An application of collapsing levels to the representation theory of affine vertex algebras
Drazen Adamovic, Victor G. Kac, Pierluigi Moseneder Frajria, Paolo Papi, Ozren Perse
aa r X i v : . [ m a t h . R T ] O c t AN APPLICATION OF COLLAPSING LEVELS TO THE REPRESENTATIONTHEORY OF AFFINE VERTEX ALGEBRAS
DRAˇZEN ADAMOVI´C, VICTOR G. KAC, PIERLUIGI M ¨OSENEDER FRAJRIA, PAOLO PAPI,AND OZREN PERˇSE
Abstract.
We discover a large class of simple affine vertex algebras V k ( g ), associated to basic Liesuperalgebras g at non–admissible collapsing levels k , having exactly one irreducible g –locally finitemodule in the category O . In the case when g is a Lie algebra, we prove a complete reducibility resultfor V k ( g )–modules at an arbitrary collapsing level. We also determine the generators of the maximalideal in the universal affine vertex algebra V k ( g ) at certain negative integer levels. Considering someconformal embeddings in the simple affine vertex algebras V − / ( C n ) and V − ( E ), we surprisinglyobtain the realization of non-simple affine vertex algebras of types B and D having exactly onenon-trivial ideal. Introduction
Affine vertex algebras are one of the most interesting and important classes of vertex algebras.Categories of modules for simple affine vertex algebra V k ( g ), associated to a simple Lie algebra g ,have mostly been studied in the case of positive integer levels k ∈ Z ≥ . These categories enjoy manynice properties such as: finitely many irreducibles, semisimplicity, modular invariance of characters(cf. [26], [31], [33], [41]).In recent years, affine vertex algebras have attracted a lot of attention because of their connectionwith affine W –algebras W k ( g , f ), obtained by quantum Hamiltonian reduction (cf. [21], [23], [34],[35]). Since the quantum Hamiltonian reduction functor H f ( · ) maps any integrable b g –module tozero (cf. [12], [34]), in order to obtain interesting W –algebras, one has to consider affine vertexalgebras V k ( g ), for k / ∈ Z ≥ .It turns out that for certain non-admissible levels k (such as negative integer levels), the associatedvertex algebras V k ( g ) have finitely many irreducibles in category O (cf. [15], [17], [40]), and theircharacters satisfy certain modular-like properties (cf. [14]). These affine vertex algebras then give C –cofinite W –algebras W k ( g , f ), for properly chosen nilpotent element f (cf. [36], [38]).In this paper, we classify irreducible modules in the category KL k (i.e. the category of g –locallyfinite V k ( g )–modules in O k (see Subsection 2.3)) for a large family of collapsing levels k . Recall from [4]that a level k is called collapsing if the simple W –algebra W k ( g , θ ), associated to a minimal nilpotentelement e − θ , is isomorphic to its affine vertex subalgebra V k ( g ♮ ) (see Definition 2.2 and (2.7)). Inthe present paper we keep the notation of [4]. In particular, the highest root is normalized by thecondition ( θ, θ ) = 2. We discover a large family of vertex algebras having one irreducible module inthe category KL k , which in a way extends the results on Deligne series from [15]. Part (1) is proventhere in the Lie algebra case. Theorem 1.1.
Assume that the level k and the basic simple Lie superalgebra g satisfy one of thefollowing conditions: (1) k = − h ∨ − and g is one of the Lie algebras of exceptional Deligne’s series A , G , D , F , E , E , E , or g = psl ( m | m ) ( m ≥ ), osp ( n + 8 | n ) ( n ≥ ), spo (2 | , F (4) , G (3) (for bothchoices of θ ); (2) k = − h ∨ / and g = osp ( n + 4 m + 8 | n ) , n ≥ , m ≥ . (3) k = − h ∨ / and g = D m , m ≥ . (4) k = − and g = E . Then V k ( g ) is the unique irreducible V k ( g ) –module in the category KL k . We also prove a complete reducibility result in KL k (cf. Theorem 5.9, Theorem 5.7): Theorem 1.2.
Assume that g is a Lie algebra and k ∈ C \ Z ≥ . Then KL k is a semi-simple categoryin the following cases: • k is a collapsing level. • W k ( g , θ ) is a rational vertex operator algebra. It is interesting that in some cases we have that KL k is a semi-simple category, but there can existindecomposable but not irreducible V k ( g )–modules in the category O . In order to prove Theorem 1.2we modified methods from [28] and [20] in a vertex algebraic setting. In particular we prove that thecontravariant functor M M σ from [20] acts on the category KL k (cf. Lemma 3.6). Then for theproof of complete reducibility in KL k it is enough to check that every highest weight V k ( g )–modulein KL k is irreducible (cf. Theorem 5.5).Representation theory of a simple affine vertex algebra V k ( g ) is naturally connected with the struc-ture of the maximal ideal in the universal affine vertex algebra V k ( g ). In the second part of paper wepresent explicit formulas for singular vectors which generate the maximal ideal in V − ℓ ( D ℓ ) (whichis case (3) of Theorem 1.1) and V − ( D ℓ ). In the second case, we show that the Hamiltonian reductionfunctor H θ ( · ) gives an equivalence of the category of g –locally finite V − ( D ℓ )–modules KL − andthe category of modules for a rational vertex algebra V ℓ − ( A ). Singular vectors in V k ( g ) for certainnegative integer levels k have also been constructed in [2].We also apply our results to study the structure of conformally embedded subalgebras of somesimple affine vertex algebras.As in [6], for a subalgebra k of a simple Lie algebra g , we denote by e V ( k, k ) the vertex subalgebraof V k ( g ) generated by x ( − , x ∈ k . If k is a reductive quadratic subalgebra of g , then we say that e V ( k, k ) is conformally embedded in V k ( g ) if the Sugawara-Virasoro vectors of both algebras coincide.We also say that k is conformally embedded in g at level k if e V ( k, k ) is conformally embedded in V k ( g ).We are able to prove that in the cases listed in Theorem 1.3 below, e V ( k, k ) is not simple. Onthe other hand, we show that V − / ( C ) contains a simple subalgebra V − ( B ) ⊗ V − / ( A ) (seeCorollary 7.4). For the conformal embedding of D × A into E at level k = −
4, we show that e V ( − , D × A ) = V − ( D ) ⊗ V − ( A ) where V − ( D ) is a quotient of the universal affine vertexalgebra V − ( D ) by two singular vectors of conformal weights two and three (cf. (9.6)). Moreover, V − ( D ) has infinitely many irreducible modules in the category of g –locally finite modules, which weexplicitly describe. All of them appear in V − ( E ) as submodules or subquotients. Theorem 1.3.
Let V k ( D ℓ ) , V k ( B ℓ ) , be the vertex algebras defined in (6.3) , (7.1) , (9.6) . Consider thefollowing conformal embeddings: (1) D ℓ × A into C l for ℓ ≥ at level k = − . (2) B ℓ × A into C l +1 for ℓ ≥ at level k = − . (3) D × A into E at level k = − .Then, • e V ( − , D ℓ × A ) = V − ( D ℓ ) ⊗ V − ℓ ( A ) in case (1), • e V ( − , B ℓ × A ) = V − ( B ℓ ) ⊗ V − ℓ − / ( A ) in case (2), • e V ( − , D × A ) = V − ( D ) ⊗ V − ( A ) in case (3).Moreover, the algebras V k ( D ℓ ) , V k ( B ℓ ) , are non-simple, with a unique non-trivial ideal. The decompositions of the embeddings above is still an open problem, and will be a subject of ourforthcoming papers.
Acknowledgement.
We would like to thank Maria Gorelik, Tomoyuki Arakawa and Anne Moreaufor correspondence and discussions. Preliminaries
We assume that the reader is familiar with the notion of vertex (super)algebra (cf. [18], [25], [32])and of simple basic Lie superalgebras (see [30]) and their affinizations (see [31] for the Lie algebracase).Let V be a conformal vertex algebra. Denote by A ( V ) the associative algebra introduced in [41],called the Zhu algebra of V .2.1. Basic Lie superalgebras and minimal gradings.
