A bound on the degree of singular vectors for the exceptional Lie superalgebra E(5,10)
aa r X i v : . [ m a t h . R T ] J un A BOUND ON THE DEGREE OF SINGULAR VECTORS FOR THEEXCEPTIONAL LIE SUPERALGEBRA E (5 , DANIELE BRILLIA
BSTRACT . We use the language of Lie pseudoalgebras to gain information about therepresentation theory of the simple infinite-dimensional linearly compact Lie superalgebraof exceptional type E (5 , . This technology allows us to prove that the degree of singularvectors in minimal Verma modules is ≤ . A few technical adjustments allow us to refinethe bound, proving that the degree must always be ≤ and it is actually, except for afinite number of cases, ≤ .
1. I
NTRODUCTION
Infinite dimensional linearly compact simple Lie superalgebras over C were classifiedby Kac in [K]. Besides Lie algebras in Cartan’s list [C], the complete list consists of tenfamilies and five exceptions, denoted by E (1 , , E (2 , , E (3 , , E (3 , , E (4 , and E (5 , .In [KR1] Kac and Rudakov started the study of representations of these exceptional super-algebras following the approach developed by Rudakov for the "non super" case, establish-ing the language of generalized Verma modules and reducing the problem to the descriptionof the so-called degenerate modules and the study of singular vectors .They completed the classification of degenerate Verma modules and singular vectors for E (3 , in [KR1, KR2, KR3]. In the meanwhile they started to investigate the cases of E (3 , and E (5 , in [KR4]. For E (3 , they could apply most of the arguments from[KR2] and found all the degenerate Verma modules; they also described many degenerate E (5 , -modules and conjectured there were no others.Afterwards Rudakov in [R] related the problem with the study of morphism between Vermamodules. He defined a degree of such a morphism and classified all the morphisms of de-gree (which correspond to the degenerate modules found in [KR4]) but he also obtainedmorphisms of degree and as products of morphisms of degree and found morphismsof degree and . He then conjectured there were no morphisms of higher degree and thathis list was exhaustive.Cantarini and Caselli in [CC] developed some combinatorial aspects of morphisms be-tween Verma modules for E (5 , that allowed them, in particular, to confirm part ofRudakov’s conjecture. Indeed, they showed that the morphisms found by Rudakov up todegree were the only ones.We briefly recall the definitions of generalized Verma module and singular vector. E (5 ,
10) = ⊕ i ≥− L i is equipped with a Z -grading of depth 2 consistent with the Liesuperalgebra structure. In particular, L is a Lie algebra isomorphic to sl . If we takean sl -module V and allow L + = ⊕ i> L i to act trivially on it, we obtain a module over Date : June 22, 2020. L ≥ = ⊕ i ≥ L i . We can then consider the induced L -module T ( V ) = U ( L ) ⊗ U ( L ≥ ) V where U ( L ) is the universal enveloping algebra of L . T ( V ) is called generalized Verma module . One says it is minimal if V is an irreducible sl -module. If T ( V ) is minimal but not irreducible, it is said to be degenerate .It is possible to define a grading on T ( V ) = ⊕ p ≥ T p ( V ) compatible with that of L , mean-ing that L i · T p ( V ) ⊂ T p − ( V ) .A singular vector is an element of T ( V ) that is killed by L + .The notions of degeneracy of a minimal Verma module, existence of (non constant) singu-lar vectors and (positive degree) morphisms are all equivalent (see, for instance, [CC, Prop.3.5]).Basically, if one has a positive degree singular vector in a minimal Verma module T ( V ) ,the L -submodule it generates is a proper submodule, thus T ( V ) is degenerate. On the otherway, a L -morphism of positive degree p maps constant vectors, which are automaticallysingular, to singular vectors of positive degree p .Despite a visible fair amount of understanding of these objects, an explicit bound on thedegree of singular vectors (or equivalently morphisms) in the literature is only implicitlyconjectured. This article aims to fill this gap and provide such a bound, with the hope thatthe techniques developed will allow to improve the result and help "attack" these conjec-tures from "above" as well as from "below".The idea is the following: the even part of E (5 , is S (5) , the Lie algebra of zero-divergence vector fields in five indeterminates. This algebra is isomorphic to the annihila-tion algebra of a Lie pseudoalgebra . Lie pseudoalgebras are Lie algebras in a pseudotensorcategory , hence their name (see [BD, BDK1]). The analogs of Verma modules are called tensor modules in this language. A structural correspondence between a Lie pseudoalgebraand its annihilation algebra guarantees a close bond between their representation theories(see [BDK1, Section 13]). (Finite) pseudoalgebras have the upside of having a developedtheory ([BDK1, BDK2, BDK3, BDK4, D]) and allow one to talk about singular vectorsand their degrees in quite a "manageable" way. For istance, we know that the degree ofsingular vectors in tensor modules for Lie pseudoalgebras of type S is at most , (Theorem4.1, see [BDK2, Section 7] for more details).The fact that the pseudoalgebraic structure associated with the even part of E (5 , hassuch an immediate bound, together with the finiteness of the odd degree, prevent vectorswith high enough degree from being singular.The paper is organized as follows: in Section 2 we recall the basic definitions about E (5 , , generalized Verma modules and singular vectors and set up the notation. InSection 3 we recall the basic notions about Lie pseudoalgebras, annihilation algebras andthe correspondence between their representation theory. Section 4 is dedicated to a briefsummary of results about representation theory of primitive Lie pseudoalgebras of type W and S . In Section 5 we build a finite filtration of S (5) -submodules on a generalized Vermamodule and realize their quotients as tensor modules, which allows us to apply the resultsstated in the previous section and prove our first result. Finally, in Section 6 we use a veryeasy technical lemma to refine the bound. BOUND ON THE DEGREE OF SINGULAR VECTORS FOR E (5 ,
2. E(5,10), V
ERMA MODULES AND NOTATION
We deal with the simple linearly compact Lie superalgebra of exceptionaly type L = E (5 , and we will use the geometric construction provided in [CK, 5.3]Let d = ( C ) ∗ and let { ∂ , . . . , ∂ } and { x , . . . , x } be bases for respectively d and d ∗ .We can realize the even part of L as zero-divergence vector fields in the indeterminates x , . . . , x , L (0) = S (5) = n D = X i =1 f i ∂ i | f i ∈ C [[ x , . . . , x ]] , div ( D ) = 0 o , and the odd part as closed 2-forms in the same indeterminates L (1) = d Ω (5) = n ω = X i
Given the fact that [ L , L n ] ⊆ L n , we can view L n as an sl ( d ) -module; in particular, L − ∼ = d and L − ∼ = V d ∗ =: s .It is useful to describe also L as a sl ( d ) -module: it is the highest weight representation in d ∗ ⊗ V d ∗ , (see [CK, Section 4.3]) and it is generated by the highest weight vector x ξ .Notice that L j = L j for j ≥ and that L − ∼ = d ⊕ s is a finite dimensional Lie superalgebrawhose superbracket is non trivial only when restricted to the odd part, where is given by(2.1).This grading extends to the universal enveloping algebra U ( L ) , and in particular to U ( L − ) . In the latter case, as common practice, the sign of the degree is inverted in or-der to have a grading over N .We will use this grading to study generalized Verma modules, which we will introducehere following [R](see also [KR1, KR2, CC]).Given a sl ( d ) ∼ = L -module V , we can extend it to a L ≥ -module by letting L + acttrivially on it; we can then consider the induced L -module T ( V ) = U ( L ) ⊗ U ( L ≥ ) V where the action is given by left multiplication. Definition 2.1.
