K ) = ( D | D ) = 0 . We write δ and Λ for the elements of e h ∗ orthogonal to h ∗ and dual to K and D ,respectively. We have the (real) root systems b ∆ re = { α + nδ | n ∈ Z , α ∈ ∆ } = b ∆ re + ⊔ ( − b ∆ re + ) , b ∆ re + = { α + nδ | α ∈ ∆ + , n > } ⊔ { α + nδ | α ∈ ∆ + , n > } , and the affine Weyl group c W , generated by reflections r α with α ∈ b ∆ re . For α ∈ h ∗ the translation t α : e h ∗ ! e h ∗ is defined by t α ( λ ) = λ + λ ( K ) α − (cid:20) ( α | λ ) + | α | λ ( K ) (cid:21) δ. For α ∈ Q ∨ we have t α ∈ c W and in fact c W ∼ = W ⋉ t Q ∨ , where t Q ∨ := { t α : α ∈ Q ∨ } and W is the Weyl group of g . The extended affine Weyl group, which is the groupof isometries of b ∆, is f W = W ⋉ t P , where t P := { t α : α ∈ P } .Let O k be the category O of e g at level k ([25]). The simple objects of O k arethe irreducible highest weight representations L ( λ ) for λ ∈ e h ∗ with λ ( K ) = k . Fora weight λ ∈ b h ∗ the corresponding integral root system is b ∆( λ ) = { α ∈ b ∆ re | h λ, α ∨ i ∈ Z } , where α ∨ = 2 α/ ( α | α ) as usual.A weight λ ∈ b h ∗ is said to be admissible (equivalently, we said that L ( λ ) is admissible ) if • λ is regular dominant, that is, h λ + ˆ ρ, α ∨ i / ∈ Z for all α ∈ b ∆ re + , JUSTINE FASQUEL • Q b ∆ re = Q b ∆( λ ).Here ˆ ρ = ρ + h ∨ Λ , with ρ = P α ∈ ∆ + α/
2. A complex number k is said to be admissible if λ = k Λ is an admissible weight. Proposition 3.1 ([27]) . A complex number k is admissible if and only if it is ofthe form k = − h ∨ + pq , with p, q ∈ Z > , ( p, q ) = 1 and either p > h ∨ if ( r ∨ , q ) = 1 , or p > h if ( r ∨ , q ) = r ∨ .Here r ∨ is the lacity of g , that is, r ∨ = 1 if g has type A ℓ , D ℓ , E , E , E , r ∨ = 2 if g has type B ℓ , C ℓ , F and r ∨ = 3 if g has type G . For k an admissible number, let P r k be the set of weights λ ∈ h ∗ such that b λ = λ + k Λ is admissible and there exists y ∈ f W such that b ∆( b λ ) = y ( b ∆( k Λ )).The weights of P r k are said principal admissible if ( r ∨ , q ) = 1 and coprincipaladmissible if ( r ∨ , q ) = r ∨ .3.1. Affine vertex algebras.
Given any k ∈ C , let V k ( g ) = U ( b g ) ⊗ U ( g [ t ] ⊕ C K ) C k , where C k is the one-dimensional representation of g [ t ] ⊕ C K on which g [ t ] acts by0 and K acts as a multiplication by the scalar k . There is a unique vertex algebrastructure on V k ( g ) such that | i is the image of 1 ⊗ V k ( g ) and x ( z ) := ( x ( − | i )( z ) = X n ∈ Z ( xt n ) z − n − for all x ∈ g , where we regard g as a subspace of V through the embedding x ∈ g ֒ ! x ( − | i ∈ V k ( g ). The vertex algebra V k ( g ) is called the universal affinevertex algebra associated with g at level k .The vertex algebra V k ( g ) is conformal by Sugawara construction provided that k is non-critical, that is, k = − h ∨ , with central charge c V k ( g ) = k dim g k + h ∨ . We write L g ( z ) = X n ∈ Z L n z − n − = Y ( ω g , z ) , where ω g is the Sugawara conformal vector.A V k ( g )-module is the same as a smooth b g -module of level k . For a non-criticallevel k , we consider a V k ( g )-module M as a e g -module by letting D act as thesemisimplification of − L . Let L k ( g ) be the unique simple graded quotient of V k ( g ). Then L k ( g ) ∼ = L ( k Λ ) as a representation of b g .For any graded quotient V of V k ( g ), we have R V = V /t − g [ t − ] V . In particular, R V k ( g ) ∼ = C [ g ∗ ] and, hence, X V k ( g ) = g ∗ . Furthermore, X L k ( g ) is a subvarietyof g ∗ ∼ = g , which is G -invariant and conic with G the adjoint group of g . Theassociated variety X L k ( g ) is difficult to compute in general. However, it is knownthat X L k ( g ) = { } if and only if L k ( g ) ∼ = L ( k Λ ) is an integral representation of b g ,that is, k ∈ Z > . Furthermore, we have the following result: Proposition 3.2 ([6]) . If k = − h ∨ + p/q is an admissible level for g , then X L k ( g ) is the closure of some nilpotent orbit O q which only depends on q . The nilpotent orbit O q is explicitly described in [6, Tables 2–10]. ATIONALITY OF THE EXCEPTIONAL W -ALGEBRAS W k ( sp , f subreg ) 7 The BRST reduction and W -algebras Let f be a nilpotent element of g that we embed into an sl -triple ( e, h, f ) throughthe Jacobson-Morosov theorem:[ h, e ] = 2 e, [ h, f ] = − f, [ e, f ] = h. The semisimple element x := h/ Z -gradation on g , g = M j ∈ Z g j , where g j = { y ∈ g : [ x , y ] = jy } . Set g > := L j > g j and g > := L j> g j . Wedefine similarly g and g < . In this note, we assume that g j = 0 for j Z , which is sufficient for our purpose. In other words, we assume that f is an even nilpotent element (which is the case of subregular nilpotent elements of sp ). Choosea basis { e α } α ∈ S j of each g j . One can assume that each e α is a root vector providedthat j = 0, and that for j = 0 either e α is a root vector or e α belongs to the Cartansubalgebra h . Set S = ⊔ j S j , S + = ⊔ j> S j and let { e α } α ∈ S + be the dual basis in g ∗ > to { e α } α ∈ S + .The Clifford affinization c Cl ( g > ) of g > is the Clifford algebra