For the reader’s convenience we recall herethe setting and notation of [4] regarding basic Lie superalgebras and their minimal gradings. Let g = g ¯0 ⊕ g ¯1 be a simple finite dimensional basic Lie superalgebra. We choose a Cartan subalgebra h for g ¯0 and let ∆ be the set of roots. Assume g is not osp (3 | n ). A root − θ is called minimal if it is evenand there exists an additive function ϕ : ∆ → R such that ϕ | ∆ = 0 and ϕ ( θ ) > ϕ ( η ) , ∀ η ∈ ∆ \ { θ } .Fix a minimal root − θ of g . We may choose root vectors e θ and e − θ such that[ e θ , e − θ ] = x ∈ h , [ x, e ± θ ] = ± e ± θ . Due to the minimality of − θ , the eigenspace decomposition of ad x defines a minimal Z -grading ([35,(5.1)]):(2.1) g = g − ⊕ g − / ⊕ g ⊕ g / ⊕ g , where g ± = C e ± θ . We thus have a bijective correspondence between minimal gradings (up to anautomorphism of g ) and minimal roots (up to the action of the Weyl group). Furthermore, one has(2.2) g = g ♮ ⊕ C x, g ♮ = { a ∈ g | ( a | x ) = 0 } . Note that g ♮ is the centralizer of the triple { f θ , x, e θ } . We can choose h ♮ = { h ∈ h | ( h | x ) = 0 } , as aCartan subalgebra of the Lie superalgebra g ♮ , so that h = h ♮ ⊕ C x .For a given choice of a minimal root − θ , we normalize the invariant bilinear form ( ·|· ) on g by thecondition(2.3) ( θ | θ ) = 2 . The dual Coxeter number h ∨ of the pair ( g , θ ) (equivalently, of the minimal gradation (2.1)) is definedto be half the eigenvalue of the Casimir operator of g corresponding to ( ·|· ), normalized by (2.3). Since θ is the highest root, we have that 2 h ∨ = ( θ | θ + 2 ρ ) hence(2.4) ( ρ | θ ) = h ∨ − . The complete list of the Lie superalgebras g ♮ , the g ♮ –modules g ± / (they are isomorphic and self-dual), and h ∨ for all possible choices of g and of θ (up to isomorphism) is given in Tables 1,2,3 of[35]. We reproduce them below. Note that in these tables g = osp ( m | n ) (resp. g = spo ( n | m )) meansthat θ is the highest root of the simple component so ( m ) (resp. sp ( n )) of g ¯0 . Also, for g = sl ( m | n )or psl ( m | m ) we always take θ to be the highest root of the simple component sl ( m ) of g ¯0 (for m = 4we take one of the simple roots). Note that the exceptional Lie superalgebras g = F (4) and g = G (3)appear in both Tables 2 and 3, which corresponds to the two inequivalent choices of θ , the first onebeing a root of the simple component sl (2) of g ¯0 . Table 1 g is a simple Lie algebra. g g ♮ g / h ∨ g g ♮ g / h ∨ sl ( n ) , n ≥ gl ( n − C n − ⊕ ( C n − ) ∗ n F sp (6) V C so ( n ) , n ≥ sl (2) ⊕ so ( n − C ⊗ C n − n − E sl (6) V C sp ( n ) , n ≥ sp ( n − C n − n/ E so (12) spin G sl (2) S C E E dim = 56 30Table 2 g is not a Lie algebra but g ♮ is and g ± / is purely odd ( m ≥ ). ADAMOVI´C, KAC, M ¨OSENEDER, PAPI, PERˇSE g g ♮ g / h ∨ g g ♮ g / h ∨ sl (2 | m ) , gl ( m ) C m ⊕ ( C m ) ∗ − m D (2 , a ) sl (2) ⊕ sl (2) C ⊗ C m = 2 psl (2 | sl (2) C ⊕ C F (4) so (7) spin − spo (2 | m ) so ( m ) C m − m/ G (3) G Dim = 0 | − / osp (4 | m ) sl (2) ⊕ sp ( m ) C ⊗ C m − m Table 3
Both g and g ♮ are not Lie algebras ( m, n ≥ ). g g ♮ g / h ∨ sl ( m | n ), m = n, m > gl ( m − | n ) C m − | n ⊕ ( C m − | n ) ∗ m − npsl ( m | m ) , m > sl ( m − | m ) C m − | m ⊕ ( C m − | m ) ∗ spo ( n | m ) , n ≥ spo ( n − | m ) C n − | m / n − m ) + 1 osp ( m | n ) , m ≥ osp ( m − | n ) ⊕ sl (2) C m − | n ⊗ C m − n − F (4) D (2 ,
1; 2) Dim = 6 | G (3) osp (3 |
2) Dim = 4 | In this paper we shall exclude the case of g = sl ( n + 2 | n ), n >
0. In all other cases the Liesuperalgebra g ♮ decomposes in a direct sum of all its minimal ideals, called components of g ♮ : g ♮ = M i ∈ I g ♮i , where each summand is either the (at most 1-dimensional) center of g ♮ or is a basic simple Liesuperalgebra different from psl ( n | n ). Let C g ♮i be the Casimir operator of g ♮i corresponding to ( ·|· ) | g ♮i × g ♮i .We define the dual Coxeter number h ∨ ,i of g ♮i as half of the eigenvalue of C g ♮i acting on g ♮i (which is 0if g ♮i is abelian).Denote by V g ( µ ) (or V ( µ )) the irreducible finite-dimensional highest weight g –module with highestweight µ . Denote by P + the set of highest weights of irreducible finite-dimensional representations of g . Since h = h ♮ ⊕ C x , we have, in particular, that µ ∈ h ∗ can be uniquely written as(2.5) µ = µ | h ♮ + ℓθ, with ℓ ∈ C . If µ ∈ P + , then, since θ ( h ♮ ) = 0, µ ( θ ∨ ) = 2 ℓ ∈ Z , so ℓ ∈ Z ≥ .2.2. Affine Lie algebras, vertex algebras, W -algebras. Let b g be the affinization of g : b g = C [ t, t − ] ⊗ g ⊕ C K ⊕ C d with the usual commutation relations. We let δ be the fundamental imaginary root. Let α = δ − θ the affine simple root. Since θ is even, hence non-isotropic, so that α ∨ = K − θ ∨ makes sense.Denote by L ( λ ) (or L g ( λ )) the irreducible highest weight b g –module with highest weight λ .Denote by V k ( g ) the universal affine vertex algebra associated to b g of level k ∈ C . We shall assumethat k = − h ∨ . Then (see e.g. [32]) V k ( g ) is a conformal vertex algebra with Segal-Sugawara conformalvector ω g . Let Y ( ω g , z ) = P L g ( n ) z − n − be the corresponding Virasoro field. Denote by V k ( g ) the(unique) simple quotient of V k ( g ). Clearly, V k ( g ) ∼ = L g ( k Λ ) as b g –modules.Denote by W k ( g , θ ) the affine W –algebra obtained from V k ( g ) by Hamiltonian reduction relative toa minimal nilpotent element e − θ . Denote by W k ( g , θ ) the simple quotient of W k ( g , θ ). Recall that thevertex algebra W k ( g , θ ) is strongly and freely generated by elements J { a } , where a runs over a basisof g ♮ , G { v } , where v runs over a basis of g − / , and the Virasoro vector ω . The elements J { a } , G { v } are primary of conformal weight 1 and 3 /
2, respectively, with respect to ω .Let V k ( g ♮ ) be the subalgebra of the vertex algebra W k ( g , θ ), generated by { J { a } | a ∈ g ♮ } . Thevertex algebra V k ( g ♮ ) is isomorphic to a universal affine vertex algebra. More precisely, letting(2.6) k i = k + ( h ∨ − h ∨ ,i ) , i ∈ I, the map a J { a } extends to an isomorphism V k ( g ♮ ) ≃ N i ∈ I V k i ( g ♮i ) . We also set V k ( g ♮ ) to be the image of V k ( g ♮ ) in W k ( g , θ ). Clearly we can write(2.7) V k ( g ♮ ) ≃ O i ∈ I V k i ( g ♮i ) , where V k i ( g ♮i ) is some quotient (not necessarily simple) of V k i ( g ♮i ).2.3. Category O and Hamiltonian reduction functor. Recall that b g -module M is in category O k if it is b h -diagonalizable with finite dimensional weight spaces, K acts as kId M and M has a finitenumber of maximal weights.There is a remarkable functor H θ from O k to the category of W k ( g , θ )-modules whose propertieswill be very important in the following. We recall them in a form suitable for our purposes (see [12]for details; there H θ is denoted by H ). Theorem 2.1. (1) H θ is exact. (2) If L ( λ ) is a irreducible highest weight b g -module, then λ ( α ∨ ) ∈ Z ≥ implies H θ ( L ( λ )) = { } .Otherwise H θ ( L ( λ )) is isomorphic to the irreducible W k ( g , θ ) -module with highest weight φ λ defined by formula (67) in [12] . Collapsing levels.Definition 2.2.
Assume k = − h ∨ . If W k ( g , θ ) = V k ( g ♮ ) , we say that k is a collapsing level . Theorem 2.3. [4, Theorem 3.3]
Let p ( k ) be the polynomial listed in Table 4 below. Then k is acollapsing level if and only if k = − h ∨ and p ( k ) = 0 . In such cases, (2.8) W k ( g , θ ) = O i ∈ I ∗ V k i ( g ♮i ) , where I ∗ = { i ∈ I | k i = 0 } . If I ∗ = ∅ , then W k ( g , θ ) = C . Table 4
Polynomials p ( k ) . g p ( k ) g p ( k ) sl ( m | n ), n = m ( k + 1)( k + ( m − n ) / E ( k + 3)( k + 4) psl ( m | m ) k ( k + 1) E ( k + 4)( k + 6) osp ( m | n ) ( k + 2)( k + ( m − n − / E ( k + 6)( k + 10) spo ( n | m ) ( k + 1 / k + ( n − m + 4) / F ( k + 5 / k + 3) D (2 , a ) ( k − a )( k + 1 + a ) G ( k + 4 / k + 5 / F (4), g ♮ = so (7) ( k + 2 / k − / G (3), g ♮ = G ( k − / k + 3 / F (4), g ♮ = D (2 ,
1; 2) ( k + 3 / k + 1) G (3), g ♮ = osp (3 |
2) ( k + 2 / k + 4 / Weyl vertex algebra.
Let M ℓ denote the Weyl vertex algebra (also called symplectic bosons)generated by even elements a ± i , i = 1 , . . . , ℓ satisfying the following λ –brackets[( a ± i ) λ ( a ± j )] = 0 , [( a + i ) λ ( a − j )] = δ i,j . Recall also that the symplectic affine vertex algebra V − / ( C ℓ ) is realized as a Z –orbifold of M ℓ (see[22]). 3. The category KL k Let k be a noncritical level. Note that the Casimir element of b g can be expressed as Ω = d + L g (0);it commutes with b g –action.Consider the category C k of modules for the universal affine vertex algebra V k ( g ), i.e. the categoryof restricted b g –modules of level k . Regard M ∈ C k as a b g –module by letting d act as − L g (0). Let KL k be the category of modules M ∈ C k such that, as b g -modules, are in O k and which admit thefollowing weight space decomposition with respect to L g (0): M = M α ∈ C M ( α ) , L g (0) | M ( α ) ≡ α Id , dim M ( α ) < ∞ . ADAMOVI´C, KAC, M ¨OSENEDER, PAPI, PERˇSE
Our definition is related but different from the one introduced in [13]. Let KL k be the category of allmodules in KL k which are V k ( g )–modules. Remark 3.1. If V k ( g ) has finitely many irreducible modules in the category KL k , one can show thatevery V k ( g ) –module M in KL k is of finite length. This happens when k is admissible (cf. [12] ) andwhen V k ( g ) is quasi-lisse (cf. [14] ). But when V k ( g ) has infinitely many irreducible modules in KL k (as in the cases considered in [39] , [11] ), then one can have modules in KL k of infinite length. Recall that there is a one-to-one correspondence between irreducible Z ≥ –graded modules for aconformal vertex algebra V (with a conformal vector ω , such that Y ( ω, z ) = P i ∈ Z L ( i ) z − i − ) andirreducible modules for the corresponding Zhu algebra A ( V ) [41]. This implies, in particular, thatthere is a one-to-one correspondence between irreducible finite-dimensional A ( V )–modules and ir-reducible Z ≥ –graded V –modules whose graded components, which are eigenspaces for L (0), arefinite-dimensional. In the case of affine vertex algebras, we have the following simple interpretation. Proposition 3.2.
Let e V k ( g ) be a quotient of V k ( g ) (not necessary simple). Consider e V k ( g ) as aconformal vertex algebra with conformal vector ω g . Then there is a one-to-one correspondence betweenirreducible e V k ( g ) in the category KL k and irreducible finite-dimensional A ( e V k ( g )) –modules. Corollary 3.3.
