The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order 2 (Part 1)
aa r X i v : . [ m a t h . QA ] F e b The irreducible weak modules for the fixed point subalgebra of thevertex algebra associated to a non-degenerate even lattice by anautomorphism of order 2 (Part 1)
Kenichiro Tanabe ∗ Department of MathematicsHokkaido UniversityKita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, [email protected]
Abstract
Let V L be the vertex algebra associated to a non-degenerate even lattice L , θ the automorphism of V L induced from the − L , and V + L the fixed point subalgebra of V L under the action of θ .In this series of papers, we classify the irreducible weak V + L -modules and show that any irreducible weak V + L -module is isomorphic to a weak submodule of some irreducible weak V L -module or to a submoduleof some irreducible θ -twisted V L -module.In this paper (Part 1), we show that when the rank of L is 1, every non-zero weak V + L -module containsa non-zero M (1) + -module, where M (1) + is the fixed point subalgebra of the Heisenberg vertex operatoralgebra M (1) under the action of θ . Mathematics Subject Classification.
Key Words. vertex algebras, lattices, weak modules.
Contents M (1) + in a weak V + L -module: the case that rank L = 1 A1-1 Computations in M (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24A1-2 The case that h α, α i 6 = 0 , / ,
1, and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A1-3 The case that h α, α i = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25A1-4 The case that h α, α i = 1 / Notation 27 ∗ Research was partially supported by the Grant-in-aid (No. 18K03198) for Scientific Research, JSPS. ertex algebra associated to an even lattice Let V be a vertex algebra, G a finite automorphism group of V , and V G the fixed point subalgebra of V under the action of G : V G = { u ∈ V | gu = u for all g ∈ G } . The fixed point subalgebras play an importantrole in the study of vertex algebras, particularly in the construction of interesting examples. For example, let V L be the vertex algebra associated to a non-degenerate even lattice L of finite rank and θ the automorphismof V L of order 2 induced from the − L . We write V ± L = { a ∈ V L | θ ( a ) = ± a } for simplicity.Then, the moonshine vertex algebra V ♮ is constructed from V +Λ and an irreducible V +Λ -module in [25] whereΛ is the Leech lattice. We remark that V +Λ is also a fixed point subalgebra of V ♮ under the action of anautomorphism of order 2.One of the main problems about V G is to describe the V G -modules in terms of V and G . If V is avertex operator algebra, it is conjectured that under some conditions on V , every irreducible V G -module isa submodule of some irreducible g -twisted V -module for some g ∈ G (cf. [14]). The conjecture is confirmedfor many examples including V + L where L is a positive definite even lattice (cf. [5, 19, 22, 23, 24, 35, 36]).In those examples, they classify the irreducible V G -modules directly by investigating the Zhu algebra, whichis an associative C -algebra introduced in [40], since [40, Theorem 2.2.1] says that for a vertex operatoralgebra V there is a one to one correspondence between the set of all isomorphism classes of irreducible N -graded weak V -modules and that of irreducible modules for the Zhu algebra associated to V , where wenote that an arbitrary V -module is automatically an N -graded weak V -module. We recall some results onthe representations of V L and V + L . It is well known that the vertex algebra V L is a vertex operator algebra ifand only if L is positive definite. It is shown in [15, Theorem 3.1] that { V λ + L | λ + L ∈ L ⊥ /L } is a completeset of representatives of equivalence classes of the irreducible weak V L -modules (see Definition 2.1 for thedefinition of a weak module), where L ⊥ is the dual lattice of L . It is also shown in [20, Theorem 3.16] thatevery weak V L -module is completely reducible. The corresponding results for θ -twisted weak V L -modules areobtained in [16]. When the lattice L is positive definite, the irreducible V + L -modules are classified in [5, 23],the fusion rules are determined in [2, 6] and it is established that V + L is C -cofinite in [4, 37] and rationalin [3, 18]. Thus, in this case it follows from [4, Theorem 4.5] that V + L is regular and every irreducible weak V + L -module is an irreducible V + L -module. Here, it is worth mentioning that if a vertex operator algebra V issimple and C -cofinite, then so is V G for any finite solvable automorphism group G of V by [31].We consider the case that L is not positive definite. In this case, vertex algebras V L and V + L are also relatedto V ♮ . In fact, in [9], Borcherds constructs the monster Lie algebra from the tensor product V ♮ ⊗ C V II , where II , is the even unimodular Lorentzian lattice of rank 2 and uses this Lie algebra to prove the moonshineconjecture stated in [10]. Moreover, let II , be the even unimodular Lorentzian lattice of signature (25 , ⊕ II , is isomorphic to II , , which implies that V II , ∼ = V Γ ⊗ V II , and hence V + II , ∼ = V +Γ ⊗ V + II , ⊕ V − Γ ⊗ V − II , . Thus, V +Γ ’s are related to each other through V + II , . Since V +Λ is a fixedpoint subalgebra of V ♮ as mentioned above and there are several interesting connections between II , andthe Niemeier lattices obtained in [7] and [11, 12] (see also [13, Chapters 24 and 26]), one may expect thatthe study of V II , and its subalgebras including V + II , provides a better understanding of the moonshinevertex algebra V ♮ and related algebras. This is one of the motivations to study representations of V + L . Asmentioned above V L is not a vertex operator algebra in this case, however, we note that the conjecture aboutrepresentations for V G above makes sense even for (weak) modules for a vertex algebra V . For V + L -modules,the irreducible V + L -modules are classified by using the Zhu algebra in [27, 38] and it is established that V + L is C -cofinite in [26] and rational in [39]. However, the study of weak V + L -modules has not progressed in spiteof the fact that V + L itself is a weak module but not a module for V + L , because of the absence of useful toollike the Zhu algebras for weak modules.The following is the main result of this series of papers, which implies that for any non-degenerateeven lattice L of finite rank, every irreducible weak V + L -module is isomorphic to a weak submodule ofsome irreducible weak V L -module or to a submodule of some irreducible θ -twisted V L -module. Namely, theconjecture above holds for irreducible weak V + L -modules. Theorem 1.1.
Let L be a non-degenerate even lattice of finite rank with a bilinear form h , i . The followingis a complete set of representatives of equivalence classes of the irreducible weak V + L -modules:(1) V ± λ + L , λ + L ∈ L ⊥ /L with λ ∈ L . ertex algebra associated to an even lattice (2) V λ + L ∼ = V − λ + L , λ + L ∈ L ⊥ /L with λ L .(3) V T χ , ± L for any irreducible ˆ L/P -module T χ with central character χ . Here, V ± λ + L = { u ∈ V λ + L | θ ( u ) = ± u } for λ + L ∈ L ⊥ /L with 2 λ ∈ L , ˆ L is the canonical centralextension of L by the cyclic group h κ i of order 2 with the commutator map c ( α, β ) = κ h α,β i for α, β ∈ L , P = { θ ( a ) a − | a ∈ ˆ L } , V T χ L is a θ -twisted V L -module, and V T χ , ± L = { u ∈ V T χ L | θ ( u ) = ± u } . Note that inTheorem 1.1, V T χ , ± L in (3) are V + L -modules, however, if L is not positive definite, then V ± λ + L in (1) and V λ + L in (2) are not V + L -modules (see (2.18)). If L is positive definite, then Theorem 1.1 is the same as [5, Theorem7.7] and [23, Theorem 5.13]. Using Theorem 1.1, we show in [33, Theorem 1.1] that every weak V + L -moduleis completely reducible for any non-degenerate even lattice L of finite rank.In this paper, Part 1 of a series of three papers, we show that when the rank of L is 1, every non-zero weak V + L -module contains a non-zero M (1) + -module, where M (1) + is the fixed point subalgebra of the Heisenbergvertex operator algebra M (1) under the action of θ (Proposition 3.13). In Part 2, we compute extensiongroups for M (1) + -modules. In Part 3, after studying intertwining operators for weak M (1) + -modules, weshow Theorem 1.1.Let us briefly explain