associated with g > [ t, t − ] ⊕ g ∗ > [ t, t − ] and the symmetric bilinear form defined by h xt m , ψt n i = δ m + n, ψ ( x ) , h xt m , yt n i = h ψt m , φt n i = 0 , for x, y ∈ g > , ψ, φ ∈ g ∗ > . We write ϕ α,m for e α t m ∈ c Cl ( g > ) and ϕ αm for e α t m ∈ c Cl ( g > ), so that c Cl ( g > ) is the associative superalgebra with • odd generators: ϕ α,m , ϕ αn , m, n ∈ Z , α ∈ S + , • relations: [ ϕ α,m , ϕ β,n ] = [ ϕ αm , ϕ βn ] = 0, [ ϕ α,m , ϕ βn ] = δ α,β δ m + n, ,where the parity of ϕ α,m and ϕ αn is reverse to e α . Since elements of g are purelyeven, this means that ϕ α,m and ϕ αn are odd.Define the charged fermion Fock space associated with g > as F ( g > ) := c Cl ( g > ) P m > α ∈ S + c Cl ( g > ) ϕ α,m + P k > α ∈ S + c Cl ( g > ) ϕ αk ∼ = V(cid:0) ϕ α,n (cid:1) n< α ∈ S + ⊗ V(cid:0) ϕ αm (cid:1) m α ∈ S + , where V ( a i ) i ∈ I denotes the exterior algebra with generators a i , i ∈ I . It is anirreducible c Cl ( g > )-module, and as C -vector spaces we have F ( g > ) ∼ = V ( g ∗ > [ t − ]) ⊗ V ( g > [ t − ] t − ) . There is a unique vertex (super)algebra structure on F ( g > ) such that the imageof 1 is the vacuum | i and Y ( ϕ α, − | i , z ) = ϕ α ( z ) := X n ∈ Z ϕ α,n z − n − , α ∈ S + ,Y ( ϕ α | i , z ) = ϕ α ( z ) := X n ∈ Z ϕ αn z − n , α ∈ S + . We call this vertex algebra the free superfermion vertex algebra associated with c Cl ( g > ). JUSTINE FASQUEL
The following construction of the W -algebra associated with g and f is due toKac, Roan and Wakimoto [26, 28, 2] (see also [11] for a survey). Let k ∈ C and set C ( g , f, k ) = V k ( g ) ⊗ F ( g > ) . We now define a differential d (0) on C ( g , f, k ). The vertex algebra F ( g > ) has thecharge decomposition: F ( g > ) = M m F m ( g > ) , where charge ϕ α ( z ) = − charge ϕ α ( z ) = 1 for α ∈ S + . Letting charge V k ( g ) = 0,this induces the charge decomposition: C ( g , f, k ) = M m C m , C m := C m ( g , f, k ) . Following [26], we set d ( z ) = X α ∈ S + : e α ( z ) ϕ α ( z ) : − X α,β,γ ∈ S + c γα,β ϕ α ( z ) ϕ β ( z ) ϕ γ ( z ) + X α ∈ S + ( f | e α ) ϕ α ( z ) , where [ e α , e β ] = P γ c γα,β e γ . The field d ( z ) does not depend on the choice of thebasis. One has [ d ( z ) , d ( w )] = 0 and therefore d = 0 since d ( z ) is odd. Moreover,[ d (0) , C m ] ⊂ C m − . Thus ( C ( g , f, k ) , d (0) ) is a Z -graded homology complex. Thezero-th homology of this complex is a vertex algebra, denoted by W k ( g , f ): W k ( g , f ) := H ( C ( g , f, k ) , d (0) ) , that we briefly write W k ( g , f ) = H f ( V k ( g )) . This construction is a particular case of BRST reduction which is usually referredto as the Drinfeld-Sokolov reduction of V k ( g ). The vertex algebra W k ( g , f ) is calledthe (universal) W -algebra associated with g and f at the level k . Its simple gradedquotient will be denoted by W k ( g , f ).Provided that k = − h ∨ , define the conformal field by: L ( z ) = L g ( z ) + ddz x ( z ) + L ch ( z ) , where L ch ( z ) = − X α ∈ S + m α : ϕ α ∂ϕ α : + X α ∈ S + (1 − m α ) : ∂ϕ α ϕ α : . Here m α is defined by letting m α = j is e α ∈ g j . The central charge of W k ( g , f ) is: c W k ( g ,f ) = dim g − k + h ∨ | ρ − ( k + h ∨ ) x | . It is known [15] that the associated variety of W k ( g , f ) is the Slodowy slice S f := f + g e , where g e is the centralizer of e in g . Moreover by [6], the associatedvariety of H f ( L k ( g )) is X L k ( g ) ∩ S f . Since W k ( g , f ) is a quotient of H f ( L k ( g )),provided that H f ( L k ( g )) = 0, we deduce that W k ( g , f ) is lisse whenever X L k ( g ) = O and f ∈ O , where O is some nilpotent orbit of g . Indeed, we have O ∩ S f = { f } if f ∈ O . In particular, we get the following result, useful for our purpose. Proposition 4.1 ([6]) . Assume that k = − h ∨ + p/q is admissible for g and pick f ∈ O q , with O q the associated variety of L k ( g ) (cf. Proposition 3.2). Then W k ( g , f ) is lisse. ATIONALITY OF THE EXCEPTIONAL W -ALGEBRAS W k ( sp , f subreg ) 9 Generating fields of W k ( g , f ) . Given a ∈ g j , introduce the following fieldsof C ( g , f, k ) of conformal weight 1 − j : J a ( z ) = a ( z ) + X α,β ∈ S + c αβ ( a ) : ϕ α ( z ) ϕ ∗ β ( z ) : , where c αβ ( a ) is defined by [ a, e β ] = P α ∈ S c βα ( a ) e α . Those fields play an importantrole in the structure of the W -algebra W k ( g , f ).Let g f be the centralizer of f in g , and set for j ∈ Z , g fj := g f ∩ g j . By thetheory of sl , we have g f = M j g fj . (3) Theorem 4.2 ([28, Theorem 4.1]) . For each a ∈ g fj , j , there exists a field J { a } ( z ) of W k ( g , f ) of conformal weight − j with respect to L such that J { a } ( z ) − J a ( z ) is a linear combination of normally order products of the fields J b ( z ) , where b ∈ g − s , s < j and their derivatives.Let { a i } i ∈ I be a basis of g f compatible with the graduation (3) . Then W k ( g , f ) is strongly generated by the fields { J { a i } } i ∈ I . In practice, to construct J { a } ( z ) from J a ( z ), with a ∈ g f , we write a linearcombination as in