Integral forms in vertex operator algebras which are invariant under finite groups

0.19 MB·English

by  Robert L. Griess Jr

January 15, 2012
Integral forms in vertex operator algebras
2
which are invariant under finite groups
1
0
2

n
a Chongying Dong
J
Department of Mathematics,

7
University of California,
1
Santa Cruz, CA 95064 USA

]
T [email protected]
R
. and
h
t
a
m Robert L. Griess Jr.
[ Department of Mathematics,

1 University of Michigan,
v Ann Arbor, MI 48109-1043 USA
1
[email protected]
1
4
3 Abstract
.
1
0 For certain vertex operator algebras (e.g., lattice type) and given
2 finite group of automorphisms, we prove existence of a positive def-
1
inite integral form invariant under the group. Applications include
:
v
an integral form in the Moonshine VOA which is invariant under the
i
X Monster, and examples in other lattice type VOAs.
r
a
Contents
1 Introduction 2
2 Preliminaries 3
3 Integral forms of lattice vertex algebras 4
3.1 About the automorphism group of V and a torus normalizer . 9
L
3.2 Initial pieces of the standard integral form for V . . . . . . . 10
E8
1
4 About a given rational form and cvcc1 12
2
4.1 An application to V+ . . . . . . . . . . . . . . . . . . . . . . 12
EE8
5 Extending an integral form of a subVOA to the full VOA 14
5.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Final Remarks 16
6.1 About the dual . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 Examples generated by degree 1 . . . . . . . . . . . . . . . . . 17
6.3 Adjusting a rational form to an integral form . . . . . . . . . . 17
A Appendix: Background 18
B About involutions 18
C Appendix: about tensors 19
D Appendix: about elementary abelian 2-subgroups in theMon-
ster 19
1 Introduction
Given a finite subgroup G of Aut(V), where V is a vertex operator algebras
(VOA), one would like to study possible G-invariant integral forms of V. We
give methods to do so in a few cases. An application is a Monster-invariant
form within the Moonshine VOA [7]. (The Monster is the largest sporadic
finite simple group [9].)
AnintegralformofaVOAcouldbeconsidered ananalogueoftheintegral
structure spanned by a Chevalley basis in a Lie algebra or of an integral
structure in an enveloping algebra. An integral form of a VOA allows one to
create VOAs over other commutative rings of coefficients, e.g., finite fields,
algebraic number fields. When such a form is invariant under a finite group,
we get an infinite series of integral representations and actions of the group
as automorphisms of finite rank algebras over Z (such algebras are typically
nonassociative). The form may be reduced modulo primes to create modular
representations of that group.
An integral form R of a vertex operator algebra V with a nondegenerate
symmetric invariant bilinear form ( , ) is a vertex algebra over Z satisfying
· ·
2
(R,R) Q. Our main examples are lattice vertex operator algebras [1],
≤
[7] and their orbifolds and subalgebras. There is also a positive definite
hermitian form on lattice vertex operator algebras [1] although the form is
not invariant in the sense of [6]. Hermitian forms invariant by a finite group
also give integral representations. Some of our examples satisfy the stronger
condition (R,R) Z.
≤
2 Preliminaries
Definition 2.1. Let F be a field of characteristic 0. A rational form of an
F-vector space is a rational subspace for which any basis is an F-basis of the
complex subspace.
Definition 2.2. An integral form in a rational vector U space of dimension
m is a free abelian subgroup of (U,+) of rank m (so a basis of a lattice is a
basis of U as a vector space). If U has a symmetric bilinear form, the integral
form is an integral (resp., rational) lattice if the integral form is an integral
(resp., rational) lattice with respect to the form.
Let F be a field of characteristic 0. An integral form, etc. for an F-vector
space is an integral form, etc. for a rational form of the F-vector space.
For a VOA, we use the usual symbols 1 for vacuum element and ω for
the (principal) Virasoro element.
Definition 2.3. Suppose that V is a VOA (over the complex numbers) with a
nondegenerate symmetric bilinear form. An integral VOA form (abbreviated
IVOA) for V is an abelian subgroup R of (V,+) such that R is a VA over
Z, there exists a positive integer s so that sω R, for each n, R := R V
n n
∈ ∩
is an integral form of V , (R,R) Q. Since R is a VOA, 1 R. For each
n
≤ ∈
degree n, R has finite rank, whence there is an integer d(n) > 0 so that
n
d(n) (R ,R ) Z.
n n
· ≤
Remark 2.4. (i) If dim(V ) = 1, the invariant property of the bilinear form
0
requires that for any u,v R , 1(L(1)nu) v 1 Z1.