the basic idea to show Theorem 1.1 in the case that rank L = 1. We write L = Z α and we further assume that h α, α i 6 = 2 to simplify the argument. When h α, α i = 2, we only need to changethe set of generators for V + L in the following argument. Let V be a vertex algebra and ( M, Y M ) a weak V -module. For u ∈ V and v ∈ M , we write the expansion of Y M ( u, x ) v by Y M ( u, x ) v = P i ∈ Z u i vx − i − anddefine ǫ ( u, v ) ∈ Z ∪ {−∞} by u ǫ ( u,v ) v = 0 and u i v = 0 for all i > ǫ ( u, v ) (1.1)if Y M ( u, x ) v = 0 and ǫ ( u, v ) = −∞ if Y M ( u, x ) v = 0. It is known that the vertex operator algebra M (1)associated to the Heisenberg algebra is a subalgebra of V L and the fixed point subalgebra M (1) + = M (1) h θ i under the action of θ is a subalgebra of V + L . The irreducible M (1) + -modules are classified in [22, 24] and thefusion rules for M (1) + -modules are determined in [1, 6]. The vertex operator algebra M (1) + is generatedby the conformal vector (Virasoro element) ω and a homogeneous element H (or J ) of weight 4, and V + L is generated by M (1) + and E = e α + θ ( e α ) (see (3.2)). We can find some relations for ω, H in M (1) + and for ω, H, E in V + L with the help of computer algebra system Risa/Asir[32] (Lemma 3.5). Let v be anon-zero element of a weak V + L -module M . If ǫ ( ω, v ) ≥
1, then taking suitable actions of the relations on v and using the commutation relations, we obtain relations in Span C { ω iǫ ( ω,v ) H jǫ ( H,v ) v | i, j ∈ Z ≥ } or inSpan C { ω iǫ ( ω,v ) H jǫ ( H,v ) E ǫ ( E,v ) v | i, j ∈ Z ≥ } with the help of computer algebra system Risa/Asir (Lemmas3.6 and 3.7). Using these relations in M , we can get a simultaneous eigenvector u ∈ M for ω and H with ǫ ( ω, u ) ≤ ǫ ( H, u ) ≤
3, namely we have an irreducible module C u for the Zhu algebra A ( M (1) + )of M (1) + (Lemmas 3.7, 3.10, 3.11, and 3.12). Thus, by [21, Theorem 6.2] we obtain a non-zero M (1) + -submodule of M (Proposition 3.13). Moreover, we have some conditions on ǫ ( E, u ) (Lemmas 3.10, 3.11,and 3.12). Using results of extension groups for M (1) + -modules (Part 2), we have an irreducible M (1) + -module K in the M (1) + -submodule of M generated by C u (Part 3). Since V + L is a direct sum of irreducible M (1) + -modules, for any irreducible M (1) + -submodule N of V + L , the V + L -module structure of M inducesan intertwining operator I ( , x ) : N × K → M (( x )) for weak M (1) + -modules. By using results of extensiongroups for M (1) + -modules (Part 2), the same argument as above shows that there exists an M (1) + -modulewhich is a direct sum of irreducible M (1) + -modules in the image of I ( , x ) (Part 3). Thus, we obtain a weakirreducible V + L -submodule W of M which is isomorphic to a submodule of a θ -twisted irreducible V L -moduleor is a direct sum of pairwise non-isomorphic irreducible M (1) + -modules (Part 3). In the latter case, by astandard argument we can determine the possible weak V + L -module structures for such an M (1) + -module(Part 3) and hence we obtain the desired result. For general L , we divide our analysis into four cases based onthe norm of an element of C ⊗ Z L and carry out the procedure above by an enormous amount of computation.Complicated computation has been done by a computer algebra system Risa/Asir[32]. Throughout thispaper, the word “a direct computation” means a direct computation with the help of Risa/Asir.The organization of the paper is as follows. In Section 2 we recall some basic properties of the vertexalgebra M (1) associated to the Heisenberg algebra and the vertex algebra V L associated to a non-degenerateeven lattice L . In Section 3 under the condition that the rank of L is 1, we construct an irreducible module ertex algebra associated to an even lattice M (1) + in a non-zero weak V + L -module M . Thus, there is a non-zero M (1) + -module in M . In Appendix A1 we put computations of a k b for some a, b ∈ V + L and k = 0 , , . . . to find the commutationrelation [ a i , b j ] = P ∞ k =0 (cid:0) ik (cid:1) ( a k b ) i + j − k . In Notation we list some notation. We assume that the reader is familiar with the basic knowledge on vertex algebras as presented in [8, 25, 28,29].Throughout this paper, p is a non-zero complex number, N denotes the set of all non-negative integers, Z denotes the set of all integers, L is a non-degenerate even lattice of finite rank d with a bilinear form h , i ,( V, Y, ) is a vertex algebra. Recall that V is the underlying vector space, Y ( , x ) is the linear map from V ⊗ C V to V (( x )), and is the vacuum vector. Throughout this paper, we always assume that V has anelement ω such that ω a = a − for all a ∈ V . For a vertex operator algebra V , this condition automaticallyholds since V has the conformal vector (Virasoro element). For i ∈ Z , we define Z i = { j ∈ Z | j > i } . (2.1)The notion of a module for V has been introduced in several papers, however, if V is a vertex operator algebra,then the notion of a module for V viewed as a vertex algebra is different from the notion of a module for V viewed as a vertex operator algebra (cf. [28, Definitions 4.1.1 and 4.1.6]). To avoid confusion, throughoutthis paper, we refer to a module for a vertex operator algebra defined in [28, Definition 4.1.6] as a module and to a module for a vertex algebra defined in [28, Definition 4.1.1] as a weak module . The reason why weuse the terminology “weak module” is that when V is a vertex operator algebra, a module for V viewed as avertex algebra is called a weak V -module (cf. [29, p.157], [20, p.150], and [4, Definition 2.3]). We write downthe definition of a weak V -module: Definition 2.1. A weak V -module M is a vector space over C equipped with a linear map Y M ( , x ) : V ⊗ C M → M (( x )) a ⊗ u Y M ( a, x ) u = X n ∈ Z a n ux − n − (2.2)such that the following conditions are satisfied:(1) Y M ( , x ) = id M .(2) For a, b ∈ V and u ∈ M , x − δ ( x − x x ) Y M ( a, x ) Y M ( b, x ) u − x − δ ( x − x − x ) Y M ( b, x ) Y M ( a, x ) u = x − δ ( x + x x ) Y M ( Y ( a, x ) b, x ) u. (2.3)For n ∈ C and a weak V -module M , we define M n = { u ∈ V | ω u = nu } . For a ∈ V n ( n ∈ C ), wt a denotes n . For a vertex algebra V which admits a decomposition V = ⊕ n ∈ Z V n and a subset U of a weak V -module, we define Ω V ( U ) = n u ∈ U (cid:12)(cid:12)(cid:12) a i u = 0 for all homogeneous a ∈ V and i > wt a − . o . (2.4)For a vertex algebra V which admits a decomposition V = ⊕ n ∈ Z V n , a weak V -module N is called N -graded if N admits a decomposition N = ⊕ ∞ n =0 N ( n ) such that a i N ( n ) ⊂ N (wt a − i − n ) for all homogeneous a ∈ V , i ∈ Z , and n ∈ Z ≥ , where we define N ( n ) = 0 for all n <
0. For a triple of weak V -modules M, N, W , u ∈ M, v ∈ W , and an intertwining operator I ( , x ) from M × W to N , we write the expansion of I ( u, x ) v by I ( u, x ) v = X i ∈ C u i vx − i − = X i ∈ C I ( u ; i ) vx − i − ∈ N { x } . (2.5) ertex algebra associated to an even lattice I ( , x ) is contained in N (( x )), namely I ( , x ) : M × W → N (( x )). For a subset X of W , M · X denotes Span C { a i u | a ∈ M, i ∈ Z , u ∈ X } ⊂ N. (2.6)For an intertwining operator I ( , x ) : M × W → N (( x )), u ∈ M , and v ∈ W , we define ǫ I ( u, v ) = ǫ ( u, v ) ∈ Z ∪ {−∞} by u ǫ I ( u,v ) v = 0 and u i v = 0 for all i > ǫ I ( u, v ) (2.7)if I ( u, x ) v = 0 and ǫ I ( u, v ) = −∞ if I ( u, x ) v = 0. If V is a simple vertex algebra and ( M, Y M ) is a weak V -module, then it follows from [28, Corollary 4.5.15] that Y M ( a, x ) u = 0 , namely ǫ Y M ( a, u ) ∈ Z (2.8)for any non-zero a ∈ V and any non-zero u ∈ M . We will frequently use the following easy formula: Lemma 2.2.
Let
M, W, N be three weak V -modules and I ( , x ) : M × W → N (( x )) an intertwining operator.For a ∈ V , b ∈ M , m ∈ Z ≥− , k ∈ Z ≤− , and n ∈ Z , ( a k b ) n = X i ≤ mi + j − k = n (cid:18) − i − − k − (cid:19) a i b j + X i ≥ m +1 i + j − k = n (cid:18) − i − − k − (cid:19) b j a i − m X i =0 (cid:18) − i − − k − (cid:19) ∞ X l =0 (cid:18) il (cid:19) ( a l b ) n + k − l = X i ≤ mi + j − k = n (cid:18) − i − − k − (cid:19) a i b j + X i ≥ m +1 i + j − k = n (cid:18) − i − − k − (cid:19) b j a i + ( − k ∞ X l =0 (cid:18) l − k − − k − (cid:19)(cid:18) m − kl − k (cid:19) ( a l b ) n + k − l . (2.9) Proof.