the theorem and try to find the coefficients so that the field J { a } ( z ) is d (0) -closed.5. Generators of W k ( sp , f ) and OPEs From now on, g refers to the simple Lie algebra sp that we may realize as the setof 4-size square matrices x such that x t J + J x = 0, where J is the anti-diagonalmatrix given by J := − − . We make the standard choice that h is the set of diagonal matrices of g . Nilpotentorbits of g = sp are parametrized by the partitions of 4 such that the number ofparts of each odd number is even (see, for instance, [14, Theorem 5.1.3]). Thusthere are four nilpotent orbits in g = sp corresponding to the following partitions:(4), (2 ), (2 , ), (1 ). They correspond, respectively, to the principal (or regular),the subregular, the minimal and the zero nilpotent orbits of g , with respectivedimensions 8, 6, 4, 0.Write Π = { α , α } a set of simple roots for the root system ∆ of ( g , h ) such that α is a long root and α is short. Then ∆ + = { α , α , η, θ } , with η := α + α and θ := α + 2 α the highest positive root. The centralizer of e − η is four-dimensionalgenerated by e − η , e − α , e − θ , h , where h i := α ∨ i ∈ ( h ∗ ) ∗ ∼ = h , for i = 1 ,
2. Hence f := e − η = f subreg belongs to the subregular nilpotent orbit of g . Setting e := e η and h := [ e, f ] =2 h + h we get the sl -triple ( e, h, f ) of g . The nilpotent element f is even and wehave: g = g − ⊕ g ⊕ g . Moreover, g f = g f − ⊕ g f , with g f − = C f ⊕ C e − α ⊕ C e θ and g f = C h . Remark . The smallest Levi subalgebra of sp containing f = e − η has semisimpletype A with basis h , h , e ± η (it is the centralizer in sp of h ), and, hence, f hasLevi type since it is principal in it.Using Theorem 4.2 and the computer program Mathematica (there is a Mathe-matica package [34] which provides a computer program for the OPE calculations),we obtain that the vertex algebra W k ( g , f ) is strongly generated by the fields J ( z ), G ± ( z ) and L ( z ), provided that k = −
3, where: J ( z ) = J { h } ( z ) = 12 J h ( z ) ,G + ( z ) = J { e − α } ( z ) = J e − α ( z ) + 12 (: J h ( z ) J e α : +( k + 2) ∂J e α ( z )) ,G − ( z ) = J { e − θ } ( z ) = J e − θ ( z ) + 12 (: J h ( z ) J e − α ( z ) : + : J h ( z ) J e − α ( z ) : +( k + 2) ∂J e − α ( z )) ,L ( z ) = J { f } ( z ) = 1(3 + k ) ( − J f ( z ) + 12 (: J e α ( z ) J e − α ( z ) : + : J h ( z ) : + : J h ( z ) J h ( z ) :+ (5 + 2 k ) ∂J h ( z ))+ : J ( z ) : +(1 + k ) ∂J ( z )) . These fields satisfy the following OPE’s : J ( z ) J ( w ) ∼ (2 + k )( z − w ) ,J ( z ) G ± ( w ) ∼ ± z − w ) G ± ( w ) ,L ( z ) L ( w ) ∼ c k z − w ) + 2( z − w ) L ( w ) + 1( z − w ) ∂L ( w ) ,L ( z ) G ± ( w ) ∼ z − w ) G ± ( w ) + 1( z − w ) ∂G ± ( w ) ,L ( z ) J ( w ) ∼ z − w ) J ( w ) + 1( z − w ) ∂J ( w ) ,G ± ( z ) G ± ( w ) ∼ ,G + ( z ) G − ( w ) ∼ − k )(2 + k ) ( z − w ) − k )(2 + k )( z − w ) J ( w )+ 1( z − w ) (cid:18) (2 + k )(3 + k ) L ( w ) − (3 + 2 k ) : J ( w ) : − k )(2 + k )2 ∂J ( w ) (cid:19) + 1( z − w ) (cid:18) (3 + k ) : L ( w ) J ( w ) : + (3 + k )(2 + k )2 ∂L ( w ) − : J ( w ) : − (3 + 2 k ) : J ( w ) ∂J ( w ) : − (5 + 4 k + k )2 ∂ J ( w ) (cid:19) , where c k := − k + 6 k )3 + k . The field L ( z ) = P n ∈ Z L n z − n − is a conformal vector of W k ( g , f ) with centralcharge c k . It gives J ( z ), G + ( z ) and G − ( z ) the conformal weights 1, 2 and 2, ATIONALITY OF THE EXCEPTIONAL W -ALGEBRAS W k ( sp , f subreg ) 11 respectively. Following (2) we write J ( z ) = X n ∈ Z J n z − n − , G ± ( z ) = X n ∈ Z G ± n z − n − , L ( z ) = X n ∈ Z L n z − n − . The monomials J n . . . J n j L m . . . L m t G − p . . . G − p g G + q . . . G + q h | i , with n . . . n j − m . . . m t − p . . . p g − q . . . q h −
2, form a PBW basis of the vertex algebra W k ( g , f ).From the OPEs, we deduce the commutation relations for all m, n ∈ Z :[ J m , J n ] =(2 + k ) mδ m + n, , [ J m , G ± n ] = ± G ± m + n , [ L m , L n ] = c k
12 ( m − m ) δ n + m, + ( m − n ) L m + n , [ L m , G ± n ] =( m − n ) G ± m + n , [ L m , J n ] = − nJ m + n , [ G + m , G − n ] = − (1 + k )(2 + k ) m − m ) δ m + n, + (2 + k )(3 + k )2 ( m − n ) L m + n + (cid:18) k )(2 + k )2 ( m + 1)( n + 1) − (5 + 4 k + k )2 ( m + n + 1)( m + n + 2) (cid:19) J m + n − (3 + 2 k )( m + 1)( J ) m + n + (3 + k )( LJ ) m + n − ( J ) m + n − (3 + 2 k )( J∂J ) m + n , where X n ∈ Z ( J ) n z − n − = : J ( z ) : , X n ∈ Z ( LJ ) n z − n − = : L ( z ) J ( z ) : , X n ∈ Z ( J∂J ) n z − n − = : J ( z ) ∂J ( z ) : . The twist ψ -action over simple W k ( sp , f subreg ) -modules We continue to assume that g = sp and f = f subreg = e − η and we keep thenotation of the previous section. We assume for the moment that k is any non-critical level for g , that is, k = − M be an irreducible positive energy representation of W k ( g , f ), with M top = M χ , χ ∈ C (see Sect. 2). Using the commutation relations of Sect. 5, we noticethat the submodule N of M generated by all J -eigenvectors of M is stable by L n , G ± n , J n , n ∈ Z . Hence, N = M because M is simple. Lemma 6.1.