Assume that g is a simple basic Lie superalgebra and e V k ( g ) is a quotient of V k ( g ) such that the trivial module C is the unique finite-dimensional irreducible A ( e V k ( g )) –module. Then e V k ( g ) = V k ( g ) .Proof. Assume that e V k ( g ) is not simple. Then it contains a non-zero graded ideal I = e V k ( g ) withrespect to L g (0): I = M n ∈ Z ≥ I ( n + n ) , L g (0) | I ( r ) = r Id , I ( n ) = 0 . Since I = e V k ( g ), we have that n >
0, otherwise ∈ I .We can consider I ( n ) as a finite-dimensional module for g and for the Zhu algebra A ( e V k ( g )).Since the Casimir element C g of g acts on I ( n ) as the non-zero constant 2( k + h ∨ ) n , we concludethat C g acts by the same constant on any irreducible g –subquotient of I ( n ). But any irreduciblesubquotient of I ( n ) is an irreducible finite–dimensional A ( e V k ( g ))–module, and therefore it is trivial.This implies that C g acts non-trivially on a trivial g –module, a contradiction. (cid:3) Take the Chevalley generators e i , f i , h i , i = 0 , . . . , ℓ , of the Kac–Moody Lie algebra b g such that e i , f i , h i , i = 1 , . . . , ℓ , are the Chevalley generators of g . Let σ be the Chevalley antiautomorphism of b g defined by e i f i , f i e i , h i h i , d d ( i = 0 , . . . , ℓ ) . Assume that M is from the category O of non-critical level k . Then M admits the decompositioninto weight spaces M = L µ ∈ Ω( M ) M µ , where Ω( M ) is the set of weights of M and dim M µ < ∞ for every µ ∈ Ω( M ). For a finite-dimensional vector spaces U , let U ∗ denote its dual space. Thenwe have the contravariant functor M M σ [20] acting on modules from the category O . Here M σ = L µ ∈ Ω( M ) M ∗ µ is the b g –module uniquely determined by h yw ′ , w i = h w ′ , σ ( y ) w i , y ∈ b g , w ′ ∈ M σ , w ∈ M. It is easy to see that M admits the decomposition M = M α ∈ C M ( α ) , L g (0) | M ( α ) ≡ α Id(3.1)such that : • for any α ∈ C we have M ( α − n ) = 0 for n ∈ Z sufficiently large; • for any µ ∈ Ω( M ) there exist α ∈ C such that M µ ⊂ M ( α ). Proposition 3.4.
Assume that a module M is in the category O k . Then M is in the category KL k if and only if M is g -locally finite.Proof. If M is in KL k then it admits a decomposition as in (3.1). Since the spaces M ( α ) are g –stableand finite-dimensional, M is g –locally finite.Let us prove the converse. If M is a highest weight module which is g –locally finite, then clearlyall eigenspaces for L g (0) are finite-dimensional. Assume now that M is an arbitrary g –locally finitemodule in the category O k . Take α ∈ C such that M ( α ) = { } . Then from [20, Proposition 3.1] wesee that M has an increasing filtration (possibly infinite) { } = M ⊂ M ⊂ · · · ⊂ M (3.2)such that for every j ∈ Z > , M j /M j − ∼ = e L ( λ j ) is a highest weight V k ( g )–module with highest weight λ j , which is g –locally finite. Let h λ j denotes the lowest conformal weight of e L ( λ j ). Since the factors M i /M i − ( i ≤ j ) of M j are highest weight modules, their L g (0)–eigenspaces are finite-dimensional.This implies that the L g (0)–eigenspaces of M j is finite-dimensional. By using the properties of thecategory O one sees the following: • There exists a finite subset { d , · · · , d s } ⊂ C such that α ∈ S si =1 ( d i + Z ≥ ). • For d ∈ C there exist only finitely many subquotients e L ( λ j ) in (3.2) such that h λ j = d .This implies that there is j ∈ Z > such that α < h λ j for j ≥ j . Therefore M ( α ) ⊂ M j . This provesthat M ( α ) is finite-dimensional. (cid:3) Remark 3.5.
We will use several times the following fact, which is a consequence of the previousproposition: for any k / ∈ Z ≥ and any irreducible highest weight module L ( λ ) in the category KL k ,one has λ ( α ∨ ) / ∈ Z ≥ . Since σ ( L g (0)) = L g (0), if M is in the category KL k , then M σ is also in the category KL k . Thenext result shows that this functor acts on the category KL k . In the proof we find an explicit relationof M σ with the contragradient modules, defined for ordinary modules for vertex operator algebras[24]. Lemma 3.6. (1)
Assume that M is a V k ( g ) –module in the category O . Then M σ is also a V k ( g ) –module in thecategory O . (2) Assume that M is a V k ( g ) –module in the category KL k . Then M σ is also in KL k .Proof. Assume that M is a V k ( g )–module in the category O . Take the weight decomposition M = L µ ∈ Ω( M ) M µ , and set M c = L µ ∈ Ω( M ) M ∗ µ . By applying the same approach as in the construction ofthe contragredient module from [24, Section 5], we get a V k ( g )–module ( M c , Y M c ( · , z )), with vertexoperator map h Y M c ( v, z ) w ′ , w i = h w ′ , Y M ( e zL g (1) ( − z − ) L g (0) v, z ) w i , (3.3)where w ′ ∈ M c , w ∈ M . The b g –action on M c is uniquely determined by h x ( n ) w ′ , w i = −h w ′ , x ( − n ) w i ( x ∈ g ) . As a vector space M c = M σ , but we have different actions of b g . (Note that, in general, M c can beoutside of the category O .)Take the Lie algebra automorphism h ∈ Aut ( g ) such that e i
7→ − f i , f i
7→ − e i , h i
7→ − h i ( i = 1 , . . . , ℓ ) . Then h can be lifted to an automorphism of V k ( g ). Since the maximal ideal of V k ( g ) is unique,then it is h –invariant, thus h is also an automorphism of V k ( g ). Then we can define a V k ( g )–module( M ch , Y M ch ( · , z )) where M ch := M c , Y M ch ( v, z ) = Y M c ( hv, z ) . ADAMOVI´C, KAC, M ¨OSENEDER, PAPI, PERˇSE On M ch we have h e i ( n ) w ′ , w i = h w ′ , f i ( − n ) w ih f i ( n ) w ′ , w i = h w ′ , e i ( − n ) w ih h i ( n ) w ′ , w i = h w ′ , h i ( − n ) w i where i = 1 , . . . , ℓ . This implies that M ch = M σ . This proves the assertion (1).Assume now that M is in the category KL k . Then all L g (0)–eigenspaces are finite-dimensional,thus M c = M µ ∈ Ω( M ) M ∗ µ = M α ∈ C M ( α ) ∗ . This implies the V k ( g )–module ( M c , Y M c ( · , z )) coincides with the contragredient module [24], realizedon the restricted dual space L α ∈ C M ( α ) ∗ , with the vertex operator map (3.3). Since the L g (0)–eigenspaces of M c are finite-dimensional, we conclude that M c and M σ = M ch are V k ( g )–modules in KL k . Claim (2) follows. (cid:3) Constructions of vertex algebras with one irreducible module in KL k viacollapsing levels By [4], if k is a collapsing level, then either W k ( g , θ ) = C , W k ( g , θ ) = M (1), or W k ( g , θ ) = V k ′ ( a )for a unique simple component a of g ♮ . Here the level k ′ is computed with respect to the invariantbilinear form of a normalized so that the minimal root has squared length 2. For a = sl ( m | n ), m ≥ sl ( m ). For a = osp ( m | n ) we write spo ( n | m )vs. osp ( m | n ) to specify the choice of the minimal root. In all other cases the minimal root of a isunique.To simplify notation define V k ′ ( g ♮ ) to be as follows: V k ′ ( g ♮ ) = C if W k ( g , θ ) = C ; in this case we set k ′ = 0; M (1) if W k ( g , θ ) = M (1); in this case we set k ′ = 1; V k ′ ( a ) otherwise.In Table 5 we summarize all the relevant data.Assume that k / ∈ Z ≥ and that:(1) k is a collapsing level for g ;(2) V k ′ ( g ♮ ) is the unique irreducible V k ′ ( g ♮ )–module in the category KL k ′ .Assume that L ( b Λ) is an irreducible V k ( g )-module in the category KL k . Set µ = b Λ | h . By Proposition3.4 we have µ ∈ P + , hence, by (2.5), the weight µ has the form µ = µ ♮ + ℓθ with ℓ ∈ Z ≥ , where µ ♮ = µ | h ♮ .Since k / ∈ Z ≥ , by Theorem 2.1, H θ ( L ( b Λ)) is a non-trivial irreducible module for W k ( g , θ ). Since L ( b Λ) is a quotient of the Verma module M ( b Λ), then, by exactness of H θ , H θ ( L ( b Λ)) is the quotientof a Verma module for W k ( g , θ ) = V k ′ ( g ♮ ) hence it is an irreducible highest weight module. By [35,(6.14)] its highest weight as V k ( g ♮ )-module is b Λ ♮ with b Λ ♮ ( K ) = k ′ and b Λ ♮ | h ♮ = µ ♮ . Therefore H θ ( L ( b Λ)) = L g ♮ ( b Λ ♮ ) . In particular H θ ( L ( b Λ)) is in the category KL k ′ .Moreover, under the identification of the centralizer g f of f in g with g ⊕ g / via ad ( f ) (seeExample 6.2 of [35]), we get that x acts on H θ ( L ( b Λ)) via J { f } , and J { f } is the conformal vector of W ( k, θ ) (see the proof of Theorem 5.1 of [35]). Since the level is collapsing we know, by Proposition4.1 of [4], that the conformal vector of W k ( g , θ ) coincides with the Segal-Sugawara vector conformal Table 5
Values of k and k ′ . g V k ′ ( g ♮ ) k k ′ sl ( m | n ), m = n, m > , m − = n V k ′ ( sl ( m − | n ))) n − m n − m +22 sl (3 | n ), n = 3 , n = 1 , n = 0 V k ′ ( sl (1 | n ))) n −
32 1 − n sl (3) C − sl (2 | n ), n = 2 , n = 1 , n = 0 V k ′ ( sl ( n ))) n − − n sl (2 |
1) = spo (2 | C − sl ( m | n ), m = n, n + 1 , n + 2 , m ≥ M (1) − psl ( m | m ) , m ≥ C − spo ( n | m ) , m = n, n + 2 , n ≥ V k ′ ( spo ( n − | m )) m − n − m − n − spo (2 | m ) , m ≥ V k ′ ( so ( m )) m −
64 4 − m spo (2 | V k ′ ( sl (2)) − spo (2 | C − spo ( n | m ) , m = n + 1 , n ≥ C − / osp ( m | n ) , m = n, m = n + 8 , m ≥ V k ′ ( osp ( m − | n )) n − m +42 8 − m + n osp ( m | n ) , n = m,
0; 4 ≤ m ≤ V k ′ ( osp ( m − | n )) n − m +42 m − n − osp ( m | n ) , m = n + 4 , n + 8; m ≥ V k ′ ( sl (2)) − m − n − osp ( n + 8 | n ) , n ≥ C − D (2 , a ) V k ′ ( sl (2)) a − a a D (2 , a ) V k ′ ( sl (2)) − a − − aa F (4) V k ′ ( D (2 ,
1; 2)) − F (4) C − / F (4) V k ′ ( so (7)) − F (4) C − E V k ′ ( sl (6)) − − E C − E V k ′ ( so (12)) − − E C − E V k ′ ( E ) − − E C − F V k ′ ( sp (6)) − − F C − / G V k ′ ( sl (2)) − G C − G (3) V k ′ ( G ) − G (3) C − G (3) V k ′ ( osp (3 | − G (3) C − ω g ♮ of V k ′ ( g ♮ ) hence, by (6.14) of [35] again, we obtain that the ( ω g ♮ ) acts on the lowest componentof H θ ( L ( b Λ)) by cI with c = ( µ + 2 ρ, µ )2( k + h ∨ ) − µ ( x ) . (4.1)Now condition (2) implies that µ ♮ = 0, so µ = ℓθ and( µ + 2 ρ, µ )2( k + h ∨ ) − µ ( x ) = ( ℓθ + 2 ρ, ℓθ )2( k + h ∨ ) − ℓ = 0 . By using formula (2.4), we get(4.2) 2 ℓ + (2 h ∨ − ℓ k + h ∨ ) − ℓ = ℓ − ( k + 1) ℓk + h ∨ = 0 . • Consider first the case k = − h ∨ / g = D n , n ≥ g = osp ( n +4 m +8 | n ), n ≥ ℓ + ( h ∨ − ℓh ∨ + 2 = 0 . (4.3) We get ℓ = 0 or 2 ℓ + h ∨ − • Next we consider the case k = − h ∨ / −
1. We get6 ℓ + h ∨ ℓ h ∨ − . (4.4) We conclude that ℓ = 0 or ℓ = − h ∨ .By using the above analysis and properties of Hamiltonian reduction, we get the following lemma,which extends a result of [15] for Lie algebras to the super case. Lemma 4.1.