The first expression follows from( a k b ) n = X i< i + j − k = n (cid:18) − i − − k − (cid:19) a i b j + X i ≥ i + j − k = n (cid:18) − i − − k − (cid:19) b j a i = X i ≤ mi + j − k = n (cid:18) − i − − k − (cid:19) a i b j + X i ≥ m +1 i + j − k = n (cid:18) − i − − k − (cid:19) b j a i − X ≤ i ≤ mi + j − k = n (cid:18) − i − − k − (cid:19) [ a i , b j ] . (2.10)The last expression follows from the fact that P mi =0 (cid:0) − i − − k − (cid:1)(cid:0) il (cid:1) = ( − k +1 (cid:0) l − k − − k − (cid:1)(cid:0) m − kl − k (cid:1) for l ∈ Z ≥ .We recall the vertex operator algebra M (1) associated to the Heisenberg algebra and the vertex algebra V L associated to a non-degenerate even lattice L . Let h be a finite dimensional vector space equipped witha non-degenerate symmetric bilinear form h , i . Set a Lie algebraˆ h = h ⊗ C [ t, t − ] ⊕ C C (2.11)with the Lie bracket relations[ β ⊗ t m , γ ⊗ t n ] = m h β, γ i δ m + n, C, [ C, ˆ h ] = 0 (2.12)for β, γ ∈ h and m, n ∈ Z . For β ∈ h and n ∈ Z , β ( n ) denotes β ⊗ t n ∈ b H . Set two Lie subalgebras of h : b h ≥ = M n ≥ h ⊗ t n ⊕ C C and b h < = M n ≤− h ⊗ t n . (2.13) ertex algebra associated to an even lattice β ∈ h , C e β denotes the one dimensional b h ≥ -module uniquely determined by the condition that for γ ∈ h γ ( i ) · e β = (cid:26) h γ, β i e β for i = 00 for i > C · e β = e β . (2.14)We take an b h -module M (1 , β ) = U ( b h ) ⊗ U ( b h ≥ ) C e β ∼ = U ( b h < ) ⊗ C C e β (2.15)where U ( g ) is the universal enveloping algebra of a Lie algebra g . Then, M (1) = M (1 ,
0) has a vertexoperator algebra structure with the conformal vector ω = 12 dim h X i =1 h i ( − h ′ i ( − (2.16)where { h , . . . , h dim h } is a basis of h and { h ′ , . . . , h ′ dim h } is its dual basis. Moreover, M (1 , β ) is an irreducible M (1)-module for any β ∈ h . The vertex operator algebra M (1) is called the vertex operator algebra associatedto the Heisenberg algebra ⊕ = n ∈ Z h ⊗ t n ⊕ C C .Let L be a non-degenerate even lattice. We define h = C ⊗ Z L and denote by L ⊥ the dual of L : L ⊥ = { γ ∈ h | h β, γ i ∈ Z for all β ∈ L } . Taking M (1) for h , we define V λ + L = ⊕ β ∈ λ + L M (1 , β ) for λ + L ∈ L ⊥ /L . Then, V L admits a unique vertex algebra structure compatible with the action of M (1) andfor each λ + L ∈ L ⊥ /L the vector space V λ + L is an irreducible weak V L -module which admits the followingdecomposition: V λ + L = M n ∈h λ,λ i / Z ( V λ + L ) n where ( V λ + L ) n = { a ∈ V λ + L | ω a = na } . (2.17)Note that if L is positive definite, then dim C ( V λ + L ) n < + ∞ for all n ∈ λ + L and ( V λ + L ) h λ,λ i / i = 0 forsufficiently small i ∈ Z . If L is not positive definite, thendim C ( V λ + L ) n = + ∞ (2.18)for all n ∈ h λ, λ i / Z , which implies that V λ + L is not a V L -module. For α ∈ h , we write E ( α ) = e α + θ ( e α ) (2.19)Let ˆ L be the canonical central extension of L by the cyclic group h κ i of order 2 with the commutatormap c ( α, β ) = κ h α,β i for α, β ∈ L : 0 → h κ i → ˆ L − → L → . (2.20)Then, the − L induces an automorphism θ of ˆ L of order 2 and an automorphism, by abuse ofnotation we also denote by θ , of V L of order 2. In M (1), we have θ ( h ( − i ) · · · h n ( − i n ) ) = ( − n h ( − i ) · · · h n ( − i n ) (2.21)for n ∈ Z ≥ , h , . . . , h n ∈ h , and i , . . . , i n ∈ Z > . For a weak V L -module M , we define a weak V L -module( M ◦ θ, Y M ◦ θ ) by M ◦ θ = M and Y M ◦ θ ( a, x ) = Y M ( θ ( a ) , x ) (2.22)for a ∈ V L . Then V λ + L ◦ θ ∼ = V − λ + L for λ ∈ L ⊥ . Thus, for λ ∈ L ⊥ with 2 λ ∈ L we define V ± λ + L = { u ∈ V λ + L | θ ( u ) = ± u } . (2.23)Next, we recall the construction of θ -twisted modules for M (1) and V L following [25]. Set a Lie algebraˆ h [ −
1] = h ⊗ t / C [ t, t − ] ⊕ C C (2.24) ertex algebra associated to an even lattice C, ˆ h [ − α ⊗ t m , β ⊗ t n ] = m h α, β i δ m + n, C (2.25)for α, β ∈ h and m, n ∈ / Z . For α ∈ h and n ∈ / Z , α ( n ) denotes α ⊗ t n ∈ b h . Set two Lie subalgebrasof ˆ h [ − b h [ − ≥ = M n ∈ / N h ⊗ t n ⊕ C C and b h [ − < = M n ∈ / N h ⊗ t − n . (2.26)Let C tw denote a unique one dimensional b h [ − ≥ -module such that h ( i ) · tw = 0 for h ∈ h and i ∈
12 + N ,C · tw = tw . (2.27)We take an b h [ − M (1)( θ ) = U ( b h [ − ⊗ U ( b h [ − ≥ ) C u ζ ∼ = U ( b h [ − < ) ⊗ C C u ζ . (2.28)We define for α ∈ h , α ( x ) = X i ∈ / Z α ( i ) x − i − (2.29)and for u = α ( − i ) · · · α k ( − i k ) ∈ M (1), Y ( u, x ) = ◦◦ i − d i − dx i − α ( x )) · · · i k − d i k − dx i k − α k ( x )) ◦◦ . (2.30)Here, for β , . . . , β n ∈ h and i , . . . , i n ∈ / Z , we define ◦◦ β ( i ) · · · β n ( i n ) ◦◦ inductively by ◦◦ β ( i ) ◦◦ = β ( i ) and ◦◦ β ( i ) · · · β n ( i n ) ◦◦ = (cid:26) ◦◦ β ( i ) · · · β n ( i n ) ◦◦ β ( i ) if i ≥ ,β ( i ) ◦◦ β ( i ) · · · β n ( i n ) ◦◦ if i < . (2.31)Let h [1] , . . . , h [dim h ] be an orthonormal basis of h . We define c mn ∈ Q for m, n ∈ Z ≥ by ∞ X m,n =0 c mn x m y n = − log( (1 + x ) / + (1 + y ) / x = ∞ X m,n =0 c mn dim h X i =1 h [ i ] ( m ) h [ i ] ( n ) x − m − n . (2.33)Then, for u ∈ M (1) we define a vertex operator Y M (1)( θ ) by Y M (1)( θ ) ( u, x ) = Y ( e ∆ x u, x ) . (2.34)Then, [25, Theorem 9.3.1] shows that ( M (1)( θ ) , Y M (1)( θ ) ) is an irreducible θ -twisted M (1)-module. Set asubmodule P = { θ ( a ) a − | a ∈ ˆ L } of ˆ L . Let T χ be the irreducible ˆ L/P -module associated to a centralcharacter χ such that χ ( κ ) = −
