Coxeter elements and McKay’s E8 observation on the
Monster simple group: Odd order case
Ching Hung Lam
Department of Mathematics
National Kung Hung University
Tainan, Taiwan
Masahiko Miyamoto
Department of Mathematics
University of Tsukuba
Tsukuba, 305, Japan
June 15, 2005
1
Introduction
Let’s explain McKay’s E8 -observation about the monster simple group M briefly. The
product of any two 2A-involutions of M falls into one of nine conjugacy classes
1A, 2A, 3A, 4A, 5A, 6A, 4B, 2B, and 3C (by ATLAS’s notation).
Here, the first numeric characters denote the order of elements in the conjugacy class and
the second upper case characters denote alphabetical order of the size of centralizer of
elements. The orders of these nine conjugacy classes is corresponding to the multiplicities
of fundamental roots in a primitive isotropic element of E8 -diagram Ê8 including the
repetition level. Roughly speaking, this correspondence is the McKay’s E8 -observation.
It suggests some relation between 2A-involutions of the monster simple group and a root
lattice of type E8 .
By assigning the conjugacy classes to the corresponding node in an extended E8 diagram, we have
1A − 2A − 3A − 4A − 5A − 6A − 4B − 2B
|
3C
The monster simple group was first realized as an automorphism group of the monstrous Griess algebra B in the theory of groups [Gr], then it was reconstructed as an
\
automorphism group of the moonshine VOA V \ = ⊕∞
n=0 Vn as an answer to the moonshine conjecture [B],[FLM],[Mi3]. As an automorphism of V \ , every 2A-involution has
a special face. It is a τ -involution (or Miyamoto-involution) τe defined by a conformal
element e with length 12 , [C],[Mi1]. Namely, there is a 1-1 correspondence between 2Ainvolutions and conformal elements in V \ of length 12 . An interesting point is that we can
also observe such elements in many situations, such as lattice VOAs and so they act as
intermediaries between these subjects and the monster simple group [Mi3].
1
Different from one element, McKay’s E8 -observation suggests a wider connection
among the monster simple group, Lie algebras and lattices, that is, not only an involution, but also a dihedral group.
In this paper, we will study the following dihedral groups and show relations among
them.
(1) An element of M inverted by a 2A-involution.
(2) An automorphism ρH of V√2E8 defined by a sublattice H, which is inverted by a special
τ -involution.
(3) A Coxeter element inverted by a product of reflections.
(4) exp(ad(B)) for a diagonal matrix B, which is inverted by A → −tA for A ∈ sl(n, C).
We abuse notation for root system, its Dynkin diagram and its root lattice. For example, E8 denotes
√ a root system of type E8 , its Dynkin diagram, and its root lattice. We also
use notation 2P to denote a lattice with double the inner product of P for any lattice P .
For the case (2), there is a progress in the research of the McKay’s E8 -observation. The
first author, Yamada and Yamauchi [LYY] found an interesting
phenomenon in a lattice
√
√
√
VOA V 2E8 . A lattice VOA V 2E8 constructed from 2E8 contains a special conformal
element e√2E8 with length 12 such that τe√2E acts on the weight one space (V√2E8 )1 as −1,
8
by a result in [DLMN].
For each node pX of an extended Dynkin diagram Ê8 , the Dynkin subdiagram Ê8 −
{pX} spanns a sublattice HpX of the E8 -root lattice E8 . For example, for pX = 5A, H5A
is a root lattice of type A4 + A4 . Then E8 /HpX is a cyclic group of order p. Using this
factor group, we can define a map
√
√
√
ϕ : 2E8 → 2E8 / 2HpX ∼
= Z/pZ
and an automorphism of V√2E8 of order p;
ρpX (m ⊗ eα ) = e2π
√
−1ϕ(α)/p
m ⊗ eα for α ∈
√
2E8 and m ∈ M(1),
which is inverted by τe√2E , see the next section for m ⊗ eα in a lattice VOA.
8
They found that as long as we restrict products into the weight two spaces, e√2E8 and
ρpX (e√2E8 ) satisfy the same relation as e, f in the monstrous Griess algebra satisfying
τe τf ∈ pX. Therefore, they expected that a subVOA V A(e√2E8 , ρpX (e√2E8 )) generated
by e√2E8 , ρpX (e√2E8 ) is isomorphic to V A(e, f ) as VOAs.
The main purpose of this paper is to show this fact is true for odd p.
The main theorem
VA(e√2E8 , ρpX (e√2E8 )) ∼
= VA(e, fpX ) as VOAs for p odd.