Let V be a sl ( d ) -module. The L -module T ( V ) is called generalized Vermamodule .If V is a finite-dimensional and irreducible sl ( d ) -module, we call T ( V ) minimal .A minimal Verma module is called non-degenerate if it is irreducible, degenerate other-wise. Remark 2.1.
Let us notice that, as vector spaces, T ( V ) ∼ = U ( L − ) ⊗ V . We will often usethis isomorphism omitting the subscript U ( L ≥ ) on the tensor product.When V = V ( λ ) is an irreducible sl ( d ) -module of highest weight λ , we may use thenotation T ( V ) = T ( λ ) . A dominant weight λ for sl ( d ) will be expressed in terms of aquadruple [ a , a , a , a ] ∈ N where λ = a ω + · · · + a ω and ω , . . . , ω are the funda-mental weights of sl ( d ) .One should pay special attention because, since we set d = ( C ) ∗ , all the highest weightsmodules are the duals of the sl usual ones. For example, in our notation the sl standardrepresentation, which has highest weight [1 , , , , will be d ∗ .The grading of U ( L − ) induces one on T ( V ) .Poincaré-Birkhoff-Witt Theorem is still true in the superalgebra setting (see [M, 6.1]), sofixed an ordered basis { ∂ . . . , ∂ , ξ , . . . , ξ } of L − , we can choose as a P BW -basis for U ( L − ) the monomials ∂ ( I ) ξ K where, ∂ ( I ) = ∂ i i ! · · · ∂ i i ! , I = ( i , . . . , i ) ∈ N (2.4) ξ K = ξ k · · · ξ k , K = ( k , . . . , k ) ∈ { , } . (2.5)The basis elements are, by definition of the grading, homogeneous of degree p = 2 | I | + | K | ,where | I | = i + · · · + i and | K | = k + · · · k ; they generate the homogeneous subspaces U p ( L − ) . We can thus equip T ( V ) with a grading T p ( V ) = U p ( L − ) ⊗ V . BOUND ON THE DEGREE OF SINGULAR VECTORS FOR E (5 , We should notice that this grading and the grading of L are compatible, by which we meanthat L n T p ( V ) ⊆ T p − n ( V ) . (2.6)We will call the elements of T p ( V ) homogeneous vectors of degree p ; in particular, we willcall the degree ones constant .For istance, T ( V ) = C ⊗ V , T ( V ) = s ⊗ V , T ( V ) = d ⊗ V + V ( s ) ⊗ V , T ( V ) = ds ⊗ V + V ( s ) ⊗ V , etc.If v ∈ S n ( d ) V m ( s ) ⊗ V , if we want to keep track of the even and odd degrees, we will saythat v has degree ( n | m ) .Degeneracy of Verma modules can be reformulated in terms of singular vectors . Definition 2.2.
Let T ( V ) be a Verma module. v ∈ T ( V ) is called a singular vector if L v = 0 .The space of singular vectors will be denoted by sing T ( V ) . Example 2.1.
Any constant vector v ∈ T ( V ) is singular, since in that case L v ∈ T − ( V ) = 0 . Remark 2.2.
Take v ∈ sing T ( V ) and z ∈ L . Then, for any y ∈ L , yv = 0 and so y ( zv ) = [ y, z ] v + z ( yv ) = [ y, z ] v = 0 where the last identity follows from the fact that [ L , L ] ⊆ L and the singularity of v .In other terms, sing T ( V ) is a L -submodule of T ( V ) . In particular, since T ( V ) = L p T p ( V ) as L -modules, homogeneous components of a singular vector are singular.We will always assume that a singular vector is homogeneous.The same holds for the weight components of a singular vector. So we will, wheneverpossible, assume that a singular vector is also a weight vector. Proposition 2.1.
A minimal Verma module T ( V ) is degenerate if and only if it containsnon constant singular vectors.Proof. Assume = v ∈ sing T p ( V ) for some p > . Since L − and L can only respec-tively rise the degree of v or preserve it and since L + .v = 0 by assumption, L v is an L -submodule of T ( V ) , which is proper because it only contains vectors of degree ≥ p .Vice versa, let W ⊂ T ( V ) be a non trivial proper L -submodule and take = w ∈ W .Since the action of L lowers strictly the degree of homogeneous components of w , weknow that eventually ( L ) n w = 0 for some finite n ≥ ; thus we can assume withoutloss of generalization that w is singular. Now, if w was constant, by irreducibility of V we would have L w = C ⊗ V and therefore, by iterated action of L − , we would obtain L w = T ( V ) . But T ( V ) ) W ⊇ L w = T ( V ) , a contradiction. Hence w is a non constantsingular vector. (cid:3) The proof of the proposition shows vividly how singular vectors "detect" degeneracy ofminimal Verma modules.
Example 2.2.
Let V = V ([0 , , , ∼ = d and let v = X i ξ i ⊗ ∂ i ∈ T ( d ) .A generic element of L is of the form y = x h ξ kl + x k ξ hl for some h, k, l ∈ { , . . . , } . DANIELE BRILLI
To check if v is singular, we can carry out the computation: y · v = X i x h [ ξ kl , ξ i ] ⊗ ∂ i + x k [ ξ hl , ξ i ] ⊗ ∂ i = X i ε ( kl i ) ⊗ ( x h ∂ ( kl i ) ) ∂ i + ε ( hl i ) ⊗ ( x k ∂ ( hl i ) ) ∂ i = X i − ε ( kl i ) ⊗ e h ( kl i ) ∂ i − ε ( hl i ) ⊗ e k ( hl i ) ∂ i = X i − ε ( kl i ) ⊗ δ hi ∂ ( kl i ) − ε ( hl i ) ⊗ δ ki ∂ ( hl i ) = − ε ( kl h ) ⊗ ∂ ( kl h ) − ε ( hl k ) ⊗ ∂ ( hl k ) = − ε ( kl h ) ⊗ ∂ ( kl h ) + ε ( kl h ) ⊗ ∂ ( kl h ) = 0 . So v ∈ sing T ( V ) .
