Table Of ContentORBIFOLD CONSTRUCTION OF HOLOMORPHIC VERTEX
OPERATOR ALGEBRAS ASSOCIATED TO INNER
AUTOMORPHISMS
5
1
CHING HUNG LAM AND HIROKI SHIMAKURA
0
2
g Abstract. In this article, we construct three new holomorphic vertex operator alge-
u
bras of central charge 24 using the Z2-orbifold construction associated to inner auto-
A
morphisms. Their weight one subspaces has the Lie algebra structures D7,3A3,1G2,1,
24 E7,3A5,1, and A8,3A22,1. In addition, we discuss the constructions of holomorphic vertex
operator algebras with Lie algebras A5,6C2,3A1,2 and D6,5A21,1 from holomorphic vertex
A] operator algebras with Lie algebras C5,3G2,2A1,1 and A24,5, respectively.
Q
.
h
t
a 1. Introduction
m
[
The classification of holomorphic vertex operator algebras (VOAs) of central charge 24
2
is one of the fundamental problems in vertex operator algebras and mathematical physics.
v
4 In 1993, Schellekens [Sc93] obtained a partial classification by determining possible Lie
9
0 algebra structures for the weight one subspaces of holomorphic VOAs of central charge
5
24. There are 71 cases in his list but only 39 of the 71 cases were known explicitly at that
0
.
1 time. It is also an open question if the Lie algebra structure of the weight one subspace
0
will determine the VOA structure uniquely when the central charge is 24.
5
1
In [La11, LS12, LS15], a special class of holomorphic VOAs, called framed VOAs, of
:
v
central charge 24 were studied and classified. In particular, it was shown in [LS15] that
i
X
thereexistexactly56holomorphicframedVOAsofcentralcharge24andtheyareuniquely
r
a
determined by the Lie algebra structures of their weight one subspaces. On the other
hand, a Z -orbifold theory associated to lattice VOAs has been developed by Miyamoto
3
[Mi13] and as an application, a holomorphic VOA whose weight one subspace has the
Lie algebra structure E G3 was constructed. By using the similar methods, several
6,3 2,1
other holomorphic VOAs have been constructed in [SS]. Recently, van Ekeren, M¨oller
and Scheithauer [EMS] announced that they have obtained a mathematically rigorous
proof for Schellekens’ list. They also claimed that they have established the Z -orbifold
n
construction for general elements of arbitrary orders. In particular, they claimed that
2010 Mathematics Subject Classification. Primary 17B69.
C.H. Lam was partially supported by NSC grant 100-2628-M-001005-MY4of Taiwan.
H. Shimakura was partially supported by JSPS KAKENHI Grant Numbers 23540013 and 26800001,
and by Grant for Basic Science Research Projects from The Sumitomo Foundation.
1
they can construct two holomorphic VOAs of central charge 24 such that their weight one
subspaces have the Lie algebras structures E C A and A2 . By the results and the
6,4 2,1 2,1 4,5
announcement above, there are 10 remaining Lie algebras in Schellekens’ list for which
the corresponding holomorphic VOAs of central charge 24 have not been constructed yet.
The main purpose of this article is to construct new holomorphic VOAs of central
charge 24 by using the Z -orbifold construction associated to inner automorphisms. More
2
precisely, three new VOAs are constructed, and two new VOAs can be constructed from
unestablished holomorphic VOAs of central charge 24. The main theorem is the following
(see Theorems 6.9, 7.7, 8.6, 9.6 and 10.10 for details):
Theorem 1.1. (1) There exist strongly regular, holomorphic VOAs of central charge 24
with Lie algebras D A G , E A and A A2 .
7,3 3,1 2,1 7,3 5,1 8,3 2,1
(2) If there exists a strongly regular, holomorphic VOA of central charge 24 with Lie
algebra C G A , then there exists a strongly regular, holomorphic VOA of central
5,3 2,2 1,1
charge 24 with Lie algebra A C A .