1. We set V T χ L = M (1)( θ ) ⊗ T χ . (2.35) ertex algebra associated to an even lattice V T χ L admits an irreducible θ -twisted V L -module structure compatiblewith the action of M (1). We define the action of θ on V T χ L by θ ( h ( − i ) · · · h n ( − i n ) u ) = ( − n h ( − i ) · · · h n ( − i n ) u (2.36)for n ∈ Z ≥ , h , . . . , h n ∈ h , i , . . . , i n ∈ / Z > , and u ∈ T χ . We set V T χ , ± L = { u ∈ V T χ L | θ ( u ) = ± u } . (2.37)We recall the Zhu algebra A ( V ) of a vertex operator algebra V from [40, Section 2]. For homogeneous a ∈ V and b ∈ V , we define a ◦ b = ∞ X i =0 (cid:18) wt ai (cid:19) a i − b ∈ V (2.38)and a ∗ b = ∞ X i =0 (cid:18) wt ai (cid:19) a i − b ∈ V. (2.39)We extend (2.38) and (2.39) for an arbitrary a ∈ V by linearity. We also define O ( V ) = Span C { a ◦ b | a, b ∈ V } .Then, the quotient space A ( V ) = M/O ( V ) , (2.40)called the Zhu algebra of V , is an associative C -algebra with multiplication (2.39) by [40, Theorem 2.1.1]. M (1) + in a weak V + L -module: thecase that rank L = 1 In this section, under the condition that the rank of L is 1, we shall show that there exists an irreducible A ( M (1) + )-module in an arbitrary non-zero weak V + L -module.Throughout this section, p is a non-zero complex number, h is a one dimensional vector space equippedwith a non-degenerate symmetric bilinear form h , i , h, α ∈ h such that h h, h i = 1 and h α, α i = p, (3.1) M, N, W are weak M (1) + -modules, and I ( , x ) : M (1 , α ) × W → N (( x )) is a non-zero intertwining operator.We define ω = 12 h ( − ,H = 13 h ( − h ( − − h ( − ,J = h ( − − h ( − h ( − + 32 h ( − = − H + 4 ω − − ω − ,E = E ( α ) = e α + θ ( e α ) . (3.2)Since 0 = ω − h ( − − ω − h ( − + 3 h ( − (3.3)and [ ω , a i ] = − ia i − for all a ∈ M (1) and i ∈ Z , h ( j ) ∈ Span n ω − i · · · ω − i m h ( − k ) (cid:12)(cid:12)(cid:12) m ∈ Z ≥ , i , . . . , i m ∈ Z > and k = 1 , , o (3.4) ertex algebra associated to an even lattice j ∈ Z . For i, j ∈ Z , a direct computation shows that[ ω i , ω j ] = ( i − j ) ω i + j − + δ i + j − , i ( i − i − , (3.5)[ ω i , J j ] = (3 i − j ) J i + j − , (3.6)[ ω i , H j ] = (3 i − j ) H i + j − + i ( i − i + j − ω i + j − + − (cid:18) i (cid:19) δ i + j − , , (3.7)[ ω i , E j ] = (( − p i − j ) E i + j − , (3.8)[ h ( i ) , ω j ] = h ( i + j − , (3.9)[ h ( i ) , H j ] = (cid:0) i ( i + j − i + j − (cid:18) i (cid:19)(cid:1) h ( i + j − , (3.10)and h ( − h ( − = ω ω,h ( − h ( − = H + 13 ω ω,h ( − h ( − = − H + 13 ω ω,h ( − h ( − = − ω H + 112 ω ω,h ( − h ( − = 13 ω − + 45 ω − H + − ω ω − ω + − ω H + 145 ω ω. (3.11)It follows from (3.4), (3.7), and (3.9)–(3.11) that M (1) + is spanned by the elements ω − i · · · ω − i m H − j · · · H − j n where m, n ∈ Z ≥ and i , . . . , i m , j , . . . , j n ∈ Z > , which is already shown in [17, Theorem 2.7].We have J E = 2(2 p − ω E, J E = ( p − p E, J i E = 0 for i ≥ , (3.12)and H E = 13 ω E, H i E = 0 for i ≥ . (3.13)If p = 1 /
2, then J E = 2 p (4 p − p − ω − E + 8 p + 52 p − ω E,H E = 2 p p − ω − E + − p − ω E (3.14)and if p = 2 , /
2, then J E = 2( p − p − p − ω − E + 4(4 p − p + 4)( p − p − ω ω − E + − p − p − ω E,H E = 2 pp − ω − E + − p (2 p − p − ω ω − E + 2(2 p − p − ω E. (3.15) ertex algebra associated to an even lattice p = 2, then by Lemma 2.2 and (A1.5) in Appendix A1 for m ≥ ω i , ( H E ) j ]= (3 i − j )( H E ) i + j − + 8 (cid:18) i (cid:19)(cid:0) X k ≤ m ω k E i + j − − k + X k ≥ m +1 E i + j − − k ω k + (( m + 1)( i + j − − (cid:18) m + 12 (cid:19) ) E i + j − (cid:1) − (cid:18) i (cid:19) ( i + j − i + j − E i + j − − (cid:18) i (cid:19) ( i + j − E i + j − + 12 (cid:18) i (cid:19) E i + j − . (3.16)If p = 1 /
2, then by (A1.10) in Appendix A1,[ ω i , ( H E ) j ] = ( 54 i − j )( H E ) i + j − + (cid:18) i (cid:19) ( i + j − E i + j − + (cid:18) i (cid:19) E i + j − . (3.17)For n ∈ Z , m ∈ Z ≥− , and k ∈ Z < , using Lemma 2.2 and (3.8), we expand each of ( ω k E ) n and ( ω − E ) n sothat the resulting expression is a linear combination of elements of the form ω i · · · ω i r E l ω j · · · ω j s (3.18)where r, s ∈ Z ≥ , l ∈ Z , i , . . . , i r ≤ m , and j , . . . , j s ≥ m + 1 as follows:( ω k E ) n = X i ≤ m (cid:18) − i − − k − (cid:19) ω i E n + k − i + X i ≥ m +1 (cid:18) − i − − k − (cid:19) E n + k − i ω i + ( − k (( − n − k ) (cid:18) m − k − k (cid:19) + (cid:18) − k − k − (cid:19)(cid:18) m − k − k (cid:19) p E n + k − (3.19) ertex algebra associated to an even lattice ω − E ) n = X i< ,j< ,k = n − i − j − ω i ω j E k + 2 X i< , ≤ j,k = n − i − j − ω i E k ω j + X ≤ i, ≤ j,k = n − i − j − E k ω j ω i = X i< ,j< ,k = n − i − j − ω i ω j E k + 2 X i< , ≤ j ≤ m,k = n − i − j − (cid:0) ω i ω j E k − (( − p j − k ) ω i E j + k − (cid:1) + 2 X i< ,m +1 ≤ j,k = n − i − j − ω i E k ω j + X ≤ i,j ≤ m,k = n − i − j − (cid:16) ω j ω i E k + ((1 − p j + k )((1 − p i + j + k − E j + k − + ((1 − p j + k ) ω i E j + k − + ((1 − p i + k ) ω j E i + k − (cid:17) + X m +1 ≤ i, ≤ j ≤ m,k = n − i − j − (cid:0) ((1 − p j + k ) E j + k − ω i + ω j E k ω i (cid:1) + ( m + 1)((1 − p m + n − m − E n − + ( m + 1) ω m E n − m − + X ≤ i ≤ m,m +1 ≤ j ( i,j ) =( m +1 , ,k = n − i − j − E k ( j − i ) ω i + j − + X ≤ i ≤ m,m +1 ≤ j,k = n − i − j − ((1 − p i + k ) E i + k − ω j + ω i E k ω j (cid:1) + X m +1 ≤ i,j,k = n − i − j − E k ω j ω i . (3.20)For i ∈ Z and a subset X of a weak M (1) + -module K , h ω i i X denotes the subspace of K spanned by theelements ω ji u, j ∈ Z ≥ , u ∈ X .The following lemmas follow from (3.5)–(3.8), (3.16), and (3.17). Lemma 3.1.
Let u be an element of a weak M (1) + -module ( K, Y K ) with ǫ ( ω, u ) = ǫ Y K ( ω, u ) ≥ .(1) For any i ≥ , j > ǫ ( ω, u ) , and k ≥ , ω j ω iǫ ( ω,u ) u = ω j ω iǫ ( ω,u ) J ǫ ( J,u ) u = 0 ,ω ik J ǫ ( J,u ) u = J ǫ ( J,u ) ω ik u. (3.21) (2) For any v ∈ h ω ǫ ( ω,u ) i{ u, J ǫ ( J,u ) u } , ǫ ( ω, v ) ≤ ǫ ( ω, u ) . Lemma 3.2.
Let u ∈ M with ǫ ( ω, u ) = ǫ I ( ω, u ) ≥ and let a be one of E, H E , or H E .(1) For any i ≥ , j > ǫ ( ω, u ) , and k ≥ , ω j ω iǫ ( ω,u ) a ǫ ( a,u ) u = 0 ,ω ik a ǫ ( a,u ) u = a ǫ ( a,u ) ω ik u. (3.22) (2) For any v ∈ h ω ǫ ( ω,u ) i a ǫ ( a,u ) u , ǫ ( ω, v ) ≤ ǫ ( ω, u ) . By using the commutation relation [ H i , E j ] = P ∞ k =0 (cid:0) ik (cid:1) ( H k E ) i + j − k for i, j ∈ Z , the following resultfollows from Lemma 2.2 and (3.13)–(3.15). Lemma 3.3.
Assume p = 1 / . Let u ∈ W with ǫ ( ω, u ) ≥ and ǫ ( H, u ) ≤ ǫ ( ω, u ) + 1 . Then, ǫ ( H E, u ) ≤ ǫ ( E, u ) + 2 ǫ ( ω, u ) + 1 . ertex algebra associated to an even lattice p = 1 /
2, then a direct computation shows that0 = ω − E − ω E, (3.23) H E = − ω − E + 43 ω ( H E ) , (3.24)0 = 8 ω − E + 12 H − E + 3 ω − ( H E ) + 4 ω ω − E − ω ( H E ) . (3.25) Lemma 3.4.