3. P
RELIMINARIES ON L IE P SEUDOALGEBRAS
In this section we briefly give the definition of finite Lie pseudoalgebras and of theirannihilation algebras and recall their main features following [BDK2, Section 2]. For amore detailed exposition, see also [BDK1].First of all we need a few preliminaries on Hopf algebras.Let H be a cocommutative Hopf algebra with coproduct ∆ , counit ǫ and antipode S .We will use the Sweedler notation (see [S]), for istance given h ∈ H : ∆( h ) = h (1) ⊗ h (2) ;(∆ ⊗ id )∆( h ) = ( id ⊗ ∆)∆( h ) = h (1) ⊗ h (2) ⊗ h (3) ;( id ⊗ S )∆( h ) = h (1) ⊗ h ( − , etc.Throughout the paper, H = U ( d ) will be the universal enveloping algebra of a Lie algebra d of dimension n .In this case the coproduct is given by ∆( ∂ ) = ∂ ⊗ ⊗ ∂ and the antipode by S ( ∂ ) = − ∂ for ∂ ∈ d .Let ∂ , . . . , ∂ n be a basis of d and take the PBW basis of H , { ∂ ( I ) } I ∈ N n as in (2.4).With this choice of basis is easy to show that ∆( ∂ ( I ) ) = X J + K = I ∂ ( J ) ⊗ ∂ ( K ) . (3.1)Moreover it lets us define an increasing filtration on U ( d ) F p H = span C { ∂ ( I ) | | I | ≤ p } . (3.2)Now let X = H ∗ := Hom ( H, C ) and let { x I } be a dual basis of { ∂ ( I ) } , i.e. h x I , ∂ ( J ) i = δ JI . In particular we indicate the duals of the basis elements of d ⊂ H , { ∂ i } , with { x i } ,which provides a basis of d ∗ ⊂ X . Alternatively, one could have done the following: notice that v is an highest weight vector, thereforerealize that it is sufficient to check that the lowest weight vector of L , x ξ , acts trivially on v , which isclearly easier (or at least shorter) (see [CC, Ch.3]). BOUND ON THE DEGREE OF SINGULAR VECTORS FOR E (5 , X can be viewed as an H -bimodule with left and right actions given respectively by h hx, f i = h x, S ( h ) f i ; (3.3) h xh, f i = h x, f S ( h ) i , for x ∈ X, h, f ∈ H. (3.4)Properties of H reflect "dually" on X ([BDK1, Section 2]): • cocommutativity of H implies commutativity of X ; • from (3.1) follows easily that x I x J = x I + J ; • using the above, one can identify X with the ring of former power series O n = C [[ x , . . . , x n ]] ; • setting F p X = ( F p ( H )) ⊥ provides a decreasing filtration on X .Considering the { F p X } as a fundamental system of neighborhoods one can define a topol-ogy on X that makes it linearly compact ([BDK1, Chapter 6]) (while considers the discretetopology on H ). This makes the action of H (and in particular of d ) on X continuous. Definition 3.1. A Lie ( H -)pseudoalgebra L is a left H -module endowed with a map, called pseudobracket [ · ∗ · ] : L ⊗ L → ( H ⊗ H ) ⊗ H L which has the following properties: H-bilinearity: [ f a ∗ gb ] = (( f ⊗ g ) ⊗ H a ∗ b ] ∀ a, b ∈ L, g, f ∈ H ; Skew-commutativity: [ b ∗ a ] = − ( σ ⊗ H a ∗ b ] ∀ a, b ∈ L ; Jacobi: [ a ∗ [ b ∗ c ]] − (( σ ⊗ ⊗ H b ∗ [ a ∗ c ]] = [[ a ∗ b ] , ∗ c ] ∀ a, b, c ∈ L .where σ : H ⊗ H → H ⊗ , f ⊗ g g ⊗ f is the permutation of factors and the compositionof pseudobrackets in the Jacobi identity are suitable defined in H ⊗ ⊗ H L .A Lie pseudoalgebra is called finite if it is finitely generated as a module over H .The name derives from the fact that this is an algebra in a pseudotensor category as in-troduced in[BD] (see also [BDK1, Chapter 3]). Example 3.1.
Take a Lie algebra g . One can define the current Lie H -pseudoalgebra Cur g = H ⊗ g as free H -module with pseudobracket defined as [(1 ⊗ a ) ∗ (1 ⊗ b )] = (1 ⊗ ⊗ H [ a, b ] for a, b ∈ g and extended then by H -bilinearity.We will focus on what are called primitive Lie pseudoalgebras, which we will define inthe next section.
Definition 3.2. A representation of L , or L -module , is a left H -module M with an H -bilinear map ∗ : L ⊗ M → ( H ⊗ H ) ⊗ H M such that [ a ∗ b ] ∗ m = a ∗ ( b ∗ m ) − (( σ ⊗ ⊗ H b ∗ ( a ∗ m )) ∀ a, b ∈ L , m ∈ M. DANIELE BRILLI An L -module is called finite if it is finitely generated as an H -module.A subspace N ⊂ M is an L -submodule if L ∗ N ⊂ ( H ⊗ H ) ⊗ H N .An L -module is irreducible if it does not contain any nontrivial proper submodules.The most important tool in the study of Lie pseudoalgebras (and their representations)is the annihilation algebra.Define A ( L ) := X ⊗ H L , where as before X = H ∗ and the right action of H on X is(3.4). Definition 3.3.
Given L a Lie H -pseudoalgebra, its annihilation algebra is the Lie algebra L = A ( L ) with Lie bracket [ x ⊗ H a, y ⊗ H b ] = X ( xf i )( yg i ) ⊗ H l i where [ a ∗ b ] = X ( f i ⊗ g i ) ⊗ H l i . (3.5) H acts on X ⊗ H L by (3.3) on the first factor. In this way d ⊂ U ( d ) = H acts on L byderivations. The semidirect sum L e = d ⋊ L is called extended annihilation algebra .If L is finite and L is a finite-dimensional subspace that generates L as a left H -module,we can define a filtration on L induced by the one of X , F p L = { x ⊗ H a ∈ L | x ∈ F p X and a ∈ L } , which satisfies: [ F n L , F p L ] ⊆ F n + p − l L and d ( F p L ) ⊆ F p − L , (3.6)where l depends only on the choice of L . We can correct the l shift by setting L p = F p + l L so that [ L p , L n ] ⊆ L p + n . In particular, L is a Lie algebra.We can carry on the filtration to L e setting L ep = L p .An L e -module V is called conformal if any v ∈ V belongs to some ker p V := { v ∈ V | L p v = 0 } . The next result [BDK2, Proposition 2.1] we state will be crucial for our purposes.
Proposition 3.1.