Let M be an irreducible positive energy representation of W k ( g , f ) ,with M top = M χ , χ ∈ C . Suppose that M top is finite dimensional. Then, there isa vector v ∈ M such that L v = χv , J v = ξv for some ξ ∈ C , and such that thebelow relations hold: J n v = 0 for n > ,L n v = 0 for n > ,G + n v = 0 for n > ,G − n v = 0 for n > . Moreover, M = L ( a,d ) ∈ C d ∈ χ + Z > M a,d , where M a,d = { m ∈ M : J m = am, L m = dm } , dim M ξ,χ = 1 and M top = M χ is spanned by the vectors ( G +0 ) i v for i > . Proof.
Since J and L commute, the action of J preserve each M n , n ∈ Z > .Moreover J is semisimple over M and each M n , then M can be written as M = M ( a,d ) ∈ C d ∈ χ + Z > M a,d . As M top is finite dimensional, there is an vector v ∈ M top such that J v = ξv , ξ ∈ C and ξ − n is not an eigenvalue of J in M top for all n ∈ Z > . The relationsof the lemma result from the following equations. Let n ∈ Z , J J n v = ξJ n v, L J n v = ( χ − n ) J n v,J L n v = ξL n v, L L n v = ( χ − n ) L n v,J G ± n v = ( ξ ± G ± n v, L G ± n v = ( χ − n ) G ± n v. We explain the relation G − n v = 0, n >
0, the others are obtained similarly. Since M is a positive energy representation of W k ( g , f ), for n > χ − n is not an eigenvalueof L and G − n v = 0. Besides, G − v ∈ M top and the choice of v implies that ξ − J in M top . Hence G − v = 0. The vectors ( G +0 ) i v , i >
0, arethe only ones attached to the eigenvalue χ for L and J ( G +0 ) i v = ( ξ + i ) G +0 v for i >
0. As a consequence they span M top and M ξ,χ = C v . (cid:3) For ( ξ, χ ) ∈ C , let L ( ξ, χ ) be the irreducible representation of W k ( g , f ) gen-erated by a vector v = | ξ, χ i satisfying the relations of Lemma 6.1. According toLemma 6.1, | ξ, χ i is uniquely defined up to nonzero scalar, so the notation is legit-imate. Zhu’s correspondence ensures that such L ( ξ, χ ) does exist and is unique upto isomorphism of W k ( g , f )-modules (see, for example, [11]).Since G − G +0 | ξ, χ i is in L ( ξ, χ ) χ,ξ , it is proportional to | ξ, χ i , and we easily obtainthat G − G +0 | ξ, χ i = g ( ξ, χ ) | ξ, χ i , where g ( ξ, χ ) = − ξ (cid:0) k + k − ξ + 6 χ + 2 kχ (cid:1) . Hence for i > G − ( G +0 ) i | ξ, χ i = i h i ( ξ, χ )( G +0 ) i − | ξ, χ i , where h i ( ξ, χ ) = 1 i i − X m =0 g ( ξ + m, χ )= (2 ξ + i − − − i + i − k − k − ξ + 2 iξ + 2 ξ − χ − kχ ) . Proposition 6.2.
Suppose that L ( ξ, χ ) top is n -dimensional. Then h n ( ξ, χ ) = 0 .Proof. If dim L ( ξ, χ ) top = n then ( G +0 ) n | ξ, χ i = 0 and ( G +0 ) n − | ξ, χ i 6 = 0. It resultsfrom (4) that h n ( ξ, χ ) = 0. (cid:3) Following the ideas of [5], we introduce the twist-action ψ ([30]). Let define∆( − J, z ) := z − J exp ∞ X m =1 ( − m +1 − J m mz m ! . ATIONALITY OF THE EXCEPTIONAL W -ALGEBRAS W k ( sp , f subreg ) 13 For a ∈ W k ( g , f ), ∆( − J, z ) a = z − J ∞ X n =0 X n n ! a ! , where X = ∞ P m =1 ( − m +1 − J m kz m and z − J is defined by z − J a = z − c a if J a = ca . Set X n ∈ Z ψ ( a ( n ) ) z − n − := Y (∆( − J, z ) a, z ) = ∞ X n =0 Y ( z − J X n n ! a, z ) . For any W k ( g , f )-module M , the space ψ ( M ) denotes the W k ( g , f )-module ob-tained by twisting the action of W k ( g , f ) as a ( n ) ψ ( a ( n ) ). The following relationsare obtained by applying the ψ -action to the generators of W k ( g , f ): ψ ( J n ) = J n − (2 + k ) δ n, ,ψ ( L n ) = L n − J n + (2 + k )2 δ n, ,ψ ( G + n ) = G + n − ,ψ ( G − n ) = G − n +1 . Proposition 6.3.
Assume that dim L ( ξ, χ ) top = i and dim ψ ( L ( ξ, χ )) top = j . Then ψ ( L ( ξ, χ )) ≃ L ( ξ + i + j − − k , χ − ξ − i − j + 7 + 2 k ) . Proof.
For all m > ψ ( J )( G +0 ) m | ξ, χ i = ( ξ + m − (2 + k ))( G +0 ) m | ξ, χ i ,ψ ( L )( G +0 ) m | ξ, χ i = ( χ − ( ξ + m ) + (2 + k )2 )( G +0 ) m | ξ, χ i . Since the smallest eigenvalue associated with the ψ ( L )-action is attached to thevector ( G +0 ) i − | ξ, χ i , we get ψ ( L ( ξ, χ )) ≃ L ( ξ + ( i − − (2 + k ) , χ − ξ − ( i −
1) + (2 + k )2 )and ψ ( L ( ξ, χ )) ≃ ψ ( L ( ξ + ( i − − (2 + k ) , χ − ξ − ( i −
1) + (2 + k )2 )) ≃ L ( ξ + ( i −
1) + ( j − − k ) , χ − ξ − i − − ( j −
1) + 2(2 + k )) . (cid:3) Remark . For all m, n ∈ Z > , ψ ( J )( G + − ) m ( G +0 ) n | ξ, χ i = ( ξ + n + m − k ))( G + − ) m ( G +0 ) n | ξ, χ i ,ψ ( L )( G + − ) m ( G +0 ) n | ξ, χ i = ( χ − ξ − n − m + 2(2 + k ))( G + − ) m ( G +0 ) n | ξ, χ i . Proposition 6.5.