∈ m n≥0 n! 2wtu−2n−1 ∈ d(m)
(ii) If R is generated over Z by a set of homogeneous elements, then
P
R = R is homogeneous.
n∈Z n
TheoLrem 2.5. We use the notations of (2.3). We assume that the finitely
generated V is generated over C by R˜ := R + R + + R and that the
0 1 d
finite group G leaves invariant the rational vector spac·e·s·panned by R˜.
3
There exists an integral form S of V such that
(a) S R and for all n, R : S is finite (where S := S V );
n n n n
≤ | | ∩
(b) S is G-invariant;
Proof. For all n, define U to be the Q-span of R .
n n
Since G is finite, the G-invariant abelian group S := gR has the
g∈G
property that for all n, S : R is finite. Therefore S is a lattice in U .
n n n n
| | T
Since G Aut(V), gR is an integral form of V, for all g G, and so S
≤ (cid:3) ∈
is also an integral form.
3 Integral forms of lattice vertex algebras
Let L be a positive definite even lattice with a basis γ ,...,γ and L∗ the
1 d
{ }
dual lattice of L. We will follow the setting of [7] for lattice vertex operator
algebra V and the Heisenberg vertex operator subalgebra M(1) of V .
L L
For any α L∗ we set
∈
α( n)
E−( α,z) := exp − zn = s zn.
α,n
− n
!
n>0 n≥0
X X
Let M(1) be the Z-span of s s s for α L and n 0. Then
Z α1,n1 α2,n2··· αk,nk i ∈ i ≥
M(1) is a subset of M(1) which is a vertex operator subalgebra of V . We
Z L
will prove that M(1) is, in fact, an IVOA.
Z
We will denote the set of partitions of postive integers by . For each
P
α L∗ and λ = (λ ,λ ,...,λ ) with λ λ λ > 0 we define
1 2 k 1 2 k
∈ ∈ P ≥ ≥ ··· ≥
vectors by the k k determinant
×
s (α) = det(s )
λ α,λi+j−i
where s is understood to be zero if n < 0. Then s (α) M(1) for any
α,n λ Z
∈
α L and λ .
∈ ∈ P
Lemma 3.1. We have E−(α,z) =
(1z0 +α( 1)z1 + 1α( 1)2z2 + 1α( 1)3z3 + )
− 2 − 6 − ··· ×
(1z0 + 1α( 2)z2 + 1 1 α( 2)2z4 + 1 1 α( 2)3z6 + )
2 − 222 − 623 − ··· ×
(1z0 + 1α( 3)z3 + 1 1 α( 3)2z6 + 1 1 α( 3)3z9 + ) .
3 − 232 − 633 − ··· ×···
The series E−(α,z) begins: 1z0 + α( 1)z1 + (1α( 2)+ 1α( 1)2)z2 +
− 2 − 2 −
4
(1α( 3)+ 1α( 1)α( 2)+ 1α( 1)3)z3 + .
3 − 2 − − 6 − ···
It follows that s = 1,s = α( 1),s = 1α( 2) + 1α( 1)2 and
α,0 α,1 − α,2 2 − 2 −
s = 1α( 3)+ 1α( 1)α( 2)+ 1α( 1)3.
α,3 3 − 2 − − 6 −
Lemma 3.2. M(1) has a Z-basis
Z
s s s
α1,n1 α2,n2··· αk,nk
where α γ ,...γ , n n n and k 0.
i 1 d 1 2 k
∈ { } ≥ ··· ≥ ≥
Proof. Since γ ,...,γ is a basis of L we see that M(1) is spanned by
1 d Z
{ }
s s s
α1,n1 α2,n2··· αk,nk
for α γ ,..., γ and k 0. It is enough to show that each s lies
i ∈ {± 1 ± d} ≥ −γi,n
in the Z-span of
s s s
α1,n1 α2,n2··· αk,nk
where α γ ,...γ and k 0. Note that E−( α,z)E−(α,z) = 1 for
i 1 d
∈ { } ≥ −
α L. Then for any n 0
∈ ≥
n
s s = δ .