Any module V over a Lie pseudoalgebra L has a natural structure of aconformal L e -module, given by the action of d on V and by ( x ⊗ H a ) v = X h x, S ( f i g i ( − ) i g i (2) v i where a ∗ v = X ( f i ⊗ g i ) ⊗ H v i (3.7) for a ∈ L , x ∈ X , v ∈ V .Conversely, any conformal L e -module V has a natural structure of an L -module given by a ∗ v = X I ∈ N n ( S ( ∂ ( I ) ) ⊗ ⊗ H (( x I ⊗ H a ) · v ) (3.8) Moreover, V is irreducible as a module over L if and only if it is irreducible as a moduleover L e (or L ). BOUND ON THE DEGREE OF SINGULAR VECTORS FOR E (5 ,
4. P
RIMITIVE L IE PSEUDOALGEBRAS OF TYPE W AND
SWe now define the Lie pseudoalgebra W ( d ) and its subalgebra S ( d ) and apply the con-structions of the previous section. Again, we will follow [BDK2, ]. Definition 4.1.
The Lie pseudoalgebra W ( d ) is the free H -module H ⊗ d with pseudo-bracket [( f ⊗ a ) ∗ ( g ⊗ b )] = ( f ⊗ g ) ⊗ H (1 ⊗ [ a, b ]) − ( f ⊗ ga ) ⊗ H (1 ⊗ b )+( f b ⊗ g ) ⊗ H (1 ⊗ a ) (4.1)Define the H -linear map div : W ( d ) → H by div ( P h i ⊗ ∂ i ) = P h i ∂ i . Then S ( d ) := { s ∈ W ( d ) | div ( s ) = 0 } is a subalgebra of the Lie pseudoalgebra W ( d ) .In [BDK1, Proposition 8.1] it is shown that S ( d ) is generated as an H -module by theelements of the form: s ab = a ⊗ b − b ⊗ a − ⊗ [ a, b ] for a, b ∈ d . (4.2)Let W = A ( W ( d )) be the annihilation algebra of W ( d ) . Since W ( d ) = H ⊗ d , W = X ⊗ H ( H ⊗ d ) ≡ X ⊗ d . The Lie bracket of W is obtained from the pseudobracket of W ( d ) by (3.5): [ x ⊗ a, y ⊗ b ] = xy ⊗ [ a, b ] − x ( ya ) ⊗ b + ( xb ) y ⊗ a for a, b ∈ d , x, y ∈ X. The action of H on W is given by h ( x ⊗ a ) = hx ⊗ a and d acts on W by derivations.Since W ( d ) is a free H -module, we can choose L = C ⊗ d and obtain the induceddecreasing filtration on WW p = F p W = F p X ⊗ H L ≡ F p X ⊗ d . W − = W and it satisfies (3.6) for l = 0 ; notice also that W / W ∼ = C ⊗ d and W / W ∼ = d ∗ ⊗ d . H can be endowed with a W ( d ) -pseudoaction given by ( f ⊗ a ) ∗ g = − ( f ⊗ ga ) ⊗ H where f, g ∈ H, a ∈ d ; this induces an action of W = X ⊗ H W ( d ) on X ⊗ H H ≡ X : ( x ⊗ a ) y = − x ( ya ) for a ∈ d , x, y ∈ X. (4.3)Using the fact that X can be identified as O n (compatibly with corresponding filtrationsand topologies) and that d acts on X by continuous derivations, we can make W act on O n = C [[ t , . . . t n ]] by continuous derivations too. This way we are defining a Lie algebrahomomorphism ϕ : W → W ( n ) where W ( n ) = Der ( O n ) = n n X i =1 f i ∂∂t i | f i ∈ C [[ t , . . . , t n ]] o W ( n ) has a natural filtration given by F p W ( n ) = n X i f i ∂∂t i | f i ∈ C [[ t , . . . , t n ]] k , k ≤ p o where C [[ t , . . . , t n ]] k is the homogeneous component of degree k .It is proven in [BDK2] that holds the following: (1) ϕ is an isomorphism of Lie algebras;(2) ϕ ( x ⊗ a ) = ϕ ( x ) ϕ ( a ) ∀ x ∈ X, a ∈ d ;(3) ϕ (1 ⊗ ∂ i ) = − ∂∂t i mod F W ( n ) ;(4) ϕ ( W p ) = F p W ( n ) ∀ p ≥ − .In what follows we will assume that dim d = n > (which is okay for us since we willapply it for n = 5 ).Let S = A ( S ( d )) = X ⊗ H S ( d ) be the annihilation algebra of the Lie pseudoalgebra S ( d ) .The Lie bracket is the one of W , since the canonical injection of S ( d ) into W ( d ) inducesa Lie algebra homomorphism ι : S ֒ → W .Explicity, if s = P h i ⊗ ∂ i ∈ S ( d ) ⊂ W ( d ) = H ⊗ d , ι ( x ⊗ H s ) = X xh i ⊗ ∂ i ∈ W ≡ X ⊗ d Choosing L = span C { s ab | a, b ∈ d } (where s ab are the ones defined in (4.2)) we get adecreasing filtration of S : S p = F p +1 S = F p +1 X ⊗ H L for p ≥ − . S − = S and it satisfies (3.6) for l = 1 .In [BDK1, Section 8.4] is proven that S ∼ = S ( n ) . But we would also like all the filtrationsand related topologies defined on these spaces to be compatible. In order to do so, wewould like to use ϕ defined before for W which behaves well related to the filtrations.This can be done but carefully.First define a map div : W → X as div ( P y i ⊗ ∂ i ) = P y i ∂ i .It is not difficult to verify that div ([ A, B ]) =
A div ( B ) − B div ( A ) ∀ A, B ∈ W (where theaction of W on X is (4.3)), which implies that S = { A ∈ W | div ( A ) = 0 } is a Lie subalgebra of M .In [BDK2, Section 3.4] is proven first that ι : S ∼ −→ S in such a way that ι ( S p ) = S ∩ W p ∀ p ≥ − [Proposition 3.5], then that φ maps S , up to a Lie algebra automorphism ψ induced by a ring automorphism of O n , to S ( n ) ⊂ W ( n ) [Proposition 3.6].Finally a Lie algebra isomorphism φ : S ∼ −→ S ( n ) ⊂ W ( n ) (4.4)such that S p maps onto S ( n ) ∩ F p W ( n ) is obtained [Corollary 3.3]. In particular we havethat S − = S − = S . Remark 4.1.