Remark 1.√For pX = 1A, 2A, 3A, 2B, the above are already known, see [Mi1],[Mi2],
[SY]. Since 2E8 ⊆ Λ, it is easy to find two conformal elements e√2E8 , ρpX (e√2E8 ) such
that
V A(e√2E8 , ρpX (e√2E8 )) ⊆ V√2E8 ⊆ VΛ ,
2
where Λ is the Leech lattice. On the other hand, from the construction of the moonshine
VOA, we may view VΛ+ ⊆ V \ . By the properties of Conway groups and the character table
0
0
in VΛ+ such that τe0 τfpX
in [Atlas], it is not hard to find two conformal elements e0 , fpX
\ ∼
belongs to pX in Aut(V ) = M. So, in VΛ we have two pairs of conformal elements
{e√2E8 , ρpx (e√2E8 )} and {e0 , f 0 pX } ⊆ V√2E8 . . What is a crucial difference between them?
The main difference is that e√2E8 and ρpX (e√2E8 ) acts on the weight one space (V√2E8 )1
(8-dimension) as −1, that is, they have the same 8-dimensional eigenspace in (VΛ )1 with
0
in (VΛ )+ have different 8-dimensional
eigenvalue −1. On the other hand, e0 and fpX
eigenspaces in (VΛ )1 with eigenvalue −1. Furthermore, τe τρpX (e) is a diagonal automor0
is an automorphism of
phism, that is, it is given as exp(v(0) ) by v ∈ CE8 , but τe0 τfpX
Λ.
\
If we can
construction,
√ construct V from the Leech lattice VOA VΛ by pX-orbifold
\
√
√
then since 2E8 ⊆ Λ, we may have V A(e 2E8 , ρpX (e 2E8 )) ⊆ V naturally. However,
we don’t know such orbifold construction except 2B-orbifold construction and it is not
hard to see from a framed VOA structure of V \ that there is no 2A-orbifold construction,
[Mi2], [Y]. However, the above fact suggests the existence of orbifold construction for a
quite large part of VΛ . From this view point, we will study McKay’s E8 observation.
2
Preliminary results
√
In this
paper,
we
mainly
consider
a
lattice
of
type
2√× (a root lattice of type S), we call
√
√
lattice
of
type
2S
and
denote
it
by
2S. Similarly, we call a vector of
it a 2-root
√
length 4 a 2-root.
2.1
A lattice VOA
First, we will briefly recall the construction of a lattice VOA VL associated with a rank n
positive definite even lattice L with an inner product h , i from [FLM].
Viewing H = C ⊗Z L as a commutative Lie algebra of rank n with an inner product,
we consider its affine Lie algebra
Ĥ = H ⊗ C[t, t−1 ] ⊕ C.
Here Lie product is given by [a ⊗ ts , b ⊗ tm ] = δs+m,0 nha, bi. Consider a special element 1
satisfying α(s)1 = 0 for all α ∈ H and s ≥ 0 and define Ĥ-irreducible module
M(1)⊗n = C[α(s) | α ∈ H, s < 0] · 1
generated from 1. (Although the original notation is M(1), but we will denote it by
M(1)⊗n to express the rank n of L). Let C{L} = ⊕α∈L Zeα denote a twisted group ring
β
∈ Zeα+β and eα eβ = (−1)hα,βi eβ eα . Usually, we first consider a central
of L, where eα eP
extension L̂ = α ±eα of L by ± and then define C{L}.
Then
VL = ⊕α∈L M(1) ⊗ eα
3
has a VOA structure, that is, for v, u ∈ VL and any integer m, VL has a bilinear product
conditions. Viewing v(m) as an element in End(V ), a genv(m) u ∈ VL satisfying
P several−m−1
∈ End(V )[[z, z −1 ]] is called a vertex operator of v and
erating function m∈Z v(m) z
denote it by Y (v, z). From now on, we denote 1 ⊗ e0 by 1 and 1 ⊗ eα by eα for α 6= 0.
The weight of elements is given by
wt(α1 (−n1 ) · · · αr (−nr )eβ ) = n1 + · · · + nr +
hβ, βi
.
2
Then VL has a N-grading:
VL = ⊕∞
n=0 (VL )n .
P
P
From the definition, we have (VL )0 = C1 and (VL )1 = α∈L Cα(−1)1 + α∈rt(L) Ceα ,
where rt(L) denotes the set of roots of L. Vertex operators for weight one elements a(−1)1
and eα are given by
P
Y (α(−1)1, z) = m∈Z α(m)z −m−1 ,
P
P
α(−m) m
α(m) −m α α
Y (eα , z) = exp( ∞
z ) exp( ∞
)e z ,
m=1
m=1 m z
m
respectively, see [FLM] for the detail.