Suppose that dim L ( ξ, χ ) top = i , dim ψ ( L ( ξ, χ )) top = j and dim ψ ( L ( ξ, χ )) top = l . (a) If k = − p/ with ( p,
3) = 1 , p > then ( ξ, χ, l ) = ( ξ ( s ) i,j , χ ( s ) i,j , l ( s ) i,j ) with s ∈ { , , } , where ξ (1) i,j = 1 − i , χ (1) i,j = 13 − i + i − j + 2 ij + 2 j + 6 k − ik − jk k ) ,l (1) i,j = 9 − i − j + 3 k,ξ (2) i,j = 7 − i − j + 2 k , χ (2) i,j = 31 − i + 2 i − j + 2 ij + j + 18 k − ik − jk + 2 k k ) ,l (2) i,j = i,ξ (3) i,j = 4 − i − j + k , χ (3) i,j = 4 + i − j + j − jk − k k ) ,l (3) i,j = 9 − i − j + 3 k. (b) If k = − p/ with ( p,
2) = 1 , p > then ( ξ, χ, l ) = ( ξ ( s ′ ) i,j , χ ( s ′ ) i,j , l ( s ′ ) i,j ) with s ∈ { , } , where ξ (1 ′ ) i,j = 1 − i , χ (1 ′ ) i,j = 13 − i + i − j + 2 ij + 2 j + 6 k − ik − jk k ) ,l (1 ′ ) i,j = 12 − i − j + 4 k,ξ (2 ′ ) i,j = 7 − i − j + 2 k , χ (2 ′ ) i,j = 31 − i + 2 i − j + 2 ij + j + 18 k − ik − jk + 2 k k ) ,l (2 ′ ) i,j = i. Proof.
By solving the system of equations h i ( ξ, χ ) = 0 ,h j ( ξ + ( i − − (2 + k ) , χ − ξ − ( i −
1) + (2 + k )2 ) = 0 ,h l ( ξ + ( i −
1) + ( j − − k ) , χ − ξ − i − − ( j −
1) + 2(2 + k )) = 0 . we find nine triples ( ξ, χ, l ) in term of i , j and k . Since l is the dimension of ψ ( L ( ξ, χ )) top , it must be a positive integer. If k = − p/
3, with ( p,
3) = 1, p >
3, the three triples described in the first part of the proposition are the onlyones among the solutions of the system corresponding to this restrictive condition.Similarly, if k = − p/
4, ( p,
2) = 1, p >
4, we find that only two triples verify thecondition. (cid:3)
Proposition 6.6. (a)
Let k = − p/ with ( p,
3) = 1 , p > then ( G + − ) p − | i belongs to the maximal ideal of W k ( g , f ) . (b) Let k = − p/ with ( p,
2) = 1 , p > then ( G + − ) p − | i belongs to themaximal ideal of W k ( g , f ) .Proof. (a) For i = j = 1, we have l (1)1 , = p −
2. Since ξ (1)1 , = χ (1)1 , = 0 and L | i = J | i = 0 the correspondence | i | ξ (1)1 , , χ (1)1 , i yields the isomorphism W k ( g , f ) ≃ L ( ξ (1)1 , , χ (1)1 , ) . Moreover ψ ( W k ( g , f )) top is at most p − h p − ( − k ) , k )) = 0 . Hence ( G + − ) p − | i = ψ (( G +0 ) p − ) | i = 0. ATIONALITY OF THE EXCEPTIONAL W -ALGEBRAS W k ( sp , f subreg ) 15 (b) The argument is the same as in the previous case with l (1 ′ )1 , = p − ξ (1 ′ )1 , = χ (1 ′ )1 , = 0. (cid:3) Assume that either k = − p/
3, with ( p,
3) = 1, p >
3, or k = − p/
4, with( p,
2) = 1, p >
4. Then k is admissible for g and by our choice of f = f subreg , wehave f ∈ O q , with q the denominator of k + 3. So by Proposition 4.1, the simplequotient W k ( g , f ) of W k ( g , f ) is lisse. As a consequence, any simple W k ( g , f )-module is positively graded with finite dimensional top degree component and sois of the form L ( ξ, χ ) for some ( ξ, χ ) ∈ C .We are now in a position to state the main result of this section. Proposition 6.7.