Take a generalized Verma module T ( V ) = U ( L − ) ⊗ V = U ( d + s ) ⊗ V .It is in particular an S (5) -module and, in view of (4.4), also an S -module. Furthermore,considering the action of d as left multiplication in U ( d + s ) , we can view T ( V ) as a S e -module. Since all these identifications are compatible with the filtrations, (2.6) implies that T ( V ) is a conformal S e -module. By Proposition 3.7, it has a natural structure of S ( d ) -module. BOUND ON THE DEGREE OF SINGULAR VECTORS FOR E (5 , Finally, we define what are called tensor modules for W ( d ) and S ( d ) (which are analogsof Verma modules) and extrapolate from [BDK2] what we need.Recall that W / W ∼ = gl ( d ) . So if we take a gl ( d ) -module V , we can allow W to acton it trivially and get a W -module. After that, we can induce so to get a W -module.But in order to correlate this with the Lie pseudoalgebra W ( d ) , we need to take account ofthe action of d . To do so, we consider the extended annihilation algebra W e .We call N W the normalizer of W p in W e . In [BDK2] is proven that it is independent of p and that W e = d ⊕ N W [Proposition 3.3]. Moreover, W acts trivially on any irreduciblefinite-dimensional conformal N -module (i.e. modules for which every element is killedby some W p ) and N W / W ∼ = d ⊕ gl ( d ) , so that we have a one-to-one correspondence be-tween irreducible finite-dimensional d ⊕ gl ( d ) -modules and irreducible finite-dimensionalconformal N W -modules [Proposition 3.4].In [BDK2, Section 3.5] totally analogous results are proven for S , whereas N S / S ∼ = d ⊕ sl ( d ) .Take a finite-dimensional d ⊕ gl ( d ) -module V ; letting W act as zero on it, we can definean action of N W and then define the W e -module T ( V ) = Ind W e N W V = W e ⊗ N W V whichcan be identified as an H -module with H ⊗ V since W e = d ⊕ N W . T ( V ) is called a tensor module for W ( d ) . Definition 4.2.
Let g and g be Lie algebras and let V i be g i -modules fro i = 1 , .We indicate with V ⊠ V the g ⊕ g -module V ⊗ V where g i only acts on the V i factor.If V is of the form Π ⊠ U , we will also indicate T ( V ) = T (Π , U ) .We can define on a tensor module T ( V ) = H ⊗ V a filtration as follows: F p T ( V ) = F p H ⊗ V for p ≥ − (4.5)which behaves nicely relatively to the filtration of W : Lemma 4.1. [BDK2, Lemma 6.3]
For every p ≥ we have:(1) d · F p T ( V ) ⊂ F p +1 T ( V ) ;(2) N W · F p T ( V ) ⊂ F p T ( V ) ;(3) W · T ( V ) ⊂ F p − T ( V ) . For S ( d ) we do the same construction:take a finite-dimensional d ⊕ sl ( d ) -module V , let S act trivially on it so that it has an actionof N S ; then consider the S e -module T S ( V ) = Ind S e N S V = S e ⊗ N S V which again can beidentified, as an H -module, with H ⊗ V .In [BDK2, Theorem 7.3] it is proven that these modules can be obtained as the restrictionof tensor modules for W ( d ) , therefore we will call them again tensor modules for S ( d ) .If V is of the form Π ⊠ U , we will also indicate T S ( V ) = T S (Π , U ) .We can define the same filtration we defined in the W ( d ) case and have the same nice be-havior relatively to the filtration of S .It makes sense now to introduce the notion of singular vectors for W ( d ) and S ( d ) . Definition 4.3.
For a W ( d ) -module V , a singular vector is an element v ∈ V such that W · v = 0 . The space of singular vectors in V in indicated by sing V .Analogously one defines singular vectors for S ( d ) -modules. Remark 4.2.
By Remark 4.1 a Verma module has a structure of S ( d ) -module. Again westretch the fact that all the identifications preserve the filtrations. Therefore, by the lastdefinition, a singular vector for E (5 , in a Verma module is also singular for S ( d ) .Focusing only on type S now, we summerize all the main results about tensor modulesfor S ( d ) and singular vectors in [BDK2, Section 7].We will indicate by Ω n the sl ( d ) -module V d ∗ with the natural action, where Ω is thetrivial module C . Theorem 4.1. • Every irreducible finite S ( d ) -module is a quotient of a tensor module; • Let Π (resp. U ) be an irreducible finite-dimensional module over d (resp. sl ( d ) ).Then the S ( d ) -module T S (Π , U ) is irreducible if and only if U is not isomorphic to Ω n for any n ≥ ; • If V = T S (Π , Ω n ) , n = 1 , then sing V ⊂ F V ; • If V = T S (Π , Ω ) , then F V ( sing V ⊂ F V . Combining the results above, one gets the following picture:if one looks for irreducible S ( d ) -modules (or, by mean of Proposition 3.1, irreducible S e -modules ) one studies tensor modules. For a tensor module, to be irreducible is equivalentto not contain positive degree singular vectors. Singular vectors can only be found whenone induces from irreducible sl ( d ) -modules Ω n and they are found in degree . except forthe case n = 1 where one finds also singular vectors of degree 2.5. B OUND ON DEGREE OF SINGULAR VECTORS
Direct computation of singular vectors is not immediate, especially in high degrees. Onething one can do is trying, as a start, to rule out as many options as possible. This can bedone, for example, by looking for restricting conditions on the degree of singular vectors,which is what we are about to do.Recall that by Theorem 4.1, we have a very important result that points in this direction inthe "pseudo" setting.Turns out, in order to get a first bound on the degree of singular vectors, it is enough to takeinto account just the even structure of L , exploiting the pseudoalgebra techniques available.To do so, we first define some subspaces of T ( V ) .For i = 0 , . . . , , let Γ i ( V ) := n v = X I,K ∂ ( I ) ξ K ⊗ v IK ∈ T ( V ) | | K | ≤ i se v IK = 0 o . (5.1)In other words, Γ i ( V ) consists of vectors with "odd degree" at most i . We may also describe Γ i ( V ) as the space generated by the PBW monomials of degree ( n | j ) with j ≤ i .It is easy to check the following properties: • Γ i ( V ) ⊆ Γ i +1 ( V ) ; BOUND ON THE DEGREE OF SINGULAR VECTORS