The weight one space (VL )1 of a VOA VL = ⊕∞
n=0 (VL )n becomes a Lie algebra with an
invariant bilinear form, where the Lie product is given by v(0) u and inner product (·, ·) is
given by (v, u)1 = v(1) u. In particular, if L is a root lattice of type S (e.g. An , Dn , En ),
then (VL )1 is a simple Lie algebra of type S, where M(1)⊗n is a Cartan subalgebra and
eα (hα, αi = 2) are roots vectors.
From the construction of lattice VOAs, we know the following:
Lemma 1. If σ is an automorphism of L̂, then we can extend σ to an automorphism of
VL naturally.
For example, if a lattice L is a root lattice of type An or type Dn , then we are able to
extend the action −1 on L to an automorphism θ of L̂ (and that of VL ).
If we restrict this automorphism to the weight one space (VL )1 , by viewing it as
sln+1 (C) or o2n (C), it coincides with
θL : A → −t A
after suitable rearrangement of roots. As a lattice VOA, it is expressed by
θL : eα → −e−α
a(−1)1 → −a(−1)1
for roots α
In this paper, for each lattice VOA VN of suitable Niemeier lattice N , we explicitly extend
θL into automorphism (which we denote by θN ) of N̂ (and so of VN ).
4
2.2
Conformal element with length 1/2
√
If v ∈ L is a
2-root, then an element of VL
e±
v =
1
1
v(−1)v(−1)1 ± (ev + e−v ) ∈ (VL)2
16
4
is a rational conformal element with length 1/2, that is,
V A(e±
v ) is a simple Virasoro
√
VOA L( 12 , 0) called the 2-dim. Ising model. If L is a 2-root lattice, then Dong, Li,
Mason and Norton show that V A(< e−
v | v ∈ rt(L) >) is a subVOA with some Virasoro
element√ω − and they have paid notice to another conformal vector ω − ω − . In particular,
if L = 2E8 , then e = ω − ω − is a rational conformal element with length 12 . We call it
”special conformal vector” in this paper. By the definition, we have
e√2E8 =
2.3
1
480
X
α(−1)2 1 +
α∈rt(E8 )
1
64
X
√
(e
√
2α
+ e−
2α
) ∈ (V√2E8 )2 .
α∈rt(E8 )
An observation by [LYY]
√
From the definition of e√2E8 , e√2E8 is orthogonal to all e−
for a ∈ rt(E8 ). Furthermore,
2a
−
from the definition ρpX , ρpX fixes all e√2a (a ∈ rt(HpX )) and so ρpX (e√2E8 ) is orthogonal
√
for a ∈ (HpX )2 . Therefore, we obtain
to all e−
2a
√
: a ∈ rt(HpX )) ⊆ V√2E8 ,
VA(e√2E8 , ρpX (e√2E8 ) ⊗ VA(e−
2a
where VA(S) denotes a subVA generated by S.
3
Coxeter elements on Lie algebra and lattice VOA
Let {y1 , ..., yn } be a fundamental root system of a lattice P . Then ξP = Ref yn · · · Ref y1
is an automorphism of P called ”a Coxeter element”. Its order is the Coxeter number. Here Ref y denotes a reflection with axis y. These elements define one conjugacy
class, which does not depend on the order of products of reflections. In particular, if
we divide {y1 , ..., yn } into two sets {y1 , ..., yk } ∪ {yk+1 , ..., yn } such that hyi , yk+j i = 0 for
Q
Q
i = 1, ..., k, j = 1, ..., n − k, then g = ki=1 Ref yi and h = n−k
j=1 Ref yk+j are involutions
which invert a Coxeter element gh.
We next extend a Coxeter element ξ of a root lattice of a Lie algebra G of type An to
an automorphism of G (we denote it by the same notation ξj. We will explain it by using
the weight one space (VL )1 of lattice VOA VL .
M(1)1 = {v(−1)1 | v ∈ CL}
is a Cartan subalgebra and ea (a roots) is a root in (VL )1 . The action of ξ on M(1) is
naturally defined by the action on L.
5
If G is a Lie algebra of type An , then a Coxeter element is
ai −→ ai+1 for i = 1, ..., n
ξAn = Rfa1 · · · Rfan :
an −→ −a1 − a2 − · · · − an
from a view point of orthonormal basis, it is given by ξAn : v1 → v2 → · · · → vn+1 → v1 .
Therefore, if we define ξAn by
ξAn : ev1 −v2 → ev2 −v3 → · · · → evn −vn+1 → evn+1 −v1 = −e−v1 +vn+1 → −e−v2 +v1 = ev1 −v2
···