Assume that k = − h ∨ − and g is one of the Lie algebras of exceptional Deligne’sseries A , G , D , F , E , E , E , or g = psl ( m | m ) ( m ≥ ), osp ( n + 8 | n ) ( n ≥ ), spo (2 | , F (4) , G (3) (for both choices of θ ).Assume that L ( λ ) is a V k ( g ) –module in the category O . Then one of the following condition holds: (1) λ ( α ∨ ) ∈ Z ≥ ; (2) ¯ λ is either or − h ∨ θ , where ¯ λ is the restriction of λ to h .Proof. By Theorem 2.1, if L ( λ ) is a V k ( g )–module for which λ ( α ∨ ) / ∈ Z ≥ , then H θ ( L ( λ )) is anirreducible W k ( g , θ ) = H θ ( V k ( g ))–module. The conditions on g exactly correspond to the cases when W k ( g , θ ) is one-dimensional (cf. [4], [15]), so the discussion that precedes the Lemma and relation(4.4) imply that ¯ λ is as in (2). (cid:3) Lemma 4.1 implies:
Theorem 4.2.
Assume that the level k and the Lie superalgebra g satisfy one of the following condi-tions: (1) k = − h ∨ − and g is one of the Lie algebras of exceptional Deligne’s series A , G , D , F , E , E , E , or g = psl ( m | m ) ( m ≥ ), osp ( n + 8 | n ) ( n ≥ ), spo (2 | , F (4) , G (3) (for bothchoices of θ ); (2) k = − h ∨ / and g = osp ( n + 4 m + 8 | n ) , n ≥ , m ≥ . (3) k = − h ∨ / and g = D m , m ≥ . (4) k = − and g = E .Then V k ( g ) is the unique irreducible V k ( g ) –module in the category KL k .Proof. If the Lie superalgebra g is as in (1), then Lemma 4.1 and Remark 3.5 imply that ¯ λ is either 0 or − h ∨ θ . Since in all cases in (1) we have that h ∨ ∈ Z ≥ , one obtains that the irreducible highest weight g –module with highest weight ¯ λ = − h ∨ θ cannot be finite-dimensional. Therefore L ( λ ) can not be a module in KL k . This proves that ¯ λ = 0 and therefore V k ( g ) is the unique irreducible V k ( g )–modulein the category KL k .Let us consider the case g = osp ( n + 4 m + 8 | n ). Then for every m ∈ Z ≥ we have: h ∨ = 4 m + 6 , (4.5) k = − h ∨ / − m + 1) , (4.6) 2 ℓ + h ∨ − = 0 ∀ ℓ ∈ Z ≥ . (4.7)We prove the claim by induction. In the case m = 0, the claim was proved in (1). Assume nowthat the claim holds for g ′ = osp ( n + 4( m −
1) + 8 , n ), and k ′ = − m .By Theorem 2.3, k = − m + 1) is a collapsing level and W k ( g , θ ) = V k ′ ( g ′ ).By inductive assumption V k ′ ( g ′ ) is the unique irreducible V k ′ ( g ′ ) in the category KL k ′ . By applying(4.3) and (4.7) we get that ℓ = 0 and therefore V k ( g ) is the unique irreducible V k ( g )–module in thecategory KL k . The assertion now follows by induction on m .(3) is a special case of (2), by taking n = 0.(4) follows from the fact that H θ ( V − ( E )) = V − ( E ) and case (1) by applying formula (4.2). (cid:3) Remark 4.3.
Theorem 4.2 can be also proved by non-cohomological methods, using explicit formulasfor singular vectors and Zhu algebra theory. As an illustration, we shall present in Theorem 8.6 adirect proof in the case of D n at level k = − h ∨ / . In the following sections we shall study some other applications of collapsing levels. We shallrestrict our analysis to the case of Lie algebras. In what follows we let ω , . . . , ω n be the fundamentalweights for g and Λ , . . . , Λ n the fundamental weights for b g .5. On complete reducibility in the category KL k In this Section we prove complete reducibility results in the category KL k when g is a Lie algebra.We start with a preliminary result, which also holds in the super setting. Lemma 5.1.
Assume that the Lie superalgebra g and level k satisfy the conditions of Theorem 4.2.Assume that M is a highest weight V k ( g ) –module from the category KL k . Then M is irreducible.Proof. By using the classification of irreducible modules from Theorem 4.2 we know that the highestweight of M is necessary k Λ , and therefore M is a Z ≥ –graded with respect to L g (0). Denote ahighest weight vector by w k Λ . We have that L g (0) v = 0 ⇐⇒ v = νw k Λ ( ν ∈ C ) . Assume that M is not irreducible. Then it contains a non-zero graded submodule N = M with respectto L g (0): N = M n ∈ Z ≥ N ( n + n ) , L g (0) | N ( r ) = r Id , N ( n ) = 0 . Since N = M , we have that n >
0, otherwise w k Λ ∈ M .We can consider N ( n ) as a finite-dimensional module for g and for the Zhu algebra A ( V k ( g )). Notethat Theorem 4.2 and Proposition 3.2 imply that any irreducible finite-dimensional A ( V k ( g ))–moduleis trivial. Since the Casimir element C g of g acts on N ( n ) as the non-zero constant 2( k + h ∨ ) n ,we conclude that C g acts by the same constant on any irreducible g –subquotient of N ( n ). But anyirreducible subquotient of N ( n ) is an irreducible finite–dimensional A ( V k ( g ))–module, and thereforeit is trivial. This implies that C g acts non-trivially on a trivial g –module, a contradiction. (cid:3) The following Lemma is a consequence of [28, Theorem 0.1].
Lemma 5.2. [28]
Assume that g is a simple Lie algebra and k is a rational number, k > − h ∨ . Then,in the category of V k ( g ) –modules, we have: Ext ( V k ( g ) , V k ( g )) = (0) . Theorem 5.3.
Assume that g is a simple Lie algebra and that the level k satisfies the conditions ofTheorem 4.2. Then any V k ( g ) –module M from the category KL k is completely reducible.Proof. Since M is in KL k we have that any irreducible subquotient of M is isomorphic to V k ( g ). M has finite length. This implies that M is Z ≥ –graded: M = M n ∈ Z ≥ M ( n ) , L g (0) | M ( r ) = r Id . Assume that M (0) = span C { w , . . . , w s } . Then by Lemma 5.1 we have that V k ( g ) w i ∼ = V k ( g ) forevery i = 1 , . . . , s . Now using Lemma 5.2 we get M ∼ = ⊕ V k ( g ) w i and therefore M is completelyreducible. (cid:3) Remark 5.4.
We expect that the previous theorem holds in the case when g is the Lie superalgebrafrom Theorem 4.2. We shall study this case in [7] . We shall now prove much more general result on complete reducibility in KL k . Theorem 5.5.