FOR E (5 , • Γ ( V ) = U ( d ) ⊗ V ; • Γ ( V ) = T ( V ) ; • Γ i ( V ) is a L (0) ∼ = S (5) -submodule of T ( V ) .Basically, we have built a finite filtration of S (5) -modules of T ( V ) .The last property follows from the fact that the action of an even element in L can onlylower the odd degree in T ( V ) . Take for example y ∈ L , v = ∂ ( I ) ξ K ⊗ v KI ∈ T ( V ) .We have y · v = y · ( ∂ ( I ) ξ K ⊗ v KI ) = [ y, ∂ ( I ) ] ξ K ⊗ v KI + ∂ ( I ) [ y, ξ K ] ⊗ v KI + ∂ ( I ) ξ K ⊗ y · v KI . Here the third term is because L + acts trivially on V ; the first term consists of elementsof degree ( | I | − | | K | ) ; lastly, the second term, once expanded the bracket and sortedeverything, can only contribute with elements of degree ( | I | | | K |− or ( | I | +1 | | K |− .In any case the odd degree cannot increase.These properties allow us to talk about quotients.Let us consider the quotients of S (5) -modules Γ i ( V ) / Γ i − ( V ) for i = 0 , . . . , (wherewe impose Γ − ( V ) = 0 ).As sl ( d ) -modules, they are isomorphic to U ( d ) ⊗ ( V i ( s ) ⊗ V ) (follows from [M, Corollary6.4.5]). The latter look a lot like tensor modules T ( V i ( s ) ⊗ V ) for S ( d ) : if the action of S (5) can be interpreted as the action of the annihilator algebra associated to pseudoactionof S ( d ) , we can put to use Proposition 3.7. This is, in fact, possible in the following way.The action of y ∈ L j on a class u · ξ K ⊗ v ∈ Γ i ( V ) / Γ i − ( V ) , where u ∈ U ( d ) , | K | = i and v ∈ V , behaves, depending on j , like: for j = − : y · uξ K ⊗ v = ( yu ) ξ K ⊗ v , since in this case y ∈ d ⊆ U ( d ) ; for j = : y · uξ K ⊗ v = [ y, u ] ξ K ⊗ v + u [ y, ξ K ] ⊗ v + uξ K ⊗ ( y · v ) ; for j > : y · uξ K ⊗ v =[ y, u ] ξ K ⊗ v + u [ y, ξ K ] ⊗ v + uξ K ⊗ ( y · v )=[ y, u ] ξ K ⊗ v where the last equality is due to the fact that y lowers the odd degree of at least , sendingthe second term to in the quotient, and acts trivially on V .Notice that in the case j = 0 , the action of y ∈ L ∼ = sl ( d ) on ξ K is actually the same asthe one on the sl ( d ) -module V i ( s ) (since the other terms that usually appear in the bracket [ y, ξ K ] once sorted are in the quotient).Summing up, we have that L (0) = S (5) acts on Γ i ( V ) / Γ i − ( V ) • by left multiplication on the H = U ( d ) factor with the negative degree part; • by the natural action of sl ( d ) on U ( d ) ⊗ ( V i ( s ) ⊗ V ) with the degree part; • trivially on V i ( s ) ⊗ V with the positive degree part.Since this is exactly the action of S on the tensor module T S ( V i ( s ) ⊗ V ) via the isomor-phism in (4.4), we can state: Theorem 5.1.
Let V be a finite-dimensional irreducible sl ( d ) -module. Then we have anisomorphism of S (5) ∼ = X ⊗ H S ( d ) -modules ϕ : Γ i ( V ) / Γ i − ( V ) ∼ −→ T S ( V i ( s ) ⊗ V ) (5.2)Take now v ∈ sing T p ( V ) . It is, in particular, a singular vector for L (0) = S (5) of degree p . We indicate the space of such vectors with sing S (5) T ( V ) .If we consider v ∈ Γ i ( V ) / Γ i − ( V ) for a suitable i = 0 , . . . , , it will still be a singularvector in what we now know is a tensor module. Therefore, by Theorem 4.1, the evendegree of v must be ≤ . Since the odd degree of a vector cannot be larger than ,these ideas, formalized, prevent vectors with a sufficiently high enough degree from beingsingular. Theorem 5.2.
Let T ( V ) be a minimal Verma module and let v ∈ sing T ( V ) . Then v hasdegree at most .Proof. We can assume that v is homogeneous of degree p .We have, if p is either even or odd: p=2n: T p ( V ) = S n ( d ) ⊗ V + S n − ( d ) V ( s ) ⊗ V + · · · + S n − ( d ) V ( s ) ⊗ V ; p=2n+1: T p ( V ) = S n ( d ) s ⊗ V + S n − ( d ) V ( s ) ⊗ V + · · · + S n − ( d ) V ( s ) ⊗ V .We study the case p = 2 n .Let ≤ m ≤ be the greatest index such that the term of v in degree ( p − m | m ) is not (i.e. the term of v in S ( p − m ) / ( d ) V m ( s ) ⊗ V ).Therefore v ∈ Γ m ( V ) and it is a combination of terms in degrees ( p − m | m ) , ( p − m + 1 | m − , . . . , ( p | . We can then consider v ∈ Γ m ( V ) / Γ m − ( V ) ∼ = U ( d ) ⊗ ( V m ( s ) ⊗ V ) . Notice that given y ∈ L , y · v ∈ T p − ( V ) and the term of degree ( p − m − | m ) can beobtained only acting with y on the term of v of degree ( p − m | m ) . Hence, if v is singularfor S (5) , so must be v .Recapitulating: if v ∈ sing T p ( V ) , v ∈ U ( d ) ⊗ ( V m ( s ) ⊗ V ) = T ( V m ( s ) ⊗ V ) is asingular vector for S (5) of (even) degree ( p − m ) / in a tensor module. By Theorem 4.1,the even degree of v must be less than or equal to , which means that ( p − m ) / ≤ ,that is p ≤ m + 4 ≤ . When p is odd the same argument holds. (cid:3) The proof actually tells us more than the statement of the theorem: we can not only es-timate the singular vectors’ degree, but we can also rule out straightforwardly most of theirreducible sl ( d ) -modules whose induced modules we expect to possibly contain singularvectors of a certain degree.We recall that in our notation Ω = d ∗ ∼ = V ([1 , , , . Similarly Ω ∼ = V ([0 , , , , Ω ∼ = V ([0 , , , and Ω ∼ = V ([0 , , , .Now apply, for example, the proof’s arguments on degree 14:let v be a singular vector of degree in a minimal Verma module T ( V ) and consider v ∈ Γ ( V ) / Γ ( V ) ∼ = U ( d ) ⊗ ( V ( s ) ⊗ V ) ∼ = U ( d ) ⊗ V . If v = 0 , this means that theterm of v of degree (2 | is not .We know by Theorem 4.1 that we can find singular vectors of (even) degree in a tensor BOUND ON THE DEGREE OF SINGULAR VECTORS FOR E (5 , module T ( V ) where V is irreducible if and only if V ∼ = Ω ∼ = V ([1 , , , .Therefore, if we assume that V ≇ d ∗ , v will necessarily be . This implies that v ∈ Γ ( V ) and we can consider v ∈ U ( d ) ⊗ ( V ( s ) ⊗ V ) , thus obtaining a singular vectorin T ( V ( s ) ⊗ V ) of (even) degree (14 − / . This cannot happen, so that the onlypossible solution is v = 0 . Iterating, we discover that v must be .We proved: Lemma 5.1. If V ≇ d ∗ ∼ = V ([1 , , , , sing T ( V ) = { } .