Let U be an A ( M (1) + ) -submodule of Ω M (1) + ( W ) , u ∈ U , and t ∈ Z such that ǫ ( E, v ) ≤ t forall v ∈ U . Then ǫ ( H E, u ) ≤ t + 3 and ǫ ( H E, u ) ≤ t + 2 .Proof. For p = 1 /
2, the result follows from Lemma 3.3 and (3.14). Assume p = 1 /
2. For i, j ∈ Z and r ∈ Z ≥ ,it follows from (3.13), (3.24), and Lemma 2.2 that[ H i , E j ] u = ( H E ) i + j u + i ( H E ) i + j − u + (cid:18) i (cid:19) ( H E ) i + j − u = −
13 ( i + 4 j )( H E ) i + j − u − (cid:18) i (cid:19) i + j − E i + j − u + − (cid:0) X k ≤ r ( − k − ω k E i + j − − k + X k ≥ r +1 ( − k − E i + j − − k ω k + ( (cid:18) r + 22 (cid:19) ( − i − j + 2) + (cid:18) r + 23 (cid:19) ) E i + j − (cid:1) u. (3.26)Using (3.26) with i = 3 and r = 1, we have ǫ ( H E, u ) ≤ t + 2. By (3.24), ǫ ( H E, u ) ≤ t + 3.A direct computations shows the following result. Lemma 3.5.
The following elements of V + L are zero: P (8) ,H = − ω − ω − ω − + 3168 ω − ω − ω − − ω − ω − − ω − ω − + 30456 ω − ω − + 2310 ω − − ω − ω − H − − ω − H − − ω − H − − ω − H − + 11868 H − + 5040 H − , (3.27) P (8) ,J = − ω − − ω − ω − + 39040 ω − ω − − ω − − ω − ω − + 497760 ω − ω − + 230360 ω − + 5024 ω − J − − ω − J − + 8939 ω − J − − ω − J − + 1572 J − + 560 J − , (3.28) P (9) = 30 J − − ω − J − + 27 ω − J − − ω − J − + 16 ω − J − + 52 ω − J − − ω − ω − J − , (3.29) P (10) ,H = 919328 ω − − ω − ω − ω − − ω − ω − + 545352 ω − ω − ω − + 520160 ω − ω − ω − − ω − ω − ω − − ω − ω − ω − ω − + 7680 ω − ω − ω − ω − + 1937712 ω − H − − ω − ω − H − − ω − ω − H − + 30720 ω − ω − ω − H − − ω − H − − ω − ω − H − − ω − H − + 234528 ω − ω − H − + 345849 ω − H − − ω − H − + 2360970 H − + 70875 H − H − + 734184 ω − ω − + 898766 ω − ω − , (3.30) ertex algebra associated to an even lattice P (10) ,J = 8192 ω − − ω − J − + 758496 ω − − ω − ω − − ω − ω − ω − − ω − ω − − ω − ω − ω − − ω − ω − ω − − ω − ω − ω − − ω − ω − ω − ω − − ω − ω − ω − ω − − ω − J − + 6272 ω − ω − J − + 360 ω − ω − J − + 152 ω − ω − J − + 1856 ω − J − + 9408 ω − ω − J − + 12656 ω − J − − ω − J − + 43320 J − + 525 J − J − + 1309248 ω − ω − + 352992 ω − ω − , (3.31) Q (4) = 2( p − −
27 + 54 p − p + 40 p ) ω − E − p ( p − − p ) ω − E − p ( p − − p )( − p ) H − E + ( − p − p + 210 p − ω ω − E + (120 p − p + 36) ω ω − E + ( − p − ω E, (3.32) Q (5 , = 3( p − p − p + 32)(10 p − p + 3) ω − E − p (3 p − p − p + 3) ω − ω − E − p − p − p − p − p + 3) H − E + 8(2 p − p − p + 8 p − ω ω − E + 24 p (8 p − ω ω − ω − E − p − p − p − p + 6) ω H − E − p − p + 29 p + 12) ω ω ω − E − p − ω ω ω ω ω E, (3.33) Q (5 , = 3( p − p − p + 32)(12 p + 16 p − p + 15) ω − E − p (3 p − p + 16 p − p + 15) ω − ω − E − p − p − p − p + 16 p − p + 15) H − E + 2(136 p − p − p + 3409 p − p + 624) ω ω − E + 12 p (20 p − p − p + 24) ω ω − ω − E − p − p − p + 21 p − p + 60) ω H − E − p − p + 29 p + 12) ω ω ω ω − E − p + 61 p − ω ω ω ω ω E, (3.34) ertex algebra associated to an even lattice Q (6) = 2(3696 p − p + 66284 p − p + 56207 p − p + 11774 p + 29190 p − ω − E − p (352 p + 2152 p − p + 7951 p − p + 6304 p − ω − ω − E − p (1584 p − p + 6456 p − p + 5214 p − p + 642) ω − ω − E + 720 p ( p − p − ω − ω − ω − E − p ( p − p − p − p + 157 p − p + 48) ω − H − E − p − p − p − (44 p − p + 62 p − p + 18) H − E + 3(1760 p − p + 1391 p + 28130 p − p + 29762 p − p + 7650) ω ω − E + 12 p (352 p − p + 2396 p − p + 1254 p − ω ω − ω − E − p − p − p + 101 p + 86 p − p + 804 p − ω H − E + 12(88 p + 1104 p − p + 3714 p − p + 2670 p − ω ω ω − E − p − p + 686 p − p + 804 p − ω ω ω ω − E − p − p − ω ω ω ω ω ω E. (3.35) If p = 2 , then we have the four following relations: ω − E − ω ω − E + ω E, (3.36)0 = 180 ω − E − ω − E + 72 H − E − ω ( H E )+ 8 ω ω − E + ω E, (3.37)0 = 9450 ω − E − ω − ( H E ) + 6750 H − E − ω ω − E − ω H − E + 297 ω ( H E ) + 128 ω ω − E + 16 ω E, (3.38)0 = 584199000 ω − E − H − E + 98941500 ω − ( H E ) − ω − ( H E )+ 34587000 H − ( H E ) − ω ω − E − ω ω − H − E + 277223400 ω H − E − ω ω − ( H E ) + 206053320 ω ω − E − ω ω − ( H E ) − ω ω − E − ω H − E + 17161013 ω ( H E ) − ω ω − E + 410400 ω E. (3.39) Lemma 3.6.
Let u be a non-zero element of a weak M (1) + -module ( K, Y K ) such that ǫ ( ω, u ) = ǫ Y K ( ω, u ) ≥ and ǫ ( ω, u ) ≤ ǫ ( ω, v ) for all non-zero v ∈ K . Then ǫ ( J, u ) = 2 ǫ ( ω, u ) + 1 , J ǫ ( ω,u )+1 u = 4 ω ǫ ( ω,u ) u, (3.40) and ǫ ( H, u ) ≤ ǫ ( ω, u ) . (3.41) Proof.
We write r = ǫ ( ω, u ) and s = ǫ ( J, u ) (3.42) ertex algebra associated to an even lattice s ∈ Z by (2.8). It follows from Lemma 3.1 (2) and the condition of u that forany non-zero v ∈ h ω r i{ u, J s u } and i ∈ Z ≥ , ω ir v = 0 . (3.43)Since the same argument as in [34, (3.23)] shows that0 = 116 P (9) s +2 r +3 u = ( − s + 2 r + 1) J s ω r u = ( − s + 2 r + 1) ω r J s u (3.44)by Lemma 3.1 (1), s = 2 r + 1 by (3.43). Since0 = P (10) ,J r +4 u = (8192 ω − − ω − J − ) r +4 u = 2048(4 ω r − J r +1 ω r ) u = 2048 ω r (4 ω r − J r +1 ) u (3.45)by Lemma 3.1 (1), (3.40) holds by (3.43). It follows from (3.2) that H i u = 0 for all i ≥ ǫ ( ω, u ) + 1 andhence ǫ ( H, u ) ≤ ǫ ( ω, u ). Lemma 3.7.
Let L be a non-degenerate even lattice of rank and M a non-zero weak V + L -module. Then,there exists a non-zero u ∈ Ω M (1) + ( M ) that satisfies one of the following conditions:(1) ǫ ( ω, u ) = ǫ ( J, u ) = ǫ ( E, u ) = − . In this case V + L · u ∼ = V + L .(2) H u = 0 .(3) ω u = u and H u = u .(4) ω u = (1 / u and H u = ( − / u. (5) ω u = (9 / u and H u = (15 / u. Proof.