α,i −α,n−i n,0
i=0
X
Since each s = 1, it follows that s is a Z-linear combination of products
α,0 −α,n
of s ’s. The Z-linear independence of these vectors follows from the C-
α,m
(cid:3)
linear independence.
Theorem 3.3. Let (V ) be the Z-span of
L Z
s s s eα
α1,n1 α2,n2··· αk,nk
where α γ ,...γ , n n n , k 0 and α L. Then (V ) is a
i 1 d 1 2 k L Z
∈ { } ≥ ··· ≥ ≥ ∈
IVOA generated by e±γi for i = 1,...,d. The natural bilinear form restricted
to (V ) is integral. Moreover, M(1) is a sub IVOA of (V ) .
L Z Z L Z
Proof. From lemma 3.2 and the construction of V , the spanning set of
L
(V ) forms a Z-basis of V . We first prove that (V ) is a vertex algebra
L Z L L Z
over Z. That is, we need to show that for basis element u,v and n Z,
∈
u v (V ) .
n L Z
∈
5
Let α,β L. Let α ,...,α ,β ,...,β L subject to the conditions
1 k 1 l
∈ ∈
α + +α = α and β + +β = β. Set
1 k 1 l
··· ···
k
α ( n)
A := exp i − wn eα V [[w , ,w ]]
n i ∈ L 1 ··· k
!
i=1 n≥1
XX
l
β ( n)
B := exp j − xn eβ V [[x , ,x ]]
n j ∈ L 1 ··· l
!
j=1 n≥1
XX
Then the coefficients in the formal power series A and B span M(1) eα and
⊗
M(1) eβ respectively as k,l,α and β vary subject to the conditions (cf.
i j
⊗
[7]). It is good enough to prove that the coefficients of Y(A,z)B lie in (V )
L Z
again.
It follows from [7] that
Y(A,z)B = ◦Y(eα1,z +w ) Y(eαk,z +w )
◦ 1 ··· k ×
Y(eβ1,x ) Y(eβl,x )◦1 (z +w x )(αi,βj).
1 ··· l ◦ i − j
1≤i≤k,1≤j≤l
Y
Since the coefficients of (z+w x )(αi,βj) are integers, it is good
1≤i≤k,1≤j≤l i− j
enough to show that the coefficients of exp αi(−n)(z +w )n preserve
Q n>0 n i
(VL)Z. Again, this is clear from the definitio(cid:16)nPof (VL)Z and Lemma(cid:17)3.2.
It is well known from the construction of V [7] that V is generated by
L L
e±γi for i = 1,....,d. Let V be a vertex operator algebra V and u,v V.
∈
Then for any n Z the vertex operator
∈
Y(u v,z) = Res (z z)nY(u,z )Y(v,z) ( z +z )Y(v,z)Y(u,z )
n z1{ 1 − 1 − − 1 1 }
involves only integer linear combinations of products u v and v u for s,t
s t s t
∈
Z. As a result, (V ) is generated by e±γi for i = 1,...,d.
L Z
Since dim(V ) = 1 and L(1)(V ) = 0, there is a unique non-degenerate
L 0 L 1
symmetric invariant bilinear form on V [17], Prop. 3.1. It is characterized
L
by the conditions below.
(1) (eα,eβ) = δ for α,β L where we have arranged for eαeβ =
α,−β
∈
ǫ(α,β)eα+β for a bimultiplicative form ǫ : L L 1 such that ǫ(α,α) =
× → h± i
( 1)(α,α)/2.
−
(2) (α(n)u,v) = (u,α( n)v) for all u,v (V ) , α L and n Z.