6. B
OUND REFINING
An extremely simple lemma will be extremely useful:
Lemma 6.1.
Let T ( V ) be a Verma module for L . If v ∈ sing T ( V ) and ξ ∈ L − ∼ = s , then ξv ∈ sing S (5) T ( V ) .Proof. Take y ∈ L . Then y · ( ξv ) = [ y, ξ ] · v + ξ ( y · v ) = 0 , where the second term is because v is singular. The same goes for the first term since [ y, ξ ] ∈ L . (cid:3) We apply this new piece of information to the case V ∼ = d ∗ .Let v ∈ sing T ( d ∗ ) . By the lemma, given any ξ ∈ s , ξv (which has now degree ), is stillsingular for the action of S (5) . Consider the term of degree (3 | and the corresponding ξv ∈ Γ ( V ) / Γ ( V ) ∼ = T ( V ( s ) ⊗ V ) . Like before, it is still singular and has even degree , which implies that ξv = 0 . We can then consider ξv ∈ Γ ( V ) / Γ ( V ) and, iterating theargument, obtain that ξv must be ∀ ξ ∈ s .We remark that L − = [ L − , L − ] which means that, for any ∂ ∈ d ∼ = L − , we can find ξ , ξ ∈ s ∼ = L − such that ∂ = [ ξ , ξ ] . This in particular implies that ∂v = 0 ∀ ∂ ∈ L − .Since the action of L − on a tensor module is simply given by left multiplication, it canonly mean that v = 0 .In conclusion, we showed that even if V ∼ = d ∗ , T ( V ) cannot have singular vectors of degree14.These ideas can be applied systematically to perform a refining of the bound in Theorem5.2. Theorem 6.1.
Let T ( V ) be a minimal Verma module and let v ∈ sing T ( V ) . Then v hasdegree ≤ . More precisely:(1) if V ≇ V ([0 , , , , singular vectors have degree at most ;(2) if V ≇ V ( λ ) where λ = [0 , , , , [0 , , , , [0 , , , , [1 , , , , [0 , , , , or [1 , , , , singular vectors have degree at most ;(3) if V ≇ V ( µ ) where µ =[0 , , , , [1 , , , , [0 , , , , [0 , , , , [0 , , , , [1 , , , , [0 , , , , [1 , , , , [0 , , , , [1 , , , , [0 , , , , [1 , , , , [0 , , , , [2 , , , , [1 , , , , or [3 , , , , singular vectors have degree at most . The proof revolves around arguments similar to the previous ones. To that end, becauseof Theorem 5.1 and Theorem 4.1, we will need to be able to determine when, given an irreducible sl ( d ) -module V , we can (or rather cannot) find a copy of V ( ω i ) in V j ( s ) ⊗ V .Recall that every irreducible sl ( d ) -module V has a highest weight vector and that V isuniquely determined by the highest weight. Here, as before, ω = [1 , , , , ω =[0 , , , , ω = [0 , , , , ω = [0 , , , and ω = [0 , , , .By Frobenius duality (keeping in mind that these are all finite-dimensional modules), V ( ω i ) ⊆ V j ( s ) ⊗ V if and only if V ( ω ) ⊆ V j ( s ) ⊗ V ⊗ V ( ω i ) ∗ se e solo se V ∗ ⊆ V j ( s ) ⊗ V ( ω i ) ∗ se e solo se V ⊆ V j ( s ∗ ) ⊗ V ( ω i ) .In the following table we list the highest weights of the irreducible representations thatappear in the decomposition of those tensor products. It was obtained using computer soft-ware "LiE" ( see [LiE] for further informations). V ( s ∗ ) ⊗ V ( ω i ) V ( s ∗ ) ⊗ V ( ω i ) V ( s ∗ ) ⊗ V ( ω i ) V ( s ∗ ) ⊗ V ( ω i ) i=0 [0,1,0,0] [1,0,1,0] [0,0,2,0],[2,0,0,1] [1,0,1,1],[3,0,0,0]i=1 [0,0,1,0],[1,1,0,0] [0,1,1,0],[1,0,0,1],[2,0,1,0] [0,0,1,1],[1,0,2,0],[1,1,0,1],[2,0,0,0],[3,0,0,1] [0,1,1,1],[1,0,0,2],[1,0,1,0],[2,0,1,1],[2,1,0,0],[4,0,0,0]i=2 [0,0,0,1],[0,2,0,0],[1,0,1,0] [0,0,2,0],[0,1,0,1],[1,0,0,0],[1,1,1,0],[2,0,0,1] [0,0,1,0],[0,1,2,0],[1,0,1,1],[1,1,0,0],[2,1,0,1],[3,0,0,0] [0,0,2,1],[0,1,0,2],[0,1,1,0],[1,0,0,1],[1,1,1,1],[2,0,0,2],[2,0,1,0],[3,1,0,0]i=3 [0,0,0,0],[0,1,1,0],[1,0,0,1] [0,0,1,1],[0,1,0,0],[1,0,2,0],[1,1,0,1],[2,0,0,0] [0,0,3,0],[0,1,1,1],[1,0,0,2],[1,0,1,0],[2,0,1,1],[2,1,0,0] [0,0,1,2],[0,0,2,0],[0,1,0,1],[1,0,2,1],[1,1,0,2],[1,1,1,0],[2,0,0,1],[3,0,1,0]i=4 [0,1,0,1],[1,0,0,0] [0,0,1,0],[1,0,1,1],[1,1,0,0] [0,0,2,1],[0,1,1,0],[1,0,0,1],[2,0,0,2],[2,0,1,0] [0,0,1,1],[1,0,1,2],[1,0,2,0],[1,1,0,1],[2,0,0,0],[3,0,0,1] Proof of Theorem 6.1.
We will outline the various steps in a schematic way. The ideas arethe same we have already discussed.