5,6 2,3 1,2
(3) If a strongly regular, holomorphic VOA of central charge 24 with Lie algebra A2
4,5
can be obtained by applying the Z -orbifold construction to the lattice VOA associated
5
to the Niemeier lattice with root lattice A6 as in 10.1, then there exists a strongly
4 §
regular, holomorphic VOA of central charge 24 with Lie algebra D A2 .
6,5 1,1
In order to prove this theorem, we choose a holomorphic VOA V and its inner auto-
morphism σ of order 2 carefully. Then, applying the Z -orbifold construction to V and
h 2
σ , we obtain a new holomorphic VOA V˜ with the desired Lie algebra. We summarize the
h
Lie algebra structures of V , (Vσh) and V˜ in Table 1, where Vσh is the set of fixed-points
1 1 1
of σ .
h
Table 1. Lie algebra structures of V , (Vσh) and V˜
1 1 1
(Original) Lie algebra V (Fixed point) Lie subalgebra (Vσh) (New) Lie algebra V˜
1 1 1
E G3 D A2 A2 G U(1) D A G
6,3 2,1 5,3 1,1 1,3 2,1 7,3 3,1 2,1
D A G D A A A U(1) E A
7,3 3,1 2,1 6,3 3,1 1,1 1,3 7,3 5,1
E A A A2 U(1) A A2
7,3 5,1 7,3 2,1 8,3 2,1
C G A A A A U(1)2 A C A
5,3 2,2 1,1 4,6 1,6 1,2 5,6 2,3 1,2
A2 A2 U(1)2 D A2
4,5 3,5 6,5 1,1
We note that van Ekeren, M¨oller and Scheithauer announced that the assumption
in Theorem 1.1 (3) is true [EMS]. We also notice that a holomorphic VOA with Lie
algebra C G A would be constructed by using Z -orbifold theory to the lattice VOA
5,3 2,2 1,1 6
associated to the Niemeier lattice with root lattice E4, which will be discussed in our
6
future article.
2
By the result in this article, there are remaining 5 cases in Schellekens’ list which have
not been constructed yet. The corresponding Lie algebras have the type C , D A ,
4,10 4,12 2,6
A , F A , and C G A .
6,7 4,6 2,2 5,3 2,2 1,1
Let us explain our construction in more detail. First, we recall the Z -orbifold con-
2
struction associated to an inner automorphism. Let V be a strongly regular, holomorphic
VOA and let h V . Assume that h is semisimple on V and the associated inner auto-
1 (0)
∈
morphism σ = exp( 2π√ 1h ) of V has order 2. Using Li’s ∆-operator introduced in
h (0)
− −
[Li96], we construct the (unique) irreducible σ -twisted V-module V(h) explicitly. It was
h
shown in [DLM96] that V V(h) has an abelian intertwining algebra structure. Hence
⊕
one can see that the subspace V˜ = Vσh (V(h))Z has a VOA structure as a simple current
⊕
extension of Vσh, where Vσh is the set of fixed-points of σh and (V(h))Z is the subspace
of V(h) with integral L(0)-weights. If V and h satisfy some conditions (see Theorem 5.4
for detail), then V˜ is of CFT-type. By a similar argument as in [Li97], we see that V˜ is
C -cofinite and holomorphic. In addition, we prove that the Lie algebras V and V˜ share
2 1 1
a common Cartan subalgebra under some assumptions.
Next we check that our choices of V and h fit the Z -orbifold construction above. Let
2
V be a strongly regular, holomorphic VOA of central charge 24 such that the Lie algebra
structure of V is one of the Lie algebra structures in column one of Table 1. Then we can
1
easily find h V so that (Vσh) has the Lie subalgebra structure in the corresponding
1 1
∈
column two in Table 1 and h satisfies the necessary conditions. Clearly, the order of σ
h
on V is 2; however, we shall show that the order of σ on V is 2, also. For this purpose,
1 h
we consider the subVOA U generated by V , which is a full subVOA of V ([DM04a]). It
1
suffices to show that the order of σ is 2 on every irreducible U-submodule of V. Recall
h
that U is the tensor product of simple affine VOAs L (k ,0) associated with simple Lie
gi i
algebras g at positive integral levels k ([DM06a]). Hence any irreducible U-module is
i i
the tensor product of irreducible L (k ,0)-modules, which are classified in [FZ92]. Since
gi i
the L(0)-weights of V are integral, there are not so many possibilities for irreducible U-
submodules of V. We can check that the order of σ is 2 for each possibility. Hence σ is
h h
of order 2 on V. Unfortunately, this argument does not work for the case (3) of Theorem
1.1. For this case, we directly check this assertion by using the explicit description of the
lattice VOA and its irreducible σ -twisted module ([Le85, DL96]).