L Z
− − ∈ ∈ ∈
6
Weshowthat((V ) ,(V ) ) Z.Letα ,...,α L∗,α,β ,...,β ,β L
L Z L Z 1 k 1 l
≤ ∈ ∈
and A,B be as before. Then we have
k l
α (n) β ( n)
(A,B) = (eα,exp i wn exp j − xn eβ) (3.1)
− n i n j
! !
i=1 n≥1 j=1 n≥1
XX XX
= (1 w x )(αi,βj)δ .
i j α,−β
−
i,j
Y
That is,
(s s eα,s s eβ)wm1 wmkxn1 xnl
α1,m1 ··· αk,mk β1,n1··· βl,nl 1 ··· k 1 ··· l
X
= (1 w x )(αi,βj)δ .
i j α,−β
−
i,j
Y
Since M(1) is a sub VOA of V , it follows immediately that M(1) is a
L Z
sub IVOA of (V ) . The proof is complete. (cid:3)
L Z
Remark 3.4. There is also a unique positive definite hermitian form ( )
· | ·
on V [1] such that
L
(1) (eα eβ) = δ for α,β L;
α,β
| ∈
(2) (α(n)u v) = (u α( n)v) for all u,v (V ) , α L and n Z.
L Z
| | − ∈ ∈ ∈
Note that the positive definite form ( ) on V is not invariant in the
L
· | ·
sense of [6] but it is useful.
As in the discussion for the bilinear form, we have
k l
α (n) β ( n)
(A B) = (eα exp i wn exp j − xn eβ)
| | n i n j
! !
i=1 n≥1 j=1 n≥1
XX XX
= (1 w x )−(αi,βj)δ .
i j α,β
−
i,j
Y
That is,
(s s eα s s eβ)wm1 wmkxn1 xnl
α1,m1 ··· αk,mk | β1,n1··· βl,nl 1 ··· k 1 ··· l
X
= (1 w x )−(αi,βj)δ .
i j α,β
−
i,j
Y
7
We will next determine the dual lattice of the integral lattice (V ) in V
L Z L
with respect to the forms ( ,) and ( ).
|
Notation 3.5. Let β ,...,β be the basis of the dual lattice L∗ which
1 d
{ }
is dual to the basis γ ,...,γ of L. Let U be the Z-span of the Z-linearly
1 d
{ }
independent vectors
s (β ) s (β ) eα
λ1 1 ··· λd d ⊗
for λ ,...,λ and α L. If L is self dual, then (V ) = U.
1 d L Z
∈ P ∈
Proposition 3.6. (i) The set
s (γ ) s (γ ) eα λ ,...,λ ,α L
{ λ1 1 ··· λd d ⊗ | 1 d ∈ P ∈ }
is a Z-basis of V ) .
L Z
(ii) U is the dual of (V ) in V with respect to both the bilinear form (,)
L Z L
and the hermitian form ( ).
|
(iii) The set
s ( β ) s ( β ) e−α λ ,...,λ ,α L
{ λ1 − 1 ··· λd − d ⊗ | 1 d ∈ P ∈ }
is the dual basis with respect to (,) and the set
s (β ) s (β ) eα λ ,...,λ ,α L
{ λ1 1 ··· λd d ⊗ | 1 d ∈ P ∈ }
is the dual basis with respect to ( ). In particular, if L is self dual, then (V )
L Z
|
is also self dual with respect to both the bilinear form (, ) and the hermitian
form ( ).
|
Proof. (i, ii) We deal only with the hermitian form and the proof for the
bilinear form is similar.
We first prove that
(sλ1(β1)···sλd(βd)⊗e−α | sλ′1(γ1)···sλ′d(γd)⊗eβ) = δ(λ1,...,λd),(λ′1,...,λ′d)δα,β
for λ ,λ′ and α,β L. Since (β ,γ ) = δ . It is enough to show that
i i ∈ P ∈ i j i,j
(sλ(βi) sλ′(γi)) = δλ,λ′
|
for λ,λ′ and i = 1,...,d. But this is clear from the bilinear form formula
∈ P
(3.1) (see Formula (4.8) of [18]).
8
We next show that the set of
s (γ ) s (γ )
λ1 1 ··· λd d
for λ ,...,λ form a basis of M(1) . Note that for α ,...,α L and
1 d Z 1 k
∈ P ∈
n ,...,n 0,
1 l
≥
(s (β ) s (β ) s s ) Z
λ1 1 ··· λd d | α1,n1··· αk,nk ∈
for λ . This implies that s s is a Z-linear combination of
i ∈ P α1,n1··· αk,nk
s (γ ) s (γ ) for λ . Consequently,
λ1 1 ··· λd d i ∈ P
s (γ ) s (γ )
λ1 1 ··· λd d
for λ ,...,λ form a basis of M(1) and
1 d Z
∈ P
s (γ ) s (γ ) eα
λ1 1 ··· λd d ⊗
for λ ,...,λ and α L form a basis of (V ) . So, (i) and (ii) are proved.