We write L = Z α . Throughout the proof of this lemma, p = h α, α i ∈ Z \ { } . For a non-zero u ∈ W with ǫ ( ω, u ) <
0, since ω u = 0, it follows from [28, Proposition 4.7.7] that V + L · u ∼ = V + L and hence ǫ ( ω, u ) = ǫ ( J, u ) = ǫ ( E, u ) = − ǫ ( ω, v ) ≥ v ∈ M . We take a non-zero u ∈ M with ǫ ( ω, u ) as small aspossible, namely 0 ≤ ǫ ( ω, u ) ≤ ε ( ω, v ) for all non-zero v ∈ M . We write r = ǫ ( ω, u ) , s = ǫ ( J, u ) , and t = ǫ ( E, u ) (3.46)for simplicity. We note that s, t ∈ Z by (2.8). Suppose r ≥
2. Then, Lemma 3.6 shows that s = 2 r + 1 and H i u = 0 for all i ≥ r + 1. By Lemma 3.2 (2), for any non-zero v ∈ h ω r i{ u, J s u, E t u, ( H E ) ǫ ( H E,u ) u } and i ∈ Z ≥ , ω ir v = 0 . (3.47)Assume p = 2. By (3.19) and (3.20) with m = r ,0 = ( ω − E ) t +2 r +2 u = ( ω ω − E ) t +2 r +2 u = ( ω ω − E ) t +2 r +2 u = ( ω E ) t +2 r +2 u (3.48)and ( ω − E ) t +2 r +2 u = ω r E t u. (3.49)Using Lemma 2.2, (3.13), (3.14), and (3.15), we expand ( H − E ) t +2 r +2 so that the resulting expression is alinear combination of elements of the form a (1) i · · · a ( l ) i l E m b (1) j · · · b ( n ) j n (3.50) ertex algebra associated to an even lattice l, n ∈ Z ≥ , m ∈ Z , and( a (1) , i ) , . . . , ( a ( l ) , i l ) ∈ { ( ω, k ) | k ≤ r } ∪ { ( H, k ) | k ≤ r } , ( b (1) , j ) , . . . , ( b ( n ) , j n ) ∈ { ( ω, k ) | k ≥ r + 1 } ∪ { ( H, k ) | k ≥ r + 1 } , (3.51)as was done in (3.19) and (3.20). Then, taking the action of the obtained expansion of ( H − E ) t +2 r +2 on u and using (3.41) and (3.48), we have ( H − E ) t +2 r +2 u = E t H r +1 u = 0 . (3.52)By (3.32), (3.48), (3.49), and (3.52),0 = Q (4) t +2 r +2 u = − p ( p − − p ) ω r E t u, (3.53)which contradicts (3.47).Assume p = 2. By Lemma 3.3, ǫ ( H E, u ) ≤ t + 2 r + 1. By (3.37), Lemma 3.6 and the results in SectionA1-3, the same argument as above shows0 = (180 ω − E − ω − E + 72 H − E − ω ( H E )+ 8 ω ω − E + ω E ) t +2 r +2 u = ( − ω r E t + 72 E t H r +1 + 63( t + 2 r + 2)( H E ) t +2 r +1 ) u = ( − ω r E t + 63( t + 2 r + 2)( H E ) t +2 r +1 ) u (3.54)and hence ( H E ) t +2 r +1 u = 0 by (3.47). By (3.38) and results in Section A1-3,0 = (9450 ω − E − ω − ( H E ) + 6750 H − E − ω ω − E − ω H − E + 297 ω ( H E ) + 128 ω ω − E + 16 ω E ) t +3 r +2 u = − ω r ( H E ) t +2 r +1 u, (3.55)which also contradicts (3.47). We conclude that r ≤ s ≥
4. By using [34, (2.29)] and (3.6), the same argument as in [34, Lemma 3.3] shows that ǫ ( ω, J s u ) ≤ J j J s u = 0 for all j ≥ s + 1. By the same argument as in [34, (3.25)],( J − J ) s +1 u = J s u (3.56)and hence 0 = P (8) ,J s +1 u = J s u = J s ( J s u ) , (3.57)which means ǫ ( J, J s u ) < s = ǫ ( J, u ). Replacing u by J s u repeatedly, we get a non-zero u ∈ M such that r ≤ s ≤
3. Thus, u ∈ Ω M (1) + ( M ) and in particular, ǫ ( H, u ) ≤
3. Deleting the terms including ω i H u ( i = 0 , , . . . ) from the following simultaneous equations0 = P (8) ,H u = − ω − ω + 3 − H ) H u and (3.58)0 = P (10) ,H u = 240 H ( −
207 + 4725 H + 4472 ω − ω + 128 ω ) u, (3.59)we have 0 = ( ω − ω − ω − H u. (3.60)By (3.58) and (3.60), the proof is complete. Remark 3.8. If p >
0, then Lemma 3.7 also follows from [3, Theorem 7.7], [23, Theorem 5.13], and [30,Theorem 2.7]. ertex algebra associated to an even lattice Remark 3.9.
As we have seen in the proof of Lemma 3.7, starting from an arbitrary non-zero element in M , we can get u in Lemma 3.7 inductively. Lemma 3.10.
Assume p = 2 , / . Let u be a non-zero element of Ω M (1) + ( W ) with I ( E, x ) u = 0 . We write t = ǫ ( E, u ) (3.61) for simplicity. We set v = ( ω − ( t + 1) p ) u. (3.62) We have E t v = ( ω − ( t + 1 − p ) p ) E t u. (3.63) (1) Assume H u = 0 . If E t v = 0 , then t = p − and ω ( E t v ) = E t v. (3.64) If u is an eigenvector of ω and v = 0 , then t = p − and ω u = p u. (3.65) (2) If ω u = u and H u = u , then t = 0 .(3) Assume ω u = (1 / u and H u = ( − / u . Then t = p/ − or ( p − / . In particular if p is aneven integer, then t = p/ − .(4) Assume ω u = (9 / u and H u = (15 / u . Then t = p/ − or ( p − / . In particular if p is aneven integer, then t = p/ − .Proof. We first expand each of Q (4) t +4 , Q (5 , t +5 , Q (5 , t +5 , and Q (6) t +6 so that the resulting expression is a linearcombination of elements of the form a (1) i · · · a ( l ) i l E m b (1) j · · · b ( n ) j n (3.66)where l, n ∈ Z ≥ , m ∈ Z , and( a (1) , i ) , . . . , ( a ( l ) , i l ) ∈ { ( ω, k ) | k ≤ } ∪ { ( H, k ) | k ≤ } , ( b (1) , j ) , . . . , ( b ( n ) , j n ) ∈ { ( ω, k ) | k ≥ } ∪ { ( H, k ) | k ≥ } , (3.67)as was done in the proof of (3.48)–(3.52) in Lemma 3.7. Then, taking the action of each expansion on u , we ertex algebra associated to an even lattice t + 1 − p ) − pω ) × ((16 p + 3) t + ( − p + 58 p ) t + 8 p + ( − ω − p + (22 ω + 42) p − ω ) E t u + 2 p ( p − p − p − E t H u, (3.68)0 = ((( t + 1 − p ) − pω )) × (cid:0) (8 p − t + (88 p + 26 p − t + ( − p + (16 ω + 375) p + ( − ω − p − t + 50 p + ( − ω − p + (184 ω + 306) p + ( − ω − p + 24 ω − (cid:1) E t u + 2( p − p − p − p + 6) t + 10 p − p + 10 p + 6) E t H u, (3.69)0 = ((( t + 1 − p ) − pω )) × (cid:0) (16 p + 61 p − t + (216 p + 326 p − p − t + ( − p + (40 ω + 349) p + ( − ω + 1772) p + ( − ω − p + 48 ω − t + 120 p + ( − ω − p + (200 ω − p + (646 ω + 1914) p + ( − ω − p + 480 ω − (cid:1) E t u + 2( p − p − (cid:0) (14 p + 21 p − p + 60) t + 24 p − p − p + 220 p + 60 (cid:1) E t H u, (3.70)0 = 5(( t + 1 − p ) − pω ) × (cid:0) (12 p − p + 6) t + (24 p + 174 p − p + 114) t + (1056 p − p + (24 ω + 2373) p + ( − ω − p + (12 ω − p + 411) t + ( − p + (352 ω + 5873) p + ( − ω − p + (1220 ω + 12088) p + ( − ω − p + (756 ω − p − ω + 414) t + 528 p + ( − ω − p + (2820 ω + 6912) p + (48 ω − ω − p + ( − ω + 7242 ω + 11310) p + (24 ω − ω − p + (3060 ω + 42) p − ω + 180 (cid:1) E t u + ( p − p − (cid:0) (1056 p − p + 1428 p − p + 1992 p − t + 792 p + (176 ω − p + ( − ω + 8666) p + (628 ω − p + ( − ω + 11014) p + (192 ω + 120) p − (cid:1) E t H u. (3.71)We also expand each of Q (4) t +4 , Q (5 , t +5 , Q (5 , t +5 and Q (6) t +6 so that the resulting expression is a linear combinationof elements of the form a (1) i · · · a ( l ) i l E m b (1) j · · · b ( n ) j n (3.72)where l, n ∈ Z ≥ , m ∈ Z , and( a (1) , i ) , . . . , ( a ( l ) , i l ) ∈ { ( ω, k ) | k ≤ } ∪ { ( H, k ) | k ≤ } , ( b (1) , j ) , . . . , ( b ( n ) , j n ) ∈ { ( ω, k ) | k ≥ } ∪ { ( H, k ) | k ≥ } . (3.73) ertex algebra associated to an even lattice u , we have0 = E t ((1 + t ) − pω ) × (cid:0) (16 p + 3) t + ( − p + 36 p + 12) t + 4 p + ( − ω − p + (22 ω + 14) p − ω + 12 (cid:1) u + E t H (cid:0) p − p + 98 p − p (cid:1) u, (3.74)0 = E t ((1 + t ) − pω ) × (cid:0) (8 p − t + (72 p + 44 p − t + ( − p + (16 ω + 166) p + ( − ω + 115) p − t + 20 p + ( − ω − p + (184 ω + 52) p + ( − ω + 112) p + 24 ω − (cid:1) u + E t H (cid:0) (24 p − p + 98 p − p + 24) t + 40 p − p + 620 p − p − p + 24 (cid:1) u, (3.75)0 = E t ((1 + t ) − pω ) × (cid:0) (16 p + 61 p − t + (176 p + 332 p + 21 p − t + ( − p + (40 ω + 106) p + ( − ω + 1088) p + ( − ω + 74) p + 48 ω − t + 48 p + ( − ω − p + (200 ω − p + (646 ω + 678) p + ( − ω + 420) p + 480 ω − (cid:1) u + E t H (cid:0) (56 p − p − p + 1064 p − p + 240) t + 96 p − p + 112 p + 2730 p − p + 280 p + 240 (cid:1) u, (3.76)0 = E t ((1 + t ) − pω ) × (cid:0) (60 p − p + 30) t + (1140 p − p + 570) t + (3520 p − p + (120 ω + 4970) p + ( − ω + 1675) p + (60 ω − p + 2505) t + ( − p + (1760 ω + 11230) p + ( − ω − p + (7180 ω + 13010) p + ( − ω + 6570) p + (3780 ω − p − ω + 4320) t + (880 p + ( − ω − p + (14340 ω + 7480) p + (240 ω − ω − p + ( − ω + 37410 ω + 2050) p + (120 ω − ω + 13280) p + (15300 ω − p − ω + 2700) (cid:1) u + E t H (cid:0) (1760 p − p + 4310 p − p + 13100 p − p + 5850 p − t + (1760 p + (352 ω − p + ( − ω + 53484) p + (3568 ω − p + ( − ω + 114333) p + (3400 ω − p + ( − ω + 22384) p + (384 ω + 2106) p − (cid:1) u. (3.77)Note that ω ’s are on the left side of E t in (3.68)–(3.71), but are on the right side of E t in (3.74)–(3.77). ertex algebra associated to an even lattice H u = 0 , E t v = ( ω − ( t + 1 − p ) / (2 p )) E t u = 0, and t = p −
2. By (3.68),0 = (2( p − p − ω − (3 t + 42 p + 58 tp + 16 t p − p − tp + 8 p )) E t v. (3.78)Then, by (3.69) and (3.70),0 = (10 p − p + 3)(2 t − p + 2)((8 p − t − p + 16 p − , p + 16 p − p + 15)(2 t − p + 2)((8 p − t − p + 16 p −
12) (3.79)and hence 0 = (2 t − p + 2)((8 p − t − p + 16 p − . (3.80)By (3.70) and (3.78),0 = (4032 p − p + 3360 p − p + 774 p − t + (11264 p − p + 71692 p − p + 82959 p − p + 12078 p − t + ( − p + 62768 p − p + 228928 p − p + 218205 p − p + 26244 p − t + 2816 p − p + 72152 p − p + 193478 p − p + 147876 p − p + 15282 p − . (3.81)If 2 t − p + 2 = 0, then t = 0 and by (3.81)0 = t (4 t + 3)(352 t + 1356 t + 2080 t + 1452 t + 385) , (3.82)a contradiction. If (8 p − t − p + 16 p −
12 = 0, then as polynomials in p , the right-hand side of (3.81)divided by (8 p − t − p + 16 p −
12 leaves a remainder of0 = (10560 p − t + (76320 p − / t + (471705 / p − / t + (3187935 / p − / t + (3012585 / p − / t + (202005 p − / t + 45900 p − . (3.83)Moreover, (8 p − t − p + 16 p −
12 divided by the right-hand side of (3.83) leaves a remainder of0 = − t + 1)(4 t + 3) (7 t + 4)(7 t + 10 t + 7)(5632 t + 40704 t + 125788 t + 212529 t + 200839 t + 107736 t + 24480) × (94556 t + 495381 t + 1010052 t + 931824 t + 326592) (3.84)and hence t = −
1, which implies p = 1 / t = p − ω u = u and H u = u . By (3.74) and (3.75), tg ( p ) = tg ( p ) = 0 where g ( p ) = 3( t + 1) ( t + 4) + 2(4 t + 7)(2 t + 5 t + 6) p − t + 45 t + 70) p + 4( t + 10) p ,g ( p ) = − t + 1)( t + 2)(3 t + 12 t + 17)+ (8 t + 60 t + 211 t + 300 t + 117) p + 2(36 t + 155 t + 292 t + 279) p − t + 224 t + 409) p + 20( t + 11) p . (3.85) ertex algebra associated to an even lattice g ( p ) and g ( p ) in p are at most 3 and 4 respectively. Assume t = 0. By(3.85), 0 = 2(10 + t ) g ( p ) − ( −
240 + 176 t + 51 t + t + 10(11 + t )(10 + t ) p ) g ( p )= − t + 1)(7 t + 212 t + 1837 t + 6152 t + 9064 t + 5840) − t + 465 t + 90 t − t − t − p + 6(61 t + 409 t + 866 t − t − p , (3.86)which is a polynomial of degree at most 2 in p . Repeating this procedure to decrease the degrees ofpolynomials in p , we finally obtain0 = ( − t )(1 + t )(2 + t ) (3 + t )( −
20 + t + 33 t ) × (433 + 235 t + 67 t + 33 t ) × ( − − t + 866 t + 409 t + 61 t ) × (5020 + 14072 t + 18476 t + 13940 t + 6272 t + 1580 t + 175 t ) . (3.87)Since t is an integer, it follows from (3.74), (3.75), and (3.87) that ( t, p ) = ( − , , (1 , , ( − , − , / t = 0. If ω u = (1 / u (resp. (9 / u ) and H u = ( − / u (resp.(15 / u ), then the same argument as above shows the results. Lemma 3.11.
Assume p = 2 . Let u be a non-zero element of Ω M (1) + ( W ) with I ( E, x ) u = 0 . We write t = ǫ ( E, u ) (3.88) for simplicity. Then, ( H E ) t +3 u = − t ) E t (36 + 72 H + 142 t + 123 t + 18 t + t + 156 ω − tω + 8 t ω − ω ) u. (3.89) We set v = ( ω − ( t + 1) u. (3.90) We have E t v = ( ω − ( t − E t u. (3.91) (1) Assume H u = 0 . If E t v = 0 , then t = 0 and ω E v = E v. (3.92) If u is an eigenvector of ω and v = 0 , then ω u = u. (3.93) (2) Let ( ζ, ξ ) ∈ { (1 , , (1 / , − / , (9 / , / } . If ω u = ζu and H u = ξu , then t = 0 . ertex algebra associated to an even lattice Proof.