h
Finally, we explain how to determine the Lie algebra structure of V˜ . A key tool is the
1
dimension formula mentioned in [Mo94]. We prove it by the following way: We check
that the character of Vσh converges to a modular function of weight 0 on the congruence
subgroup Γ (2), and we express it as a Laurent polynomial of a Hauptmodul of Γ (2).
0 0
Substituting it to certain equations about the characters of V, V˜ and Vσh and comparing
some coefficients of the q-expansions, we obtain the formula. By this formula, dimV˜ can
1
3
be determined by dim(V(h)) . Indeed, in all our cases, (V(h)) = 0. It follows from
1/2 1/2
the dimension formula directly or by establishing that (M(h)) = 0 for each possible
1/2
irreducible U-submodule M of V. In the case (3) of Theorem 1.1, we also check it
directly. By [DM04a], V˜ is semisimple, and the ratio of the dual Coxeter number and
1
the level for every simple ideal of V˜ is determined by dimV˜ . By using the fact that V
1 1 1
and V˜ share a common Cartan subalgebra H, any simple Lie subalgebra of V˜ spanned
1 1
by weight vectors for H is contained in a unique simple ideal of V˜ , and the level of the
1
simple ideal can be determined. For example, any simple ideal of (Vσh) (cf. Table 1) is
1
spanned by weight vectors for H since σ is an inner automorphism. Thus we have enough
h
data of V˜ to determine its Lie algebra structure in each case.
1
The organization of the article is as follows. In Section 2, we recall some preliminary
results about strongly regular, holomorphic VOAs. In Section 3, we recall the definition
of Li’s ∆-operator and the associated construction of the σ -twisted module for h V .
h 1
∈
We also discuss some basic properties of simple affine VOAs and their twisted modules
constructed by the ∆-operator. In Section 4, we prove the dimension formula for the Z -
2
orbifold construction associated to an inner automorphism. In Section 5, we discuss the
Z -orbifold construction associated to inner automorphisms. We also discuss some prop-
2
erties of the resulting VOAs. In Sections 6,7 and 8, we apply the construction successively
and obtain three holomorphic VOAs of central charge 24 with Lie algebras D A G ,
7,3 3,1 2,1
E A , and A A2 . Finally, in Sections 9 and 10, we will discuss the constructions of
7,3 5,1 8,3 2,1
holomorphic VOAs of central charge 24 with Lie algebras A C A and D A2 from
5,6 2,3 1,2 6,5 1,1
holomorphic VOAs with Lie algebras C G A and A2 using the similar methods,
5,3 2,2 1,1 4,5
respectively.
4
Notations
( ) the normalized Killing form on a semisimple Lie algebra
·|·
so that (αα) = 2 for long roots α.
|
the normalized symmetric invariant bilinear form on a VOA
h·|·i
so that 1 1 = 1, equivalently, a b 1 = a b for a,b V .
(1) 1
h | i − h | i ∈
a(h) the n-th mode of an element a V on the σ -twisted V-module M(h).
(n) ∈ h
α a simple root of a root system.
i
E a root vector in a simple Lie algebra with respect to root α.
α
h the dual Coxeter number of a simple Lie algebra.
∨
h the subspace of fixed-points of τ in h = C N(A6).
(0) 0 ⊗Z 4
θ the highest root with respect to a fixed set of simple roots.
L(0) the weight operator ω .
(1)
L(h)(0) the weight operator ω(h) on a σ -twisted module M(h).