Assume that level k ∈ Q , k > − h ∨ , and the simple Lie algebra g satisfy the followingproperty: Every highest weight V k ( g ) –module in KL k is irreducible . (5.1) Then the category KL k is semi-simple.Proof. We shall present a sketch of the proof and omit some standard representation theoretic argu-ments which can be found in [20] and [28]. • Since every irreducible V k ( g )-module in KL k is isomorphic to L ( λ ) for certain rational, non-critical weight λ , then [28, Theorem 0.1] implies that Ext ( L ( λ ) , L ( λ )) = (0) in the category KL k . • We prove that in the category KL k we have(5.2) Ext ( L , L ) = (0)for any two irreducible modules L and L from KL k .It remains to consider the case L = L . Take an exact sequence in KL k :0 → L ( λ ) → M → L ( λ ) → , where λ = λ . Then M contains a singular vector w λ of highest weight λ and a subsingularvector w λ of weight λ and w λ generates a submodule isomorphic to L ( λ ). Consider the case λ − λ / ∈ Q + . Then λ is a maximal element of the set Ω( M ) of weights of M , and thereforethe subsingular vector w λ in M of weight λ is a singular vector. By (5.1), it generates anirreducible module isomorphic to L ( λ ) and we conclude that M ∼ = L ( λ ) ⊕ L ( λ ).If λ − λ ∈ Q + we can use the contravariant functor M M σ and get an exact sequence0 → L ( λ ) → M σ → L ( λ ) → . Since M σ is again a V k ( g )–module in KL k (cf. Lemma 3.6) by the first case we have that M σ = L ( λ ) ⊕ L ( λ ). This implies that M = L ( λ ) σ ⊕ L ( λ ) σ = L ( λ ) ⊕ L ( λ ) . • Assume now that M is a finitely generated module from KL k . Then from [20, Proposition3.1] we see that M has an increasing filtration(5.3) (0) = M ⊆ M ⊆ · · · such that(1) for every j ∈ Z > , M j /M j − is an highest weight module in category O ;(2) for any weight λ of M , there exists r such that ( M/M r ) λ = 0. Since M is finitely generated as b g –module, we can assume that its generators are weightvectors of weights say µ , ...µ p . Since they are a finite number there certainly exists t suchthat ( M/M t ) µ i = 0 for all i = 1 , .., p . Hence the filtration (5.3) is finite and stops at M = M t .Since M is in category KL k , we have that the factors of (5.3) are in category KL k . Hence, byour assumption, they are irreducible. Therefore (5.3) is a composition series of finite length.Using assumption (5.1), relation (5.2) and induction on t we get that M ∼ = t M j =1 L ( λ j ) . • Finally, we shall consider the case when M is not finitely generated. Since M is in KL k ,it is countably generated. So M = ∪ ∞ n =1 M ( n ) such that each M ( n ) is finitely generated V k ( g )–module. By previous case M ( n ) is completely reducible, so: M ( n ) = n i M i =1 L ( λ i,n ) . (5.4) Therefore M is a sum of irreducible modules from KL k and by using classical algebraicarguments one can see that M is a direct sum of countably many irreducible modules from KL k appearing in decompositions (5.4).The claim follows. (cid:3) In order to apply Theorem 5.5, the basic step is to check relation (5.1). We have the followingmethod.
Lemma 5.6.
Let k ∈ Q \ Z ≥ . Assume that H θ ( U ) is an irreducible, non-zero W k ( g , θ ) = H θ ( V k ( g )) –module for every non-zero highest weight V k ( g ) –module U from the category KL k . Then every highestweight V k ( g ) –module in KL k is irreducible.Proof. Assume that M is a highest weight V k ( g )–module in KL k . Then H θ ( M ) is an irreducible H θ ( V k ( g ))–module. If M is not irreducible, then it contains a highest weight submodule U such that { } $ U $ M . Modules U and M/U are again highest weight modules in KL k . By the assumptionof the Lemma we have that H θ ( U ) is a non-trivial submodule of H θ ( M ). Irreducibility of H θ ( M )implies that H θ ( U ) = H θ ( M ), and therefore H θ ( M/U ) = { } , a contradiction. (cid:3) Theorem 5.7.
Assume that g is a simple Lie algebra and k ∈ C \ Z ≥ such that W k ( g , θ ) is rational.Then KL k is a semi-simple category.Proof. Assume that e L ( λ ) is a highest weight V k ( g )–module in KL k . Clearly λ ( α ∨ ) / ∈ Z ≥ and byTheorem 2.1 H θ ( e L ( λ )) = (0). Since H θ ( e L ( λ )) is non-zero highest weight module for the rational vertexalgebra W k ( g , θ ), we conclude that H θ ( e L ( λ )) is irreducible. Now assertion follows from Theorem 5.5and Lemma 5.6. (cid:3) Remark 5.8.
The previous theorem proves that the category KL k is semisimple in the following(non-admissible) cases: • g = D , E , E , E and k = − h ∨ using results from [38] . Moreover, using Theorem 5.5 and Lemma 5.6 we can prove the semi-simplicity of KL k for allcollapsing levels not accounted by Theorem 1.1. We list here only non-admissible levels, since inadmissible case KL k is semi-simple by [12]. Theorem 5.9.
The category KL k is semisimple in the following cases: (1) g = D ℓ , ℓ ≥ and k = − ; (2) g = B ℓ , ℓ ≥ and k = − ; (3) g = A ℓ , ℓ ≥ and k = − ; (4) g = A ℓ − , ℓ ≥ , k = − ℓ ; (5) g = D ℓ − , ℓ ≥ and k = − ℓ + 3 ; (6) g = C ℓ , k = − − ℓ/ ; (7) g = E , k = − ; (8) g = E , k = − ; (9) g = F , k = − .Proof. We will give a proof of relations (1) and (2) in Corollaries 6.8 and 7.7, respectively. Case (1)for ℓ = 3 will follow from Theorem 5.7. Note also that case (1) for ℓ = 3 is a special case of case (4),and that case (2) for ℓ = 2 is a special case of (6). The proof in cases (3) – (6) is similar, and it usesthe classification of irreducible modules from [10], [11], [16] and the results on collapsing levels [4].Cases (7) – (9) are reduced to cases we have already treated. Here are some details.Case (3): • [16], [4] H θ ( V − ( A ℓ )) is isomorphic to the Heisenberg vertex algebra M (1) of central charge c = 1 • By using the fact that every highest weight M (1)–module is irreducible, we see that if U isa highest weight V − ( A ℓ )–module in KL − , then H θ ( U ) is a non-trivial irreducible M (1)–module.Case (4): • [16], [4] H θ ( V − ℓ ( A ℓ − )) = V − ℓ +1 ( A ℓ − ). • For ℓ = 2, we have that every highest weight V − ℓ +1 ( A ℓ − ) = V − ( sl (2))–module e L ( λ ) in KL − with highest weight λ = − (1 + j )Λ + j Λ , j ∈ Z ≥ , is irreducible. • By induction, we see that for every highest weight V − ℓ ( A ℓ − )–module U in KL − ℓ , H θ ( U ) isa non-trivial irreducible V − ℓ +1 ( A ℓ − )–module.Case (5) • H θ ( V − ℓ +3 ( D ℓ − )) ∼ = V − ℓ +5 ( D ℓ − ). • By induction we see that for or every highest weight V − ℓ +3 ( D ℓ − )–module U in KL − ℓ +3 , H θ ( U ) is a non-trivial irreducible V − ℓ +5 ( D ℓ − )–module.Case (6) • H θ ( V − − ℓ/ ( C ℓ )) ∼ = V − / − ℓ/ ( C ℓ − ). • For ℓ = 2, we have that every highest weight V − / − ℓ/ ( C ℓ − ) = V − / ( sl (2))–module in KL − / is irreducible. • By induction, we see that for every highest weight V − − ℓ/ ( C ℓ )–module U in KL − − ℓ/ , H θ ( U )is a non-trivial irreducible V − / − ℓ/ ( C ℓ − )–module.The proof follows by applying Theorem 5.5 and Lemma 5.6.Cases (7) – (8)We have H θ ( V − ( E )) = V − ( A ) , H θ ( V − ( E )) = V − ( D ) , and these cases are settled in (3) and Theorem 1.1 (3) respectively. Case (9) follows from the factthat H θ ( V − ( F )) is isomorphic to the admissible affine vertex algebra V − ( C ) which is semisimplein KL − / (cf. [1]). (cid:3) Remark 5.10.
The problem of complete-reducibility of modules in KL k when g is a Lie superalgebrawill be also studied in [7] . An important tool in the description of the category KL k will be theconformal embedding of e V k ( g ) to V k ( g ) where g is the even part of g . Note that in the category O we can have indecomposable V k ( g )–modules in some cases listed inTheorem 5.9. See [10, Remark 5.8] for one example. The vertex algebra V − ( D ℓ ) and its quotients In this section we exploit Hamiltonian reduction and the results on conformal embeddings from[4] to investigate the quotients of the vertex algebra V − ( D ℓ ). In particular we are interested ina non-simple quotient V − ( D ℓ ) which appears in the analysis of certain dual pairs (see [6]) as wellas in the simple quotient V − ( D ℓ ). We will show that the vertex algebra V − ( D ℓ ) has infinitelymany irreducible modules in the category KL − , while by [15], V − ( D ℓ ) has finitely many irreduciblemodules in KL − . Recall that − D ℓ [4].Consider the vector(6.1) w := ( e ǫ + ǫ ( − e ǫ + ǫ ( − − e ǫ + ǫ ( − e ǫ + ǫ ( −
1) + e ǫ + ǫ ( − e ǫ + ǫ ( − . It is a singular vector in V − ( D ℓ ) (cf. [15]). Note that this vector is contained in the subalgebra V − ( D ) of V − ( D ℓ ).By using the explicit expression for singular vectors v n in V n − ℓ +1 ( D ℓ ) (see (8.1)), we have that(6.2) w := v ℓ − = (cid:16) ℓ X i =2 e ǫ − ǫ i ( − e ǫ + ǫ i ( − (cid:17) ℓ − is a singular vector in V − ( D ℓ ).For ℓ = 4 we also have a third singular vector (cf. [40]) w := ( e ǫ + ǫ ( − e ǫ − ǫ ( − − e ǫ + ǫ ( − e ǫ − ǫ ( −
1) + e ǫ − ǫ ( − e ǫ + ǫ ( − . The vertex algebra V − ( D ℓ ) for ℓ ≥ . Define the vertex algebra(6.3) V − ( D ℓ ) = V − ( D ℓ ) (cid:14) J ℓ , where J ℓ = h w , w i ( ℓ = 4) , J ℓ = h w i ( ℓ ≥ . The following proposition is essentially proven in [6].
Proposition 6.1. (1)
There is a non-trivial vertex algebra homomorphism
Φ : V − ( D ℓ ) → M ℓ where M ℓ the Weylvertex algebra of rank ℓ . (2) V − ( D ℓ ) is not simple, and L (( − − t )Λ + t Λ ) , t ∈ Z ≥ are V − ( D ℓ ) –modules.Proof. The homomorphism Φ : V − ( D ℓ ) → M ℓ was constructed in [6, Section 7]. By direct calcula-tion one proves that Φ( w ) = 0 for ℓ ≥ w ) = 0 for ℓ = 4. Finally [6, Lemma 7.1] implies that L (( − − t )Λ + t Λ ), t ∈ Z ≥ are V − ( D ℓ )–modules. Since the simple vertex algebra V − ( D ℓ ) has onlyfinitely many irreducible modules in the category O [15], we have that V − ( D ℓ ) is not simple. (cid:3) Next, we exploit the fact that in the case g = D ℓ , k = − W -algebra W k ( g , θ ), all generators G { u } at conformal weight 3 / u ∈ g − / , belong to the maximalideal (see [4] for details). This implies that there exists a non-trivial ideal I in V − ( g ) such that G { u } ∈ H θ ( I ) for all u ∈ g − / .Note also that g ♮ = A ⊕ D ℓ − , so we have that V ℓ − ( A ) ⊗ V ( D ℓ − ) is a subalgebra of W − ( D ℓ , θ ).In the case ℓ = 4 we identify D with A ⊕ A . Lemma 6.2.