Taking the ( t + 3)-th action of (3.36), the ( t + 4)-th action of (3.37), the ( t + 5)-th action of (3.38),and the ( t + 7)-th action of (3.39) on u , we have0 = t ((1 − t ) − ω ) E t u (3.94)= E t t ((1 + t ) − ω ) u, (3.95)0 = 9(4 + 7 t )( H E ) t +3 u + 72 E t H u + (36 + 298 t + 35 t + 26 t + t + 156 ω − tω + 8 t ω − ω ) E t u (3.96)= E t (cid:16) ( t + 1)( t + 17 t + 106 t + 36) + 4(2 t − t + 39) ω − ω (cid:17) u + 9(7 t + 4)( H E ) t +3 u + 72 E t H u, (3.97)0 = 9( − − t + 33 t − ω )( H E ) t +3 u + 72( −
155 + 44 t ) E t H u − t + 1576 t − t + 62 t + 2 t + 2685 ω + 32 tω − t ω + 16 t ω − ω − tω ) E t u, (3.98)= E t (cid:16) − t + 1)(2 t + 44 t − t + 1222 t + 585) − t + 992 t + 2685) ω + 768( t + 5) ω (cid:17) u + 72(44 t − E t H u + ( H E ) t +3 (cid:0) t − t − − ω (cid:1) u, (3.99) ertex algebra associated to an even lattice
230 = − t + 1097587588 t + 5494080415 t − t + 68909700044 t + 2468574039524 t + 3786872840265 t + 493804109430) E t u + 16(52787700 t + 336785348 t + 16075086171 t − t − t − ω E t u + 3(5886227459 t + 64230119866 t − t − t − H E ) t +3 u + 72(743028209 t + 17731219498 t + 23020475889 t − E t H u − t + 9549468148 t + 19688188167) ω ( H E ) t +3 u − ω ω ( H E ) t +3 u + 1536(93467197 t + 449390927 t + 1282501650) ω ω E t u + 14515200(931 t + 661) ω ω ω E t u + 67132800(931 t + 661) ω E t H u + 35590023000( H E ) t +3 H u (3.100)= − t + 1)(52787700 t + 1150375288 t + 5017275823 t − t − t + 959628223785 t + 493804109430) E t u + 16(52787700 t + 336785348 t + 663193947 t − t − t − E t ω u + 3(5886227459 t + 67153865586 t − t − t − H E ) t +3 u + 72(743028209 t + 16863155098 t + 22404159489 t − E t H u − t + 8602993948 t + 19688188167)( H E ) t +3 ω u − H E ) t +3 ω u + 1536(67073347 t + 430651577 t + 1282501650) E t ω u + 14515200(931 t + 661) E t ω u + 67132800(931 t + 661) E t H ω u + 35590023000( H E ) t +3 H u. (3.101)Note that ω ’s are on the left side of E t in (3.94), (3.96), (3.98), and (3.100), but are on the right side of E t in (3.95), (3.97), (3.99), and (3.101). By Lemma 3.3 and (3.96), we have (3.89). Deleting the terms including( H E ) t +3 u from the simultaneous equations (3.96), (3.98), and (3.100), we have0 = 360( − t + 11 t + 4 ω ) E t H u + ((1 − t ) − ω ) (cid:0) − t ( − − t + 745 t + 29 t ) − − t + 82 t ) ω − ω (cid:1) E t u, − − t + 11 t + 4 ω ) E t H u + ((1 − t ) − ω ) (cid:0) t + 745 t + (328 ω − t + ( − ω − t − ω + 240 ω (cid:1) E t u. (3.102)By (3.94), t = 0 or ((1 − t ) − ω ) E t u = 0. If t = 0 and H u = 0, then by (3.102),0 = ( ω − ω −
14 ) E t u, (3.103)which finishes (1).By using (3.95), (3.97), (3.99), and (3.101), the same argument as above shows (2). ertex algebra associated to an even lattice Lemma 3.12.
Assume p = 1 / . Let U be an A ( M (1) + ) -submodule of Ω M (1) + ( W ) , u a simultaneouseigenvector of { ω , H } in U with I ( E, x ) u = 0 . We write t = ǫ ( E, u ) (3.104) for simplicity. Then, ω u = (1 + t ) u. (3.105) If t = − , then ( H E ) t +2 u = E t (cid:0) t )(3 + 2 t ) H + (1 + t ) (cid:1) u, (3.106) and if t = − , then H u = 0 . (3.107) Proof.
Taking the ( t + 2)-th action of (3.23) on u , we have ω E t u = ( t + 12 ) E t u (3.108)and hence (3.105) holds. Taking the ( t + 3)-th action of (3.24) and the ( t + 4)-the action of (3.25) on u , wehave 0 = 8( t + 1) E t ω u − ( t + 1)(11 t + 15)( H E ) t +2 u + 4( t + 1) E t u + 12 E t H u + 3( H E ) t +2 ω u, (3.109)and hence (3.106) and (3.107) by (3.105).Combining Lemmas 3.7, 3.10, 3.11, 3.12, and [21, Theorem 6.2], we have the following result: Proposition 3.13.
Let L be a non-degenerate even lattice of rank and M a non-zero weak V + L -module.Then, there exists an irreducible A ( M (1) + ) -submodule of Ω M (1) + ( M ) . In particular, there exists a non-zero M (1) + -submodule of M . Appendix A1
In this appendix, for some a, b ∈ V L , we put the computations of a k b for k ∈ Z ≥ . For k ∈ Z ≥ not listedbelow, a k b = 0. Using these results, we can compute the commutation relation [ a i , b j ] = P ∞ k =0 (cid:0) ik (cid:1) ( a k b ) i + j − k . A1-1 Computations in M (1) ω ω = ω ω − , ω ω = 2 ω − , ω ω = 0 , ω ω = 12 , (A1.1) ω H = ω H − , ω H = 4 H − , ω H = − ω ω − ,ω H = 2 ω − , ω H = 0 , ω H = − , (A1.2) ertex algebra associated to an even lattice H H = 2 ω ω − ω − + 245 ω ω − H − + − ω ω − ω − + − ω H − + 120 ω ω − ,H H = 4 ω − ω − + 485 ω − H − + − ω ω − ω − + 1615 ω H − + 745 ω ω − ,H H = 10 ω H − + 718 ω ω − ,H H = 20 H − + 43 ω ω − ,H H = 103 ω ω − ,H H = 203 ω − ,H H = 0 ,H H = 53 . (A1.3) A1-2 The case that h α, α i 6 = 0 , / , , and Let p be a complex number with p = 0 , / , , α ∈ h such that h α, α i = p . H E = 2 pp − ω − E + − p ( p − p − ω ω − E + 2( p − p − ω E,H E = 2 p p − ω − E + − p − ω ω E,H E = 13 ω E. (A1.4) A1-3 The case that h α, α i = 2 Let α ∈ h with h α, α i = 2. ω H E = ω ( H E ) , ω H E = 4( H E ) , ω H E = 8 ω − E − ω ω E,ω H E = 6 ω E, ω H E = 12 E, (A1.5) H E = H E, H E = 43 ω − E + − ω ω E, H E = 13 ω E, (A1.6) H H E = 40965145 ω − ω − ω − E + 21041246305 ω − ( H E )+ 189445145 ω − H − E + − H − E + 588836015 ω ω − ( H E ) + 3032168324135 ω H − E + − ω ω ω − ω − E + − ω ω H − E + − ω ω ω ( H E ) + 32226464218791125 ω ω ω ω ω − E + − ω ω ω ω ω ω E, (A1.7) ertex algebra associated to an even lattice H H E = 367 ω − ( H E ) + 807 H − E + 10881575 ω ω − ω − E + 1496525 ω H − E + − ω ω ( H E ) + − ω ω ω ω − E + − ω ω ω ω ω E,H H E = 12825 ω − ω − E + 52825 H − E + − ω ( H E )+ − ω ω ω − E + − ω ω ω ω E,H H E = 27( H E ) ,H H E = 48 ω − E − ω ω E,H H E = 20 ω E. (A1.8) A1-4 The case that h α, α i = 1 / Let α ∈ h with h α, α i = 1 / ω E = ω E, ω E = 14 E, (A1.9) ω H E = ω ( H E ) , ω H E = 94 ( H E ) , ω H E = 2 ω E, ω H E = E, (A1.10) H E = − ω − E + 43 ω ( H E ) , H E = ( H E ) , H E = 13 ω E, (A1.11) H H E = − H − E + 4225 ω − ( H E ) + 12625 ω H − E + 2225 ω ω − ( H E )+ − ω ω ω − E + − ω ω ω ( H E ) ,H H E = − H − E + 52 ω − ( H E ) + − ω ω − E + 296 ω ω ( H E ) ,H H E = − ω − E + 253 ω ( H E ) ,H H E = 8( H E ) ,H H E = 4 ω E,H H E = 13 E. (A1.12) ertex algebra associated to an even lattice Notation V a vertex algebra. U a subspace of a weak V -module.Ω V ( U ) = { u ∈ U (cid:12)(cid:12)(cid:12) a i u = 0 for all homogeneous a ∈ V and i > wt a − . } (2.4). p a non-zero complex number. h a finite dimensional vector space equipped with a nondegenerate symmetric bilinearform h , i . h an element of h with h h, h i = 1. h [1] , . . . , h [ d ] an orthonormal basis of h . α an element of h with h α, α i = p . M (1) the vertex algebra associated to the Heisenberg algebra. L a non-degenerate even lattice of finite rank. d the rank of L . V L the vertex algebra associated to L . θ the automorphism of V L induced from the − L . M (1) + the fixed point subalgbra of M (1) under the action of θ . V + L the fixed point subalgbra of V L under the action of θ . K, M, N, W weak M (1) + (or V + L )-modules. I ( , x ) an intertwining operator for M (1) + . ǫ ( u, v ) u ǫ ( u,v ) v = 0 and u i v = 0 for all i > ǫ ( u, v ) if I ( u, x ) v = 0 and ǫ ( u, v ) = −∞ if I ( u, x ) v = 0, where I : M × W → N (( x )) is an intertwining operator and u ∈ M , v ∈ W (2.7). h ω i i X the space spanned by the elements ω ji u, j ∈ Z ≥ , u ∈ X . A ( V ) the Zhu algebra of a vertex operator algebra V . ω = (1 / h ( − or (1 / P di =1 h [ i ] ( − . H = (1 / h ( − h ( − − h ( − ). J = h ( − − h ( − h ( − + (3 / h ( − = − H + 4 ω − − ω − . E = E ( α ) = e α + θ ( e α ) where α ∈ h . t an integer such that t ≥ ǫ ( E, u ) or t = ǫ ( E, u ) for a given non-zero element u . Acknowledgments
The author thanks the referee for pointing out a mistake in the proof of the earlier version of Proposition3.13.
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