(1) h
L (k,0) the simple affine VOA associated with simple Lie algebra g at level k.
g
L (k,λ) the irreducible L (k,0)-module with highest weight λ.
g g
Λ the fundamental weight with respect to simple root α .
i i
M(h) the σ -twisted V-module constructed from a V-module M by Li’s ∆-operator.
h
N = N(A6) a Niemeier lattice with root lattice A6.
4 4
Π(λ,X ) the set of all weights of the irreducible module with highest weight λ
n
over the simple Lie algebra of type X .
n
ρ half of the sum of all positive roots.
σ the inner automorphism exp( 2π√ 1h ) of a VOA V associated to h V .
h (0) 1
− − ∈
τ the order 5 automorphism of the lattice VOA V defined in 10.1.
0 N(A64) §
Spec h the set of spectra of h for a semisimple element h V .
(0) (0) 1
∈
U(1) a 1-dimensional abelian Lie algebra.
Vσh the set of fixed-points of σ , which is a full subVOA of V.
h
X (the type of) a root system, a simple Lie algebra or a root lattice.
n
X (the type of) a simple Lie algebra whose type is X and level is k.
n,k n
2. Preliminary
In this section, we will review some fundamental results about VOAs.
2.1. Vertex operator algebras. Throughout this article, all VOAs are defined over the
field C of complex numbers. We recall the notion of vertex operator algebras (VOAs) and
modules from [Bo86, FLM88, FHL93].
A vertex operator algebra (VOA) (V,Y,1,ω) is a Z-graded vector space V = V
m Z m
∈
equipped with a linear map
L
Y(a,z) = a z i 1 (End(V))[[z,z 1]], a V
(i) −− −
∈ ∈
i Z
X∈ 5
and the vacuum vector 1 and the conformal vector ω satisfying a number of conditions
([Bo86, FLM88]). We often denote it by V. For a V and n Z, we often call a the
(n)
∈ ∈
n-th mode of a. Note that L(n) = ω satisfy the Virasoro relation:
(n+1)
1
[L ,L ] = (m n)L + (m3 m)δ c id ,
(m) (n) (m+n) m+n,0 V
− 12 −
where c is a complex number, called the central charge of V.
A linear automorphism of V is called an automorphism of V if it satisfies
gω = ω and gY(v,z) = Y(gv,z)g for all v V.
∈
Avertex operatorsubalgebra(orasubVOA)isagradedsubspaceofV whichhasastructure
of a VOA such that the operations and its grading agree with the restriction of those of
V and that they share the vacuum vector. When they also share the conformal vector,
we will call it a full subVOA. For an automorphism g of a VOA V, let Vg denote the set
of fixed-points of g. Clearly Vg is a full subVOA of V.
An (ordinary) V-module (M,Y ) is a C-graded vector space M = M equipped
M m C m
∈
with a linear map
L
Y (a,z) = a z i 1 (End(M))[[z,z 1]], a V
M (i) −− −
∈ ∈
i Z
X∈
satisfying a number of conditions ([FHL93]). We often denote it by M. For an automor-
phism g of V, we also consider a g-twisted V-module. For the detail, see [Li96, DLM00]
and references therein. Note that a g-twisted V-module is an (untwisted) Vg-module.
The L(0)-weight of a homogeneous vector v M is k, where L(0) = ω . Note that
k (1)
∈
L(0)v = kv if v M .
k
∈
A VOA is said to be rational if any module is completely reducible. A rational VOA
is said to be holomorphic if it itself is the only irreducible module up to isomorphism. A
VOA is said to be of CFT-type if V = C1 (note that V = 0 for all n < 0 if V = C1
0 n 0
[DM06b, Lemma 5.2]), andissaidtobeC -cofiniteifthecodimension inV ofthesubspace
2
spanned by the vectors of form u v, u,v V, is finite. A module is said to be self-dual
( 2)
− ∈
if its contragredient module is isomorphic to itself. It is obvious that a holomorphic VOA
is simple and self-dual. A VOA is said to be strongly regular if it is rational, C -cofinite,
2
self-dual and of CFT-type.