Let M be a simple W k ( g , f ) -module. (a) If k = − p/ with ( p,
3) = 1 , p > , then the W k ( g , f ) -module M isisomorphic to L ( ξ ( s ) i,j , χ ( s ) i,j ) for ξ ( s ) i,j and χ ( s ) i,j as in Proposition 6.5(a) with i p − , j p − i − and s ∈ { , , } . (b) If k = − p/ with ( p,
2) = 1 , p > , then the W k ( g , f ) -module M isisomorphic to L ( ξ ( s ′ ) i,j , χ ( s ′ ) i,j ) for ξ ( s ′ ) i,j and χ ( s ′ ) i,j as in Proposition 6.5(b) with i p − and j ( p − i − / if s = 1 or i p − and j p − i − if s = 2 .Proof. (a) Since M is a simple W k ( g , f )-module, there exist ξ, χ ∈ C such that M ∼ = L ( ξ, χ ). By Proposition 6.6, G + ( z ) p − = 0. Hence: G + ( z ) p − : def = X n ∈ Z (( G + ) p − ) n z − n − = 0 . In particular ( G +0 ) p − | ξ, χ i = (( G + ) p − ) | ξ, χ i = 0. As a consequence L ( ξ, χ ) top isat most ( p − ψ ( L ( ξ, χ )) is a simple W k ( g , f )-module, there exist 1 i, j p − s ∈ { , , } such that ξ = ξ ( s ) i,j and χ = χ ( s ) i,j . In the same way, ψ ( L ( ξ, χ )) is a simple W k ( g , f )-module and there are1 l, m p − r ∈ { , , } such that ψ ( ξ ) := ξ + ( i −
1) + ( j − − k ) = ξ ( r ) l,m and ψ ( χ ) := χ − ξ − i − − ( j −
1) + 2(2 + k ) = χ ( r ) l,m . The ψ -actionpermutes the three forms of the eigenvalues ξ and χ : ψ ( L ( ξ (1) i,j , χ (1) i,j )) ≃ L ( ξ (2) p − i − j,i , χ (2) p − i − j,i ) ,ψ ( L ( ξ (2) i,j , χ (2) i,j )) ≃ L ( ξ (3) i,p − i − j , χ (3) i,p − i − j ) ,ψ ( L ( ξ (3) i,j , χ (3) i,j )) ≃ L ( ξ (1) p − i − j,j , χ (1) p − i − j,j ) . The condition j p − i − l, m p − G + ( z ) p − = 0 and L ( ξ, χ ) top is at most ( p − ψ ( L ( ξ, χ )) and ψ ( L ( ξ, χ ))are simple W k ( g , f )-modules, ξ = ξ ( s ′ ) i,j , χ = χ ( s ′ ) i,j , ψ ( ξ ) = ξ ( r ′ ) l,m and ψ ( χ ) = χ ( r ′ ) l,m with 1 i, j, l, m p − r, s ∈ { , } . On the contrary of the first case, the ψ -action preserves the form of the eigenvalues ξ and χ : ψ ( L ( ξ (1 ′ ) i,j , χ (1 ′ ) i,j )) ≃ L ( ξ (1 ′ ) p − i − j,j , χ (1 ′ ) p − i − j,j ) ,ψ ( L ( ξ (2 ′ ) i,j , χ (2 ′ ) i,j )) ≃ L ( ξ (2 ′ ) i,p − i − j , χ (2 ′ ) i,p − i − j ) . If s = 1 then the condition p − i − j > j p − i − , and if s = 2, with thesame argument we get j p − i − (cid:3) Remark . The simple W k ( g , f )-modules L ( ξ ( s ) i,j , χ ( s ) i,j ) of the Proposition 6.7 areall mutually non-isomorphic since their highest weights are distinct. Remark . For k = − p/
3, with ( p,
3) = 1, p >
3, or k = − p/
4, with( p,
2) = 1, p >
4, the application ψ is a bijection of the set of the simple W k ( g , f )-modules L ( ξ ( s ) i,j , χ ( s ) i,j ) described in the Proposition 6.7 over itself of inverse ψ if k isprincipal admissible, and ψ otherwise. We describe below the L ( ξ ( s ) i,j , χ ( s ) i,j ) orbitsunder the ψ -action: • if k = 3 + p/ p,
3) = 1, p > L ( ξ (1) i,j , χ (1) i,j ) ψ ! L ( ξ (3) j,p − i − j , χ (3) j,p − i − j ) ψ ! L ( ξ (2) p − i − j,i , χ (2) p − i − j,i ) ψ ! L ( ξ (1) i,p − i − j , χ (1) i,p − i − j ) ψ ! L ( ξ (3) p − i − j,j , χ (3) p − i − j,j ) ψ ! L ( ξ (2) j,i , χ (2) j,i )) ψ ! L ( ξ (1) i,j , χ (1) i,j ) . • if k = 3 + p/ p,
2) = 1, p > L ( ξ (1 ′ ) i,j , χ (1 ′ ) i,j ) ψ ! L ( ξ (2 ′ ) j,p − i − j , χ (2 ′ ) j,p − i − j ) ψ ! L ( ξ (1 ′ ) p − i − j,j , χ (1 ′ ) p − i − j,j ) ψ ! L ( ξ (2 ′ ) j,i , χ (2 ′ ) j,i ) ψ ! L ( ξ (1 ′ ) i,j , χ (1 ′ ) i,j ) . Proof of the Main Theorem
The section is devoted to the proof of the Main Theorem. As explained in theintroduction, the last step uses results and technics from [9].For the moment, g is any simple Lie algebra and f is a nilpotent element of g .Let P , + := { λ ∈ h ∗ : h λ, α ∨ i ∈ Z > for all α ∈ ∆ , + } , where ∆ , + := ∆ ∩ ∆ + with ∆ the root system of ( g , h ). Note that P , + containsthe set P + = { λ ∈ h ∗ : h λ, α ∨ i ∈ Z > for all α ∈ ∆ + } . Let U ( g ) be the enveloping algebra of g and H f (?) the functor from the categoryof Harish-Chandra U ( g )-bimodules to the category of bimodules over the finite W -algebra U ( g , f ) (see [32]), which is the Zhu algebra of W k ( g , f ) [15]. By abuse ofnotation we use the same notation as for the functor in Sect. 4. We refer to [9] formore about this topic.Write J λ ⊂ U ( g ) for the annihilating ideal of the irreducible g -module L ( λ ) withhighest weight λ ∈ h ∗ . The quotient H f ( U ( g ) /J λ ) is a quotient algebra of the finite W -algebra U ( g , f ) = H f ( U ( g )). Theorem 7.1 ([9, Theorem 7.8]) . Let k = − h ∨ + p/q be an admissible numberfor g and pick f ∈ O q . Let λ ∈ P , + be such that b λ = λ + k Λ ∈ P r k . Then H f ( U ( g ) /J λ − pq x ) has a unique simple module, denoted by E J λ − pq x , and H f ( b L k ( λ )) ≃ L ( E J λ − pq x ) , where L ( E J λ − pq x ) is the irreducible Ramond twisted W k ( g , f )-module attached to E J λ − pq x (see [29, 3] ) and b L k ( λ ) is the simple b g -module with highest weight b λ . Inparticular, W k ( g , f ) ≃ H f ( V k ( g )) ≃ L ( E J − pq x ) . ATIONALITY OF THE EXCEPTIONAL W -ALGEBRAS W k ( sp , f subreg ) 17 Assume now, as in the previous sections, that g = sp and f = f subreg = e − η .Here ∆ , + = { α } , whence P , + = { λ ∈ h ∗ : h λ, α ∨ i ∈ Z > } = C ̟ + Z > ̟ , and P + = Z ̟ + Z ̟ . According Proposition 6.7 if M is a simple W k ( g , f )-module, where k = − p/ p,
3) = 1, p >
3, or k = − p/
4, with ( p,
2) = 1, p >
4, then it is isomorphicto L ( ξ ( s ) i,j , χ ( s ) i,j ) with ξ ( s ) i,j and χ ( s ) i,j as described in the Proposition 6.6. Proposition 7.2.