Degree 13:
Let v ∈ sing T ( V ) . We first consider the term of v in degree (2 | ; it can be differentfrom zero only if V appears in the decomposition of V ( s ∗ ) ⊗ V ( ω ) ∼ = V ([0 , , , ⊕ V ([1 , , , . So if V is not isomorphic to one of these two representations, the term in BOUND ON THE DEGREE OF SINGULAR VECTORS FOR E (5 , (2 | of v must be equal to , we look at next term, which is also because has degree (3 | . Iterating, we deduce that v = 0 .If V ∼ = V ([0 , , , or V ([1 , , , , take ξ ∈ s and consider the term in degree (2 | of ξv . It can be non zero only if V does not appear in V ( s ∗ ) ⊗ V ( ω ) ∼ = V ( ω ) . It followsthat it must be and, iterating, so does ξv ∀ ξ ∈ s which in turn implies, as we already saw,that v must be . In conclusion, sing T ( V ) is always . Degree 12:
Let v ∈ sing T ( V ) . We consider the term of degree (1 | ; it can be non zero only if V appears in V ( s ∗ ) ⊗ V ( ω i ) ∼ = ω i for i = 0 , . . . , . Therefore if V is not of the form V ( ω i ) , we check the term of degree (2 | which must be unless a copy of V appears in V ( s ∗ ) ⊗ V ( ω ) = V ([0 , , , ⊕ V ([1 , , , ⊕ V ([2 , , , . In the remaining cases v = 0 .Now assume V ∼ = V ([0 , , , , V ([1 , , , , V ([2 , , , or V ( ω i ) with i = 0 , . . . , .Take ξ ∈ s and consider the term of ξv of degree (2 | . It cannot be non zero if V doesnot appear in V ( s ∗ ) = V ([0 , , , ⊕ V ([1 , , , . So if V ≇ V ( ω ) ξv = 0 ∀ ξ ∈ s and again it implies that v = 0 . Therefore the only case in which we cannot rule out thepresence of singular vectors of degree is for V ∼ = V ( ω ) . Degree 11:
Let v ∈ sing T ( V ) . We consider the term of degree (1 | ; it can be differentfrom zero when V appears in V ( s ∗ ) ⊗ V ( ω i ) for i = 0 , . . . , . It happens when V has ashighest weight one belonging to the first column of the table.If we assume V is not isomorphic to any of them, we can move to the next term whichhas degree (2 | . According to the table, if V is also not isomorphic to V ([0 , , , , V ([1 , , , , V ([1 , , , , V ([2 , , , and V ([3 , , , , v = 0 .Assume V is isomorphic to one of these representations, a fundamental representation orthe trivial one; take ξ ∈ s and check the term of degree (1 | ; if V is not a fundamen-tal representation or the trivial one, we can move to the term of degree (2 | which willbe unless a copy of V appears in V ( s ∗ ) ⊗ V ( ω ) ∼ = V ([0 , , , ⊕ V ([1 , , , ⊕ V ([2 , , , . Therefore if V is not isomorphic to V ([0 , , , , V ([0 , , , , V ([0 , , , , V ([1 , , , , V ([0 , , , or V ([1 , , , , sing T ( V ) = 0 . Degree 10:
Let v ∈ sing T . The term of v with the greatest odd degree is the one of degree (0 | .In this case we cannot deduce anything, since this term is constant from the point of viewof S (5) , therefore singular.We relay again on Lemma 6.1: take ξ ∈ s and consider ξv , which has now degree . Thistime we can still look at ξv ∈ Γ / ( V )Γ ( V ) ∼ = T ( V ( s ) ⊗ V )) which has even degree . Again, we know that it is unless V appears as an irreducible of V ( s ∗ ) ⊗ V ( ω i ) , i = 0 . . . , . So if the highest weight of V does not appear in the first column of the table, ξv = 0 . We can then move to the next term, which has degree (2 | . Here we look at theirreducible modules in V ( s ∗ ) ⊗ V ( ω ) ∼ = V ([0 , , , ⊕ V ([1 , , , ⊕ V ([1 , , , ⊕ V ([2 , , , ⊕ V ([3 , , , . So if in addition we ask that V is not isomorphic to thesemodules, ξv = 0 ∀ ξ ∈ s and again we cannot have singular vectors of degree in T ( V ) . (cid:3) R EFERENCES [BD] A. Beilinson, V. Drinfeld, Chiral algebras, AMS Colloquium Publications, 51, American Math. Soci-ety, Providence, RI, (2004).[BDK1] B. Bakalov, A. D’Andrea, V. Kac,
Theory of finite pseudoalgebras , Advances in Mathematics 162(1), 1-140 (2001). [BDK2] B. Bakalov, A. D’Andrea, V. Kac,
Irreducible modules over finite simple Lie pseudoalgebras I.Primitive pseudoalgebras of type W and S , Advances in Mathematics 204 (1), 278-346 (2006).[BDK3] B. Bakalov, A. D’Andrea, V. Kac,
Irreducible modules over finite simple Lie pseudoalgebras II.Primitive pseudoalgebras of type K , Advances in Mathematics 232 (1), 188-237 (2013).[BDK4] B. Bakalov, A. D’Andrea, V. Kac,
Irreducible modules over finite simple Lie pseudoalgebras III.Primitive pseudoalgebras of type H , arXiv 2001.04104 (2020).[C] E. Cartan,
Les groupes des transformations continues, infinis, simples , Ann. Sci. École Norm. Sup. 26,93-161 (1909)[CC] N. Cantarini, F. Caselli,
Low degree morphisms of E(5,10)-generalized Verma modules , arXiv1903.11438 (2009)[CK] S.J. Cheng, V. Kac,
Structure of some Z-graded Lie superalgebras of vector fields , TransformationGroups 4, 219-272 (1999).[D] A. D’Andrea,
Irreducible modules over finite simple Lie pseudoalgebras IV. Non-primitive pseudoalge-bras , arXiv 2005.07470 (2020).[K] V. Kac,
Classification of infinite-dimensional simple linearly compact Lie superalgebras , Adv. Math.139139, 1-55 (1998)[KR1] V. Kac, A. Rudakov,
Representations of the exceptional Lie superalgebra E(3,6) I: Degeneracy con-ditions , Transformation Groups 7, 67-86 (2002).[KR2] V. Kac, A. Rudakov,
Representations of the exceptional Lie superalgebra E(3,6) II: Four series ofdegenerate modules , Comm. Math. Phys. 222, 611-661 (2001).[KR3] V. Kac, A. Rudakov,
Representations of the exceptional Lie superalgebra E(3,6) III: Classification ofsingular vectors , J. Algebra Appl, 4, 15-59 (2005).[KR4] V. Kac, A. Rudakov,
Complexes of modules over the exceptional Lie superalgebras E(3,8) andE(5,10) , Int. MAth. Res. Not. 19, 1007-1025 (2002).[M] I.M. Musson,
Lie superalgebras and enveloping algebras , American Mathematical Society 131 (2012).[R] A. Rudakov,
Morphisms of Verma modules over exceptional Lie superalgebra E(5,10) , arXiv1003.1369v1 (2012)[S] M. Sweedler, Hopf algebras, Math. Lecture Note Series, W. A. Benjamin, Inc., New York (1969).[LiE] M.A.A. van Leeuwen, A.M. Cohen, B. Lisser, "LiE, A Package for Lie Group Computations", Com-puter Algebra Nederland, Amsterdam (1992).D
IPARTIMENTO DI M ATEMATICA "G UIDO C ASTELNUOVO ", U
NIVERSITA ’ DI R OMA "L A S APIENZA ",P
IAZZALE A LDO M ORO
5, R
OMA , I, I