We have • x ( − ∈ H θ ( J ℓ ) for all x ∈ D ℓ − ⊂ g ♮ , • G { u } ∈ H θ ( J ℓ ) for all u ∈ g − / .Proof. Assume that ℓ ≥
5. Since w is a singular vector in V − ( D ℓ ), the ideal J ℓ is a highest weightmodule of highest weight λ = − + ǫ + ǫ + ǫ + ǫ . Now, the Main Theorem from [12] implies that H θ ( J ℓ ) is a non-trivial highest weight module. By formula [35, (6.14)] the highest weight is (0 , ω )and, by (4.1), the conformal weight of its highest weight vector is 1. Up to a non-zero constant, there is only one vector in W − ( D ℓ , θ ) = V ℓ − ( A ) ⊗ V ( D ℓ − ) that has these properties, namely J { e ǫ ǫ } ( − , and therefore H θ ( J ℓ ) contains all generators of V ( D ℓ − ).In the case ℓ = 4, w and w generate submodules N and N of highest weights λ = − + ǫ + ǫ + ǫ + ǫ , λ = − + ǫ + ǫ + ǫ − ǫ , respectively. Applying the same arguments as above we get that J { e ǫ ± ǫ } ( − ∈ H θ ( I ) , which implies that H θ ( J ℓ ) contains all generators of V ( D ) = V ( A ) ⊗ V ( A ).Now, claim follows by applying the action of generators of V ( D ℓ − ) to G { u } (see [4]). (cid:3) Proposition 6.3.
We have (1) H θ ( V − ( D ℓ )) = V ℓ − ( A ) . (2) H θ ( L (( − − t )Λ + t Λ )) ∼ = L A (( ℓ − − t )Λ + t Λ ) , t ∈ Z ≥ . (3) The set { L (( − − t )Λ + t Λ ) | t ∈ Z ≥ } provides a complete list of irreducible V − ( D ℓ ) –modulesfrom the category KL − .Proof. By Lemma 6.2 we see that the vertex algebra H θ ( V − ( D ℓ )) is generated only by x ( − , x ∈ A ⊂ D ♮ℓ . So there are only two possibilities: either H θ ( V − ( D ℓ )) = V ℓ − ( A ) or H θ ( V − ( D ℓ )) = V ℓ − ( A ). Moreover, for every t ∈ Z ≥ , H θ ( L (( − − t )Λ + t Λ )) must be the irreducible H θ ( V − ( D ℓ ))–module with highest weight tω with respect to A . So H θ ( L (( − − t )Λ + t Λ )) ∼ = L A (( ℓ − − t )Λ + t Λ ) , t ∈ Z ≥ . Therefore, H θ ( V − ( D ℓ )) contains infinitely many irreducible modules, whichgives that H θ ( V − ( D ℓ )) = V ℓ − ( A ). In this way we have proved claims (1) and (2).Let us now prove claim (3).Assume that L ( k Λ + µ ) ( µ ∈ P + , k = −
2) is an irreducible V k ( D ℓ )–module in the category KL k .Then H θ ( L ( k Λ + µ )) is a non-trivial irreducible V ℓ − ( A )–module. The representation theory of V ℓ − ( A ) implies that: H θ ( L ( k Λ + µ )) = L A (( ℓ − − j )Λ + j Λ ) for j ∈ Z ≥ . Since D ♮ℓ = A × D ℓ − , we conclude that µ ♮ = jω and therefore, by (2.5), µ = jω + sω = ( s + j ) ǫ + sǫ ( s ∈ Z ≥ ) . By using the action of L (0) = ω on the lowest component of H θ ( L ( k Λ + µ )) we get( µ + 2 ρ, µ )2( k + h ∨ ) − µ ( x ) = j ( j + 2)4( ℓ −
2) ( x = θ ∨ / . Since 2( k + h ∨ ) = 2( − ℓ −
2) = 4( ℓ −
2) and µ ( x ) = (2 s + j ) / µ + 2 ρ, µ ) − ( h ∨ − s + j ) = j ( j + 2) . By direct calculation we get( µ + 2 ρ, µ ) = ( s + j ) + s + h ∨ ( s + j ) + ( h ∨ − s, which gives an equation:( s + j ) + s + h ∨ ( s + j ) + ( h ∨ − s − ( h ∨ − s + j ) = j ( j + 2) . ⇐⇒ ( s + j ) + s + h ∨ ( s + j ) − ( h ∨ − s + j ) = j ( j + 2) . ⇐⇒ ( s + j )( s + j + 2) = j ( j + 2) ⇐⇒ s = 0 or s = − j − . Since µ ∈ P + we conclude that s = 0. Therefore µ = jω for certain j ∈ Z ≥ . The proof of claim (3)is now complete. (cid:3) The simple vertex algebra V − ( D ℓ ) . Next we use the fact that the simple affine W -algebra W − ( D ℓ , θ ) is isomorphic to the simple affine vertex algebra V ℓ − ( A ), for ℓ ≥ Proposition 6.4.
The set { L (( − − j )Λ + j Λ ) | j ∈ Z ≥ , j ≤ ℓ − } provides a complete list ofirreducible V − ( D ℓ ) –modules from the category KL − . Proof.
Assume that N is an irreducible V − ( D ℓ )–module from the category KL − . Then N is alsoirreducible as V − ( D ℓ )–module, and therefore N ∼ = L (( − − j )Λ + j Λ ) for certain j ∈ Z ≥ . Since H θ ( N ) must be an irreducible H θ ( V − ( D ℓ )) = W − ( D ℓ , θ ) = V ℓ − ( A )–module, we get j ≤ ℓ −
4, asdesired. (cid:3)
Now we want to describe the maximal ideal in V − ( D ℓ ). The next lemma states that any non-trivialideal in V − ( D ℓ ) is automatically maximal. Lemma 6.5.
Let { } 6 = I $ V − ( D ℓ ) be any non-trivial ideal in V − ( D ℓ ) . Then we have (1) H θ ( I ) is the maximal ideal in V ℓ − ( A ) . (2) I is a maximal ideal in V − ( D ℓ ) and I = L ( − ℓ − + 2( ℓ − ) .Proof. Assume that I is a non-trivial ideal in V − ( D ℓ ). Then I can be regarded as a V − ( D ℓ )–modulein the category KL − and therefore, by Proposition 6.3, (3), it contains a non-trivial subquotientisomorphic to L (( − − j )Λ + j Λ ) for some j ∈ Z ≥ . Since, by part (2) of the aforementionedProposition, H θ ( L (( − − j )Λ + j Λ )) = 0 for every j ∈ Z ≥ , we conclude that H θ ( I ) is a non-trivialideal in H θ ( V − ( D ℓ )) = V ℓ − ( A ). But since V ℓ − ( A ), ℓ ≥
4, contains a unique non-trivial ideal,which is automatically maximal, we have that H θ ( I ) is a maximal ideal in V ℓ − ( A ). So H θ ( V − ( D ℓ ) /I ) ∼ = V ℓ − ( A ) . Assume now that V − ( D ℓ ) /I is not simple. Then it contains a non-trivial singular vector v ′ of weight − (2 + j )Λ + j Λ for j ∈ Z > . By [12], we have that H θ ( V − ( D ℓ ) .v ′ ) is a non-trivial ideal in V ℓ − ( A )generated by a singular vector of A –weight jω . This is a contradiction. So I is the maximal ideal.Since the maximal ideal in V ℓ − ( A ) is generated by a singular vector of A –weight 2( ℓ − ω andsince the maximal ideal is simple, we conclude that I = V − ( D ℓ ) .v sing for a certain singular vector v sing of weight λ = − ℓ − + 2( ℓ − . It is also clear that this singular vector is unique, upto scalar factor. Therefore, I = L ( − ℓ − + 2( ℓ − ). (cid:3) Note that in the previous lemma we proved the existence of a singular vector which generates themaximal ideal without presenting a formula for such a singular vector. Since the vector in (6.2) hasthe correct weight, we also have an explicit expression for this singular vector: (cid:16) ℓ X i =2 e ǫ − ǫ i ( − e ǫ + ǫ i ( − (cid:17) ℓ − (1) The maximal ideal in V − ( D ℓ ) is generated by the vectors w and w for ℓ ≥ and by the vectors w , w , w for ℓ = 4 . (2) The homomorphism
Φ : V − ( D ℓ ) → M ℓ is injective. In particular, the vertex algebra V − ( D ℓ ) ⊗ V − ℓ ( A ) is conformally embedded into V − / ( C ℓ ) . (3) ch ( V − ( D ℓ )) = ch ( V − ( D ℓ )) + ch L ( − ℓ − + 2( ℓ − ) . Remark 6.7.
D. Gaiotto in [27] has started a study of the decomposition of M ℓ as a V − ( D ℓ ) ⊗ V − ℓ ( A ) –module in the case ℓ = 4 . By combining results from [6, Section 8] and results from thisSection we get that Com ( V − ℓ ( A ) , M ℓ ) ∼ = V − ( D ℓ ) . So the vertex algebra responsible for the decomposition of M ℓ is exactly V − ( D ℓ ) . Therefore in thedecomposition of M ℓ only modules for V − ( D ℓ ) can appear. In our forthcoming papers we plan toapply the representation theory of V − ( D ℓ ) to the problem of finding branching rules. Corollary 6.8.
For ℓ ≥ the category KL − is semi-simple.Proof. The assertion in the case ℓ ≥ W − ( D ℓ , θ ) = V ℓ − ( sl (2))is a rational vertex algebra.In the case ℓ = 3, we have that a highest weight V − ( D )–module M is isomorphic to e L (( − − j )Λ + j Λ ) where j ∈ Z ≥ . The irreducibility of M follows easily from the fact that H θ ( M ) is isomorphic to an irreducible V − ( sl (2))–module L A ( − − j )Λ + j Λ ). Now claim follows from Theorem 5.5 andLemma 5.6. (cid:3) The vertex algebra V − ( B ℓ ) and its quotients In this section let ℓ ≥
2. Note that k = − B ℓ [4], and that the simple affine W -algebra W − ( B ℓ , θ ) is isomorphic to V ℓ − ( A ). This implies that H θ ( V − ( B ℓ )) = V ℓ − ( A ) . Butas in the case of the affine Lie algebra of type D , we can construct an intermediate vertex algebra V so that H θ ( V ) = V ℓ − / ( A ). Remark 7.1.