Let k , ξ ( s ) i,j and χ ( s ) i,j be as in Proposition 6.6 then L ( ξ ( s ) i,j , χ ( s ) i,j ) isa simple W k ( g , f ) -module.Proof. The argument depends on whether k is principal admissible of coprincipaladmissible, but it is very similar in both cases. We only detail the case k = − p/ p,
3) = 1, p >
3. For 1 i p − j p − i −
1, set λ i,j =( j − ̟ + ( i − ̟ . We check that λ i,j ∈ P + ∩ P r k . By Theorem 7.1, H f ( b L k ( λ i,j )) ≃ L ( E J λi,j − p x ) ≃ H f, − ( b L k ( λ i,j − p x )) , where x = h = ̟ and H • f, − (?) is the “ − ” variant of the quantized Drinfeld-Sokolov reduction functor H • f defined in Sect. 4 (see [3, 9] for a construction). Thelowest L -eigenvalue h λ i,j − p x is the conformal dimension of L ( E J λi,j − p x ) givenby [9, (7.4)]: h λ i,j − p x = ( λ | λ + 2 ρ )2( k + h ∨ ) − k + h ∨ | x | + ( x | ρ )= −
15 + 3 i + 6 ij + 6 j + 6 p − ip − jp p = χ (1) i,j . Besides, using [3, (70)] the lowest J -eigenvalue of H f, − ( b L k ( λ i,j )) is( λ i,j − p x | − α ∨ − i ξ (1) i,j , since J ( z ) = J α ( z ) and b w b t − x J Rα = − J α . In conclusion(5) L ( E J λi,j − p x ) ≃ L ( ξ (1) i,j , χ (1) i,j ) . In this way L ( ξ (1) i,j , χ (1) i,j ) is a simple W k ( g , f )-module for all 1 i p − j p − i −
1. If s = 2 or 3, using the ψ -action on the module L ( ξ ( s ) i,j , χ ( s ) i,j ) italways comes down to a module L ( ξ (1) i ′ ,j ′ , χ (1) i ′ ,j ′ ). As a consequence L ( ξ ( s ) i,j , χ ( s ) i,j ) is asimple module of W k ( g , f ), too. (cid:3) Lemma 7.3.
Suppose that there is a nontrivial extension of W k ( g , f ) -modules, −! L ( ξ, χ ) ι −! M π −! L ( ξ ′ , χ ′ ) −! . Then L acts locally finitely on M . Proof.
Suppose there is a nontrivial extension(6) 0 −! L ( ξ, χ ) ι −! M π −! L ( ξ ′ , χ ′ ) −! . Since W k ( g , f ) is lisse, L := L ( ξ, χ ) and L ′ := L ( ξ ′ , χ ′ ) are L -diagonalizableand the L -eigenspaces are finite dimensional.Let m ∈ M . Since π ( m ) ∈ L ′ there exist w , . . . , w s ∈ L ′ and µ , . . . , µ s ∈ C such that L w j = µ j w j for all 1 j s and Q sj =1 ( L − µ j Id) π ( m ) = 0. Then π ( s Y j =1 ( L − µ j Id) m ) = 0 . As a consequence Q sj =1 ( L − µ j Id) m ∈ im ι . Let m ∈ L such that ι ( m ) = Q sj =1 ( L − µ j Id) m . As before, there are v , . . . , v r ∈ L and ν , . . . , ν r ∈ C suchthat Q ri =1 ( L − ν i Id) m = 0. Then ι ( r Y i =1 ( L − ν i Id) m ) = 0 = r Y i =1 ( L − ν i Id) ι ( m ) = r Y i =1 ( L − ν i Id) s Y j =1 ( L − µ j Id) m. Hence m belongs to some L -stable finite dimensional vector subspace of r M i =1 ker( L − ν i Id) ⊕ s M j =1 ker( L − µ j Id) . (cid:3) Lemma 7.4.
If there exists a nontrivial extension of W k ( g , f ) -modules −! L ( ξ, χ ) −! M −! L ( ξ ′ , χ ′ ) −! then χ and χ ′ coincide modulo Z .Proof. Suppose that there is a nontrivial extension0 −! L ( ξ, χ ) −! M −! L ( ξ ′ , χ ′ ) −! . Set as in the previous proof, L := L ( ξ, χ ) and L ′ := L ( ξ ′ , χ ′ ). For d ∈ C , let M d be the generalized L -eigenspace of M attached to the eigenvalue d . Set M [ d ] := L d ′ ∈ d + Z M d ′ . It is a W k ( g , f )-submodule of M . Then M = L d ∈ C , Re( d ) < M [ d ]is a direct sum decomposition of the W k ( g , f )-modules of M . For any d , the previ-ous decomposition induces the following exact sequence(7) 0 −! L [ d ] −! M [ d ] −! L ′ [ d ] −! . Assume χ − χ ′ / ∈ Z . Since L [ d ] = 0 if d − χ / ∈ Z , and L [ d ′ ] = 0 if d ′ − χ ′ / ∈ Z , weget that M = M [ χ ] ⊕ M [ χ ′ ]. Taking d = χ and d = χ ′ in (7) we get0 −! L [ χ ] −! M [ χ ] −! , −! M [ χ ′ ] −! L ′ [ χ ′ ] −! . Finally M = L [ χ ] ⊕ L ′ [ χ ′ ] = L ⊕ L ′ since L and L ′ are simple modules. So thesequence 0 ! L ! M ! L ′ ! (cid:3) Proposition 7.5.