Let V be a VOA of CFT-type. Then, the 0-th mode gives a Lie algebra structure on
V . Moreover, the n-th modes v , v V , n Z, define an affine representation of the
1 (n) 1
∈ ∈
Lie algebra V on V. For a simple algebra a of V , the level of a is defined to be the scalar
1 1
by which the canonical central element acts on V as the affine representation. When the
type of the root system of a is X and the level of a is k, we denote the type of a by X .
n n,k
Assume that V is self-dual. Then there exists a symmetric invariant bilinear form on
h·|·i
6
V, which is unique up to scalar ([Li94]). We normalize it so that 1 1 = 1. Then for
h | i −
a,b V , we have a b 1 = a b. For an element a V , exp(a ) is an automorphism
1 (1) 1 (0)
∈ h | i ∈
of V, called an inner automorphism. For a semisimple element h V , we often consider
1
∈
the inner automorphism σ = exp( 2π√ 1h ) associated to h.
h (0)
− −
Assume that V is semisimple. Let H be a Cartan subalgebra of V . Let ( ) be the
1 1
·|·
Killing formonV . Weidentify H withHvia ( )andnormalize( )sothat (α α) = 2for
1 ∗
·|· ·|· |
any long root α H. In this article, weights for H are defined via ( ), that is, the weight
∈ ·|·
of a vector v V for H is λ H if x v = (x λ)v for all x H. Remark that for h H,
(0)
∈ ∈ | ∈ ∈
σ acts on a vector with weight λ as the scalar multiple by exp( 2π√ 1(h λ)). The
h
− − |
following lemma is immediate from the commutator relations of n-th modes (cf. [DM06a,
(3.2)]).
Lemma 2.1. If the level of a simple algebra of V is k, then = k( ) on it.
1
h·|·i ·|·
Let us recall some results related to the Lie algebra V .
1
Proposition 2.2 ([DM06a, Theorem 1.1, Corollary 4.3]). Let V be a strongly regular,
simple VOA. Then V is reductive. Let s be a simple Lie subalgebra of V . Then V is an
1 1
integrable module for the affine representation of s on V, and the subVOA generated by s
is isomorphic to the simple affine VOA associated with s at positive integral level.
Proposition 2.3 ([DM04b, Theorem 1] and [Ma14, Section 3.3]). Let V be a strongly
regular, simple VOA and let M be a V-module. Then for any element x in a Cartan
subalgebra of V , the 0-th mode x acts semisimply on M.
1 (0)
Proposition 2.4 ([DM04a, (1.1), Theorem 3 and Proposition 4.1]). Let V be a strongly
regular, holomorphic VOA of central charge 24. If the Lie algebra V is neither 0 nor
1
{ }
abelian, then V is semisimple, and the conformal vectors of V and the subVOA generated
1
by V are the same. In addition, for any simple ideal of V at level k, the identity
1 1
h dimV 24
∨ 1
= −
k 24
holds, where h is the dual Coxeter number.
∨
3. ∆-operator, simple affine VOAs and twisted modules
In this section, we recall the twisted module constructed by Li’s ∆-operator. Moreover,
we discuss the lowest L(0)-weight of such a twisted module over simple affine VOAs.
3.1. Twisted modules constructed by Li’s ∆-operator. Let V be a vertex operator
algebra of CFT-type. Let σ be a finite order automorphism of V and let h V with
1
∈
σ(h) = h. We assume that h acts semisimply on V and that there exists a positive
(0)
integer T Z such that Spec h , the set of spectra of h on V, is contained in
>0 (0) (0)
∈
7
(1/T)Z. Then σ = exp( 2π√ 1h ) is an automorphism of V with σT = 1. Note that
h − − (0) h
σσ = σ σ since σ(h) = h.
h h
Let ∆(h,z) be Li’s ∆-operator defined in [Li96], i.e.,
∆(h,z) = zh(0) exp ∞ h(n)( z)−n .
n −
!
n=1 −
X
Proposition 3.1 ([Li96, Proposition 5.4]). Let σ be an automorphism of V of finite order
and let h V be as above such that σ(h) = h. Let (M,Y ) be a σ-twisted V-module and
1 M
∈
define (M(h),Y ( ,z)) as follows:
M(h) ·
M(h) = M as a vector space;
Y (a,z) = Y (∆(h,z)a,z) for any a V.