The formula for a singular vector of conformal weight two in V − ( B ℓ ) was given in [15, Theorem 4.2] for ℓ ≥ , and in [15, Remark 4.3] for ℓ = 2 . Note that, for ℓ ≥ , the vector σ ( w ) from [15] is equal to the vector w from relation (6.1), i.e. it is contained in the subalgebra V − ( D ) .For ℓ = 3 , we have w = ( e ǫ + ǫ ( − e ǫ ( − − e ǫ + ǫ ( − e ǫ ( −
1) + e ǫ ( − e ǫ + ǫ ( − . For ℓ = 2 , the singular vector of conformal weight two in V − ( B ) is equal to w = ( e ǫ + ǫ ( − e − ǫ ( −
1) + 12 h ǫ ( − e ǫ ( − − e ǫ − ǫ ( − e ǫ ( − . Consider the singular vector in V − ( B ℓ ) denoted by σ ( w ) in [15, Theorem 4.2] and [17, Section7]. Let us denote that singular vector by w in this paper (see Remark 7.1 for explanation).Then we have the quotient vertex algebra(7.1) V − ( B ℓ ) = V − ( B ℓ ) (cid:14) h w i . As in the case of the vertex algebra V − ( D ℓ ), we have the non-trivial homomorphism V − ( B ℓ ) → M ℓ +1 .The proof of the following result is completely analogous to the proof of Proposition 6.3 and it istherefore omitted. Proposition 7.2.
We have (1)
There is a non-trivial homomorphism
Φ : V − ( B ℓ ) → M ℓ +1 . (2) H θ ( V − ( B ℓ )) = V ℓ − / ( A ) . (3) H θ ( L (( − − t )Λ + t Λ )) ∼ = L A (( ℓ − / − t )Λ + t Λ ) , t ∈ Z ≥ . (4) The set { L (( − − t )Λ + t Λ ) | t ∈ Z ≥ } (7.2) provides a complete list of irreducible V − ( B ℓ ) –modules from the category KL − . We have the following result on classification of irreducible modules.
Proposition 7.3.
Assume that ℓ ≥ . Then the set { L (( − − j )Λ + j Λ ) | j ∈ Z ≥ , j ≤ ℓ −
3) + 1 } provides a complete list of irreducible V − ( B ℓ ) –modules from the category KL − .Proof. The proof is analogous to the proof of Proposition 6.4: it uses the exactness of the functor H θ and the representation theory of affine vertex algebras. In particular, we use the result from [8] whichgives that the set { L ( − ( ℓ − / − j )Λ + j Λ ) | j ∈ Z ≥ , j ≤ ℓ −
3) + 1 } provides a complete list of irreducible V ℓ − / ( A )–modules from the category KL ℓ − / . (cid:3) An important consequence is the simplicity of the vertex algebra V − ( B ). Corollary 7.4.
The vertex algebra V − ( B ℓ ) is simple if and only if ℓ = 2 . In particular, the set (7.2)provides a complete list of irreducible modules for V − ( B ) in KL − . Proof.
Since by Proposition 7.2, V − ( B ℓ ) has infinitely many irreducible modules in the category KL − , and, by Proposition 7.3, V − ( B ℓ ) has finitely many irreducible modules in the category KL − (if ℓ ≥ V − ( B ℓ ) cannot be simple for ℓ ≥ ℓ = 2. Assume that V − ( B ) is not simple. Then it must contain an ideal I generated by a singular vector of weight λ = ( − − j )Λ + j Λ for certain j >
0. By applying thefunctor H θ , we get a non-trivial ideal in V − / ( A ), against the simplicity of V − / ( A ). (cid:3) Next we notice that V ℓ − / ( A ) has a unique non-trivial ideal J which is generated by a singularvector of A –weight 2( ℓ − ω . The ideal J is maximal and simple (cf. [5]). By combining this withproperties of the functor H θ from [12], one proves the existence of a unique maximal ideal I (which isalso simple) in V − ( B ℓ ) such that I ∼ = L ( − ℓ − + 2( ℓ − )). Remark 7.5.
The explicit expression for a singular vector which generates I is more complicated thatin the case D , and it won’t be presented here. In [6] we constructed a homomorphism V − ( B ℓ ) ⊗ V − ℓ − / ( A ) → M ℓ +1 . The results of this sectionenable us to find the image of this homomorphism. Corollary 7.6.
We have: (1)
The vertex algebra V − ( B ℓ ) ⊗ V − ℓ − / ( A ) is conformally embedded into V − / ( C ℓ +1 ) . (2) The vertex algebra V − ( B ℓ ) for ℓ ≥ contains a unique ideal I ∼ = L ( − ℓ − + 2( ℓ − )) andch ( V − ( B ℓ )) = ch ( V − ( B ℓ )) + ch ( L ( − ℓ − + 2( ℓ − )) . Finally, we apply Theorem 5.5 and prove that KL − is a semi-simple category. Corollary 7.7. If ℓ ≥ , then every V − ( B ℓ ) –module in KL − is completely reducible.Proof. It suffices to prove that every highest weight V − ( B ℓ )–module in KL − is irreducible. Assumethat ℓ ≥
3. If M ∼ = e L ( λ ) is a highest weight module in KL − then the highest weight is λ = − (2 + j )Λ + j Λ where 0 ≤ j ≤ ℓ − j + 1. Since H θ ( L ( λ )) is a non-zero highest weight V − ℓ +7 / ( sl (2))–module, then the complete reducibility result from [8] implies that H θ ( L ( λ )) is irreducible. Theassertion now follows from Lemma 5.6. The proof in the case ℓ = 2 is similar, and it uses theclassification of irreducible V − ( B )–modules from Corollary 7.4 and the fact that every highest weight V − / ( sl (2)) = H θ ( V − ( B ))–module in KL − / is irreducible. (cid:3) On the representation theory of V − ℓ ( D ℓ )8.1. The vertex algebra V − ℓ ( D ℓ ) . Let g be a simple Lie algebra of type D ℓ . Recall that 2 − ℓ = − h ∨ / v n = (cid:16) ℓ X i =2 e ǫ − ǫ i ( − e ǫ + ǫ i ( − (cid:17) n (8.1)in V n − ℓ +1 ( D ℓ ), for any n ∈ Z > . As in [40], we consider the vertex algebra(8.2) V − ℓ ( D ℓ ) = V − ℓ ( D ℓ ) (cid:14) h v i , where h v i denotes the ideal in V − ℓ ( D ℓ ) generated by the singular vector v . We recall the followingresult on the classification of irreducible V − ℓ ( D ℓ )–modules in the category KL − ℓ . Proposition 8.1. [40](1)
The set { V ( tω ℓ ) , V ( tω ℓ − ) | t ∈ Z ≥ } provides a complete list of irreducible finite-dimensional modules for the Zhu algebra A ( V − ℓ ( D ℓ )) . (2) The set { L ((2 − t − ℓ )Λ + t Λ ℓ ) , L ((2 − t − ℓ )Λ + t Λ ℓ − ) | t ∈ Z ≥ } provides a complete list of irreducible V − ℓ ( D ℓ ) –modules from the category KL − ℓ . In the odd rank case D ℓ − , the modules from Proposition 8.1 (2) provide a complete list ofirreducible V − ℓ ( D ℓ − )–modules from the category KL − ℓ (cf. [11]). The paper [11] also containsa fusion rules result in the category KL − ℓ . Detailed fusion rules analysis will be presented elsewhere.On the other hand, Theorem 4.2 implies that in the even rank case D ℓ , V − ℓ ( D ℓ ) is the uniqueirreducible V − ℓ ( D ℓ )–module from the category KL − ℓ . In the next section we will give an expla-nation of this difference using singular vectors existing in the even rank case D ℓ .8.2. Singular vectors in V n − ℓ +1 ( D ℓ ) . In this section, we construct more singular vectors in V n − ℓ +1 ( D ℓ ). In the case n = 1, we show that the maximal submodule of V − ℓ ( D ℓ ) is gener-ated by three singular vectors. We present explicit formulas for these singular vectors.Let g be a simple Lie algebra of type D ℓ . Denote by S ℓ the group of permutations of 2 ℓ elements.Let Π ℓ = n p ∈ S ℓ | p = 1 , p ( i ) = i, ∀ i ∈ { , . . . , ℓ } o be the set of fixed-points free involutions, which is well known to have (2 ℓ − · · . . . · (2 ℓ − i = j , denote by ( i j ) ∈ S ℓ the transposition of i and j . Then, any p ∈ Π ℓ admits aunique decomposition of the form: p = ( i j ) · · · ( i ℓ j ℓ ) , such that i h < j h for 1 ≤ h ≤ ℓ , and i < . . . < i ℓ . Define a permutation ¯ p ∈ S ℓ by:¯ p (2 h −
1) = i h , ¯ p (2 h ) = j h , ≤ h ≤ ℓ. Thus, we have a well defined map p ¯ p from Π ℓ to S ℓ . Define the function s : Π ℓ → {± } as follows: s ( p ) = sign(¯ p ) , where sign( q ) denotes the sign of the permutation q ∈ S ℓ .We have: Theorem 8.2.
The vector (8.3) w n = (cid:16) X p ∈ Π ℓ s ( p ) Y i ∈{ ,..., ℓ } i
.Proof. Direct verification of relations e ǫ k − ǫ k +1 (0) w n = 0, for k = 1 , . . . , ℓ − e ǫ ℓ − + ǫ ℓ (0) w n = 0and e − ( ǫ + ǫ ) (1) w n = 0. (cid:3) Remark 8.3.
The vector w n has conformal weight nℓ and its g –highest weight equals nω ℓ = n ( ǫ + . . . + ǫ ℓ ) . In particular, for n = 1 , the vector w has conformal weight ℓ and highest weight ω ℓ = ǫ + . . . + ǫ ℓ . Example 8.4.