Suppose that either k = − p/ with ( p,
3) = 1 , p > , or k = − p/ with ( p,
2) = 1 , p > . Let W k ( g , f ) - Mod be the category of W k ( g , f ) -modules. Then Ext W k ( g ,f ) - Mod ( L ( ξ ( s ) i,j , χ ( s ) i,j ) , L ( ξ ( s ′ ) i ′ ,j ′ , χ ( s ′ ) i ′ ,j ′ )) = 0 , ATIONALITY OF THE EXCEPTIONAL W -ALGEBRAS W k ( sp , f subreg ) 19 where ξ ( s ) i,j and χ ( s ) i,j are described in the Proposition 6.7.Proof. Assume k = − p/ p,
3) = 1, p >
3. The argument for thecoprincipal case is the same. It clearly appears that for all 1 i, i ′ p − j p − i − j ′ p − i ′ −
1, the differences χ (2) i,j − χ (1) i ′ ,j ′ and χ (3) i,j − χ (1) i ′ ,j ′ are not integers. According to Lemma 7.4, any extension0 −! L ( ξ ( s ) i,j , χ ( s ) i,j ) −! M −! L ( ξ ( s ′ ) i ′ ,j ′ , χ ( s ′ ) i ′ ,j ′ ) −! , where exactly one of χ ( s ) i,j and χ ( s ′ ) i ′ ,j ′ is with the first form is trivial. Applying ψ wededuce that if s = s ′ thenExt W k ( g ,f )-Mod ( L ( ξ ( s ) i,j , χ ( s ) i,j ) , L ( ξ ( s ′ ) i ′ ,j ′ , χ ( s ′ ) i ′ ,j ′ )) = 0 . Suppose that s = s ′ . Using the ψ -action we can suppose that s = s ′ = 1. Accordingto (5), since W k ( g , f ) is lisse, it suffices to show that there is not nontrivial extension(8) 0 −! L ( E J λi,j − p x ) ι −! M π −! L ( E J λi ′ ,j ′ − p x ) −! . Set L i,j := L ( E J λi,j − p x ) and L i ′ ,j ′ := L ( E J λi ′ ,j ′ − p x ). If χ (1) i,j = χ (1) i ′ ,j ′ , since theZhu algebra Zhu( W k ( g , f )) is semisimple, the sequence(9) 0 −! ( L i,j ) top −! M top −! ( L i ′ ,j ′ ) top −! . of Zhu( W k ( g , f ))-modules is split. Applying the Zhu induction functor we get (8)splits.Let suppose χ (1) i,j > χ (1) i ′ ,j ′ . The module L ( E J λi ′ ,j ′ − p x ) is the unique simplequotient of the Verma module M i ′ ,j ′ = M ( E J λi ′ ,j ′ − p x ) := U ( W k ( g , f )) ⊗ U ( W k ( g ,f )) > E J λi ′ ,j ′ − p x . Let v + be a primitive vector of M i ′ ,j ′ and v ∈ M be such that π ( v ) is the image of v + in L i ′ ,j ′ . We have π (( L − χ (1) i ′ ,j ′ Id) v ) = ( L − χ (1) i ′ ,j ′ Id) π ( v ) = 0. Hence ( L − χ (1) i ′ ,j ′ Id) v ∈ im ι . Since im ι ≃ L i,j and χ (1) i,j > χ (1) i ′ ,j ′ , ( L − χ (1) i ′ ,j ′ Id) v = 0. As a con-sequence, it exists an injective W k ( g , f )-modules homomorphism f : M i ′ ,j ′ ! M such that the diagram commutes: M i ′ ,j ′ M L i ′ ,j ′ f Let us suppose that the sequence (8) does not split. The module f ( M i ′ ,j ′ ) is asubmodule of M . As a consequence, since L i,j ≃ ι ( L i,j ) is simple, either ι ( L i,j ) ⊂ f ( M i ′ ,j ′ ) or ι ( L i,j ) ⊕ f ( M i ′ ,j ′ ). However if ι ( L i,j ) ⊕ f ( M i ′ ,j ′ ) then the sequence0 −! L i,j −! ι ( L i,j ) ⊕ f ( M i ′ ,j ′ ) −! L i ′ ,j ′ −! ι ( L i,j ) ⊂ f ( M i ′ ,j ′ ). Let m ∈ M . Since π is surjective it exists m ∈ M i ′ ,j ′ such that π ( m ) = π ◦ f ( m ). Thus m − f ( m ) ∈ ker π . We get m ∈ ι ( L i,j ) ⊂ f ( M i ′ ,j ′ ).Therefore f is surjective. It implies that M is isomorphic to M i ′ ,j ′ as W k ( g , f )-modules. Hence [ M i ′ ,j ′ : L i,j ] = 0. By [9, Theorem 7.6] this happens only if it exists µ ∈ P , + such that [ c M k ( λ i ′ ,j ′ − p x ) : b L k ( µ − p x )] = 0, where c M k ( λ i ′ ,j ′ − p x ) is the Verma module of g with highest weight b λ i ′ ,j ′ − p x , and E J λi,j − p x is a directsummand of H Lie0 ( L ( µ − p x )) .This first condition implies that µ ∈ W ◦ λ i ′ ,j ′ and we get λ i,j ∈ W ◦ µ from thesecond one. Hence λ i,j ∈ W ◦ λ i ′ ,j ′ . Since b λ i,j and b λ i ′ ,j ′ are both dominant theyare equal, and λ i,j = λ i ′ ,j ′ contradicting χ (1) i,j > χ (1) i ′ ,j ′ .Finally if χ (1) i,j < χ (1) i ′ ,j ′ by applying the duality functor to (8) we are back to theprevious case χ (1) i,j > χ (1) i ′ ,j ′ . (cid:3) Example . Let k = − . There exist nine simple W − ( g , f )-modules. We de-scribe below the two orbits under the action of ψ : W − ( g , f ) ψ ! L (cid:18) − , (cid:19) ψ ! L (cid:18) − , (cid:19) ψ ! L (cid:18) , (cid:19) ψ ! L (cid:18) − , (cid:19) ψ ! L (cid:18) , (cid:19) ψ ! W − ( g , f ) ,L (cid:18) − , (cid:19) ψ ! L (cid:18) , (cid:19) ψ ! L (cid:18) − , (cid:19) ψ ! L (cid:18) − , (cid:19) . A vertex algebra V is said to be positive [9] if every irreducible V -modules besides V itself has positive conformal dimension. In our setting, W k ( g , f ) is positive if χ ( s ) i,j > s, i, j as in Proposition 6.7, and k, g , f as in this proposition. Weobserve the vertex algebras W − ( g , f ), W − ( g , f ), W − ( g , f ) and W − ( g , f ) arepositive.If V is unitary ([16, Sect. 2]) then it is unitary as a module of the Virasorosubalgebra generated by the conformal vector as well. This forces the conformaldimension to be nonnegative. In particular, it is a positive vertex algebra. Weexpect the following result: Conjecture 7.7.
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3) = 1 and p >
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Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painleve, F-59000 Lille, France
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