M(h) M ∈
Then (M(h),Y ( ,z)) is a σ σ-twisted V-module. Furthermore, if M is irreducible, then
M(h) · h
so is M(h).
(h)
For a σ-twisted V-module M and a V, we denote by a the operator which corre-
∈ (i)
sponds to the coefficient of z i 1 in Y (a,z), i.e.,
−− M(h)
Y (a,z) = a(h)z n 1 for a V.
M(h) (i) − − ∈
i Z
X∈
Now we will review the action of some elements of V on the σ σ-twisted V-module
h
M(h). The 0-th mode of an element x V on M(h) is given by
1
∈
(h)
(3.1) x = x + h x id.
(0) (0) h | i
Let us denote by L(h)(n) the (n + 1)-th mode of the conformal vector ω V on M(h).
∈
Then the L(0)-weights on M(h) are given by
h h
(3.2) L(h)(0) = L(0)+h + h | iid.
(0)
2
The following lemma is immediate from the equation above.
Lemma 3.2. Let M be a σ-twisted V-module whose L(0)-weights are half-integral. Let
h V such that h is semisimple on M and h h Z. Assume that the spectra of h
1 (0) (0)
∈ h | i ∈
on M are half-integral. Then the L(0)-weights of the σ σ-twisted V-module M(h) are also
h
half-integral.
3.2. Simple affine VOAs and irreducible twisted modules. In this subsection, we
recall some properties of simple affine VOAs and their modules from [Ka90, FZ92]. More-
over, we study σ -twisted modules constructed by Li’s ∆-operator.
h
Let g be a simple Lie algebra with Cartan subalgebra H. Let Φ be the set of roots of g.
Let ( ) be the Killing form on g. We identify H with H by ( ) and normalize the form
∗
·|· ·|·
so that (α α) = 2 for any long root α Φ. Let α 1 i n H be a set of simple
i
| ∈ { | ≤ ≤ } ⊂
8
roots and {Λi | 1 ≤ i ≤ n} ⊂ H the set of the fundamental weights so that 2((αΛijα|αii)) = δij.
|
For every β Φ, fix a root vector E in g associated to β.
β
∈
Let V = L (k,0) be the simple affine VOA associated with g at positive integral level
g
k. It was proved in [FZ92] that all irreducible V-modules are given by L (k,λ), where λ
g
ranges over dominant integral weights with (θ λ) k for the highest root θ. By [Ka90,
| ≤
Corollary 12.8], the lowest L(0)-weight of L (k,λ) is given by
g
(λ+2ρ λ)
(3.3) | ,
2(k +h )
∨
where ρ = n Λ and h is the dual Coxeter number. The following facts on L (k,λ) is
i=1 i ∨ g
well-known.
P
Lemma 3.3 ([FZ92, 2]). Let ℓ be the lowest L(0)-weight of L (k,λ). Then the following
g
§
hold:
(1) L (k,λ) is an irreducible g-module with highest weight λ, where V = g via the 0-th
g ℓ 1 ∼
mode;
(2) Let v L (k,λ) . Then the L(0)-weight of E ...E v is ℓ+ m n ;
∈ g ℓ β1( n1) βm( nm) j=1 j
− −
(3) L (k,λ) is spanned by
g P
E ...E v β Φ,n Z ,m Z ,v L (k,λ) .
{ β1(−n1) βm(−nm) | i ∈ i ∈ >0 ∈ ≥0 ∈ g ℓ}
Fromnow on, let hbe an element in H with Spec h (1/T)Zon V for some T Z .
(0) >0
⊂ ∈
Note that h n Qα and that the restriction of the normalized Killing form ( ) to
∈ i=1 i ·|·
n Qα is positive-definite. In addition, we assume
i=1 i L
(L3.4) (h α) 1 for all α Φ.
| ≥ − ∈
Lemma 3.4. Let M be a V-module. Let v be a vector in M(h) with L(0)-weight p,
i.e., L(h)(0)v = pv. Let u = E ...E v M(h) be a non-zero vector, where
β1( n1) βm( nm) ∈
− −
n Z . Then u is a homogeneous vector in M(h) and its L(0)-weight is greater than
i >0
∈
or equal to p. Moreover the equality holds if and only if n = 1 and (h β ) = 1 for all
i i
| −
1 i m.