1 d L Z
∈ P ∈ (cid:3)
The rest of the proposition is clear.
Remark 3.7. If L is self dual, then (V ) is a positive definite self dual
L Z
lattice by Proposition 3.6. This was pointed out in [1] already.
Remark 3.8. Let R = (V ) . It seems that finding the group structure of
L Z
U /R (3.5) for an arbitrary even lattice L will be very complicated. One
n n
can easily see that U /R is isomorphic to L∗/L.
1 1
3.1 About the automorphism group of VL and a torus
normalizer
Notation 3.9. In this article, we use the standard notation pn for an ele-
mentary abelian p-group of order pn, where p is a prime. Context will indicate
when pn means a group.
We now review the automorphism group of V and certain other groups
L
which act on (V ) preserving either the bilinear form or the hermitian form.
L Z
For this purpose we need to fix a bimultiplicative function ǫ : L L 1
such that c(α,β) := ǫ(α,β)ǫ(β,α) equals ( 1)(α,β) for α,β L×. L→et Lhˆ±bei
− ∈
the central extension of L by 1 with commutator map c and we denote
h± i
the map from Lˆ to L by¯. Then any automorphism σ of Lˆ induces an
automorphism σ¯ of L. Let O(L) be the isometry group of L and O(Lˆ) :=
9
σ Aut(Lˆ) σ¯ O(L) .Theabovegroupembeds inAut(V )andpermutes
L
{ ∈ | ∈ }
the set eα α L . This subgroup of Aut(V ) is often denoted O](L) in
L
{± | ∈ }
the literature. It has the shape O](L) = Zrank(L).O(L).
∼
Let T be the torus obtained by exponentiating a where a ranges over the
0
standardCartansubalgebraof(V ) . ThenT = H/L∗ whereH = C Land
L 1 ∼ Z
the normalizer N of T in Aut(V ) is a group of the form N = T.O⊗(Lˆ). We
L L L ∼
letN bethenormalsubgroupofAut(V )generatedbyev0 forv (V ) .Then
L L 1
∈
Aut(V ) = N N , product of subgroups, and N is the connected component
L L
ofthe identity in thealgebraicgroupAut(V ). See [5] and[2] forbackground.
L
There exists a standard lift, called θ, of the 1 isometry of L to an
−
automorphism of order 2 of V . The standard lift interchanges eα and e−α.
L
See [7] and an appendix of [11] for a general discussion of lifts. We have
C (θ) = 2rank(L).O(L).
NL ∼
Lemma 3.10. Suppose that L is an even lattice. We use the IVOA (V )
L Z
of (3.3). If S is any subgroup of O](L), the fixed point subVOA VS has the
IVOA (V )S. Note that N (S)/S acts as automorphisms of the IVOA
L Z O^(L)
(V )S. If M is a sublattice so that S fixes the subVOA V , then V (V )S
L Z M M ∩ L Z
is an IVOA in VS.
M
Notation 3.11. A special case of S in (3.10) is the fixed points of θ (3.1)
in (V ) , denoted by (V )+. Then (V )+ is an IVOA in V+.
L Z L Z L Z L
Remark 3.12. We discuss the hermitian form more. We define the anti-
linear involution σ from V to V such that σ(au) = a¯θ(u) for a C and
L L
∈
u (V ) where a¯ is the complex conjugate of a. It is easy to see that
L Z
∈
(u v) = (u,σ(v)) foru,v V .Note that N does not preserve the hermitian
L L
| ∈
form but the subgroup O](L) does since it preserves the set eα α L ,
{± | ∈ }
which generates the Q-VOA Q (V ) .
L Z
⊗
Lemma 3.13. If A,B are IVOA forms on the respective VOAs V,W, then
A B is an IVOA form on V W (graded tensor product of VOAs; see [6]).
⊗ ⊗
3.2 Initial pieces of the standard integral form for VE
8
Let L be any even integral lattice. Recall the bilinear form (,) and the
hermitian form ( ) on V (3.4). We use the usual notation for H = C L
L Z
· | · ⊗
and the subspaces H = H t−n of V .
−n L
⊗
The next lemma recalls a few calculations for our use.
10