Set n = 1 for simplicity. For ℓ = 2 we recover the singular vector w = ( e ǫ + ǫ ( − e ǫ + ǫ ( − − e ǫ + ǫ ( − e ǫ + ǫ ( −
1) + e ǫ + ǫ ( − e ǫ + ǫ ( − in V − ( D ) of conformal weight from [40] . For ℓ = 3 , the formula for the singular vector in V − ( D ) of conformal weight is more complicated. It is a sum of monomials: w = ( e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − e ǫ + ǫ ( − . Denote by ϑ the automorphism of V n − ℓ +1 ( D ℓ ) induced by the automorphism of the Dynkindiagram of D ℓ of order two such that ϑ ( ǫ k − ǫ k +1 ) = ǫ k − ǫ k +1 , k = 1 , . . . , ℓ − , (8.4) ϑ ( ǫ ℓ − − ǫ ℓ ) = ǫ ℓ − + ǫ ℓ , ϑ ( ǫ ℓ − + ǫ ℓ ) = ǫ ℓ − − ǫ ℓ . (8.5)Theorem 8.2 now implies that ϑ ( w n ) is a singular vector in V n − ℓ +1 ( D ℓ ), for any n ∈ Z > , also.The vector ϑ ( w n ) has conformal weight nℓ and its highest weight for g is 2 nω ℓ − = n ( ǫ + . . . + ǫ ℓ − − ǫ ℓ ).We consider the associated quotient vertex algebra(8.6) e V n − ℓ +1 ( D ℓ ) := V n − ℓ +1 ( D ℓ ) (cid:14) h v n , w n , ϑ ( w n ) i , where v n is given by relation (8.1) (for D ℓ ): v n = (cid:16) ℓ X i =2 e ǫ − ǫ i ( − e ǫ + ǫ i ( − (cid:17) n . In particular, for n = 1 we have the vertex algebra e V − ℓ ( D ℓ ) = V − ℓ ( D ℓ ) (cid:14) h v , w , ϑ ( w ) i . Clearly, e V − ℓ ( D ℓ ) is a quotient of vertex algebra V − ℓ ( D ℓ ) from Subsection 8.1. The associatedZhu algebra is A ( e V − ℓ ( D ℓ )) = U ( g ) (cid:14) h ¯ v, ¯ w, ϑ ( ¯ w ) i , where ¯ v = ℓ X i =2 e ǫ − ǫ i e ǫ + ǫ i , ¯ w = X p ∈ Π ℓ s ( p ) Y i ∈{ ,..., ℓ } i
We have: (1) ¯ wV ( tω ℓ ) = 0 , for t ∈ Z > . (2) ϑ ( ¯ w ) V ( tω ℓ − ) = 0 , for t ∈ Z > .Proof. (1) Let t = 1. Denote by v ω ℓ the highest weight vector of V ( ω ℓ ), and by v − ω ℓ the lowestweight vector of V ( ω ℓ ). One can easily check, using the spinor realization of V ( ω ℓ ), that there existsa constant C = 0 such that ¯ w ( v − ω ℓ ) = Cv ω ℓ . For general t ∈ Z > , the claim follows using the embedding of V ( tω ℓ ) into V ( ω ℓ ) ⊗ t . Claim (2)follows similarly. (cid:3) Theorem 8.6.
We have: (i)
The trivial module C is the unique finite-dimensional irreducible module for A ( e V − ℓ ( D ℓ )) . (ii) V − ℓ ( D ℓ ) is the unique irreducible g –locally finite module for e V − ℓ ( D ℓ ) . (iii) The vertex operator algebra e V − ℓ ( D ℓ ) is simple, i.e. V − ℓ ( D ℓ ) = V − ℓ ( D ℓ ) (cid:14) h v , w , ϑ ( w ) i . Proof. (i) Proposition 8.1 implies that the set { V ( tω ℓ ) , V ( tω ℓ − ) | t ∈ Z ≥ } provides a complete list of finite-dimensional irreducible modules for the algebra U ( g ) (cid:14) h ¯ v i = A ( V − ℓ ( D ℓ )) . Lemma 8.5 shows that V ( tω ℓ ) and V ( tω ℓ − ) are not modules for A ( e V − ℓ ( D ℓ )), for t ∈ Z > . Claim(i) follows. Claims (ii) and (iii) follow from (i) by applying Proposition 3.2 and Corollary 3.3. (cid:3) Remark 8.7.
A general character formula for certain simple affine vertex algebras at negative integerlevels has been recently presented by V. G. Kac and M. Wakimoto in [37] , (more precisely, g = A n , C n for k = − and g = D , E , E , E for k = − , − , − , ). Note that conditions (i)-(iii) of [37,Theorem 3.1] hold for vertex algebras V − b ( D n ) , n > , b = 1 , . . . , n − , too. We conjecture thatcondition (iv) of this theorem holds as well; therefore formula (3.1) in [37] gives the character formula. Conformal embedding of e V ( − , D × A ) into V − ( E )In this section, we apply the results on representation theory of V − ( D ) from previous sections tothe conformal embedding of e V ( − , D × A ) into V − ( E ). This gives us an interesting example of amaximal semisimple equal rank subalgebra such that the associated conformally embedded subalgebrais not simple.We use the construction of the root system of type E from [19], [29], and the notation for rootvectors similar to the notation for root vectors for E from [9].For a subset S = { i , . . . , i k } ⊆ { , , , , , } , i < . . . < i k , with odd number of elements (so that k = 1 , e ( i ...i k ) a suitably chosen root vector associated to the positive root12 ǫ − ǫ + X i =1 ( − p ( i ) ǫ i ! , such that p ( i ) = 0 for i ∈ S and p ( i ) = 1 for i / ∈ S . We will use the symbol f ( i ...i k ) for the root vectorassociated to corresponding negative root.Note now that the subalgebra of E generated by positive root vectors(9.1) e ǫ + ǫ , e α = e (1) , e α = e ǫ − ǫ , e α = e ǫ − ǫ , e α = e ǫ + ǫ , e α = e ǫ − ǫ and the associated negative root vectors is a simple Lie algebra of type D . There are 30 root vectorsassociated to positive roots for D : e ǫ + ǫ , e ǫ − ǫ ,e ( i ) , i ∈ { , , , } ,e ( ijk ) , i, j, k ∈ { , , , } , i < j < k,e ( i , i ∈ { , , , } ,e ( ijk , i, j, k ∈ { , , , } , i < j < k,e ± ǫ i + ǫ j , i, j ∈ { , , , } , i < j. (9.2)Furthermore, the subalgebra of E generated by e ǫ − ǫ and the associated negative root vector is asimple Lie algebra of type A . Thus, D ⊕ A is a semisimple subalgebra of E .It follows from [3], [9] that the affine vertex algebra e V ( − , D × A ) is conformally embeddedin V − ( E ). Remark that e V ( − , A ) = V − ( A ) (since V − ( A ) = V − ( A )). This implies that e V ( − , D × A ) ∼ = e V ( − , D ) ⊗ V − ( A ).It was shown in [15] that v E = ( e ǫ − ǫ ( − e ǫ + ǫ ( −
1) + e (156) ( − e (23456) ( −
1) ++ e (256) ( − e (13456) ( −
1) + e (356) ( − e (12456) ( −
1) ++ e (456) ( − e (12356) ( − (9.3)is a singular vector in V − ( E ). Moreover, V − ( E ) ∼ = V − ( E ) (cid:14) h v E i . Vectors ( e (12346) ( − s , for s ∈ Z > are (non-trivial) singular vectors for the affinization of D ⊕ A in V − ( E ) of highest weights − ( s + 4)Λ + s Λ for D (1)6 and − ( s + 4)Λ + s Λ for A (1)1 . Thus thereexist highest weight modules e L D ( − ( s + 4)Λ + s Λ ) and e L A ( − ( s + 4)Λ + s Λ ), for D (1)6 and A (1)1 , respectively such that ( e V ( − , D ) ⊗ V − ( A )) . ( e (12346) ( − s is isomorphic to e L D ( − ( s + 4)Λ + s Λ ) ⊗ e L A ( − ( s + 4)Λ + s Λ ) . This implies that L D ( − ( s + 4)Λ + s Λ ) ⊗ L A ( − ( s + 4)Λ + s Λ )are irreducible e V ( − , D × A )–modules, for s ∈ Z > .In particular, L D ( − ( s + 4)Λ + s Λ ) are irreducible ( D –locally finite) e V ( − , D )–modules, for s ∈ Z > . In the next proposition, we use the notation from (8.2), (8.3), (8.4), (8.5). Proposition 9.1.
We have: (1)
Assume that e L D ( − + 2Λ ) and e L D ( − + 2Λ ) are highest weight V − ( D ) –modules fromthe category KL − , not necessarily irreducible. Then e L D ( − + 2Λ ) ⊠ e L D ( − + 2Λ ) = 0 , where ⊠ is the tensor functor for KL − –modules. In other words, we cannot have a non-zero V − ( D ) –module M from KL − and a non-zero intertwining operator of type (cid:18) M e L D ( − + 2Λ ) e L D ( − + 2Λ ) (cid:19) . (9.4)(2) Relations w = 0 and ϑ ( w ) = 0 hold in V − ( E ) . In particular, e V ( − , D ) is not simple.Proof. For the proof of assertion (1) we first notice that the following decomposition of D –modulesholds: V D (2 ω ) ⊗ V D (2 ω ) = V D (2 ω + 2 ω ) ⊕ V D ( ω + ω + ω ) ⊕ V D (2 ω ) ⊕ V D ( ω + ω + ω ) ⊕ V D ( ω + ω ) ⊕ V D (2 ω ) . (9.5)Assume that M is a non-zero V − ( D )–module in the category KL − such that there is a non-trivialintertwining operator of type (9.4). Then the Frenkel-Zhu formula for fusion rules implies that M must contain a non-trivial subquotient whose lowest graded component appears in the decompositionof V D (2 ω ) ⊗ V D (2 ω ). But by Proposition 8.1, the D –modules appearing in (9.5) cannot be lowestcomponents of any V − ( D )–module. This proves assertion (1).Assertion (1) implies that if w = 0 and ϑ ( w ) = 0 in V − ( E ), then Y ( w , z ) ϑ ( w ) = 0 , a contradiction since V − ( E ) is a simple vertex algebra. The same fusion rules argument shows thatif ϑ ( w ) = 0 in V − ( E ), then Y ( ϑ ( w ) , z ) e (12346) ( − = 0 , which again contradicts the simplicity of V − ( E ). So, ϑ ( w ) = 0.But if w = 0, then, by Theorem 8.6 (iii), we have that e V ( − , D ) = V − ( D ). Theorem 4.2implies that e V ( − , D ) is not simple, since the simple vertex operator algebra V − ( D ) has only oneirreducible D –locally finite module, a contradiction. So w = 0 and claim (2) follows. (cid:3) Set(9.6) V − ( D ) = V − ( D ) < v , ϑ ( w ) > . Theorem 9.2.
We have: (1) e V ( − , D ) ∼ = V − ( D ) . (2) The set { L D ( − ( s + 4)Λ + s Λ ) | s ∈ Z ≥ } provides a complete list of irreducible V − ( D ) –modules.Proof. We first notice that e V ( − , D ) is a certain quotient of V − ( D )
This work was supported by the Croatian Science Foundation [grant number 2634 to D.A. andO.P.]; and the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Governmentand European Union through the European Regional Development Fund - the Competitiveness andCohesion Operational Programme [KK.01.1.1.01 to D.A. and O.P.].
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