≤ ≤
Proof. By (3.2) and [L(h)(0),E ] = (n+(h β))id, we have
β( n) |
−
m
L(h)(0)u = (n +(h β ))+p u.
i i
|
!
i=1
X
By the assumption (3.4), n +(h β ) 0 for all i, and hence the L(0)-weight of u in M(h)
i i
| ≥
is greater than or equal to p. The latter assertion is obvious. (cid:3)
Lemma 3.5. The lowest L(0)-weight of the irreducible σ -twisted V-module L (k,λ)(h) is
h g
non-negative. If the lowest L(0)-weight of L (k,λ)(h) is 0, then λ = kΛ and h = Λ for
g j j
−
some fundamental weight Λ , or λ = h = 0.
j
9
Proof. Let v be a lowest L(0)-weight vector in L (k,λ) with weight µ H. By Lemmas
g
∈
3.3 (3) and 3.4, we may assume that v has also the lowest L(0)-weight in L (k,λ)(h). If
g
λ = 0, then v C1 and µ = 0, which proves the assertion by (3.2). We assume that
∈
λ = 0. By Lemma 2.1, h h = k(h h). Hence by (3.2), the L(0)-weight of v in L (k,λ)(h)
g
6 h | i |
is
(λ+2ρ λ) k(h h)
(3.5) | +(h µ)+ | ,
2(k +h ) | 2
∨
which is equal to the lowest L(0)-weight of L (k,λ)(h). Applying 2k(λ ρ) h (λ λ)
g ∨
| ≥ |
([Ka90, Theorem 13.11]) and (λ λ) (µ µ) ([Ka90, Proposition 11.4]) to (3.5), we see
| ≥ |
that the lowest L(0)-weight of L (k,λ)(h) is non-negative since
g
(λ+2ρ λ) k(h h) (µ µ) k(h h) (µ+kh)2
(3.6) | +(h µ)+ | | +(h µ)+ | = 0.
2(k +h ) | 2 ≥ 2k | 2 2k ≥
∨
If the L(0)-weight of v in L (k,λ)(h) is 0, then the all equalities hold in (3.6). The former
g
equality holds if and only if µ = λ = kΛ for some fundamental weight Λ , and the latter
j j
equality holds when µ+kh = 0. Combining them, we obtain λ = kΛ and h = Λ . (cid:3)
j j
−
Lemma 3.6. Let X be the type of the simple Lie algebra g. The lowest L(0)-weight of
n
L (k,λ)(h) is equal to
g
(λ+2ρ λ) k(h h)
(3.7) | +min (h µ) µ Π(λ,X ) + | ,
n
2(k +h ) { | | ∈ } 2
∨
where Π(λ,X ) is the set of all weights for H of the irreducible module with the highest
n
weight λ over the simple Lie algebra of type X .
n
Proof. By Lemmas 3.3 (3) and 3.4, it suffices to consider the lowest L(h)(0)-weight of the
lowest L(0)-weight space of L (k,λ). Hence, this lemma follows from (3.5) and Lemma
g
(cid:3)
3.3 (1).
Later, we will use the following lemma:
Lemma 3.7. Let v be a vector in L (k,λ) with weight µ for H. Then v is also a weight
g
vector in L (k,λ)(h) for H and its weight is µ+kh.
g
(cid:3)
Proof. This lemma follows from (3.1) and Lemma 2.1.
3.3. Lowest L(0)-weight of a twisted module. In this subsection, we give a sufficient
condition so that the lowest L(0)-weight of the σ -twisted V-module V(h) is positive.
h
Proposition 3.8. Let V be a strongly regular, simple VOA. Assume that the Lie algebra
g = V is semisimple. Let g = t g be the decomposition into the direct sum of t simple
1 i=1 i
ideals g . Let U be the subVOA of V generated by V . Let h be an element in a (fixed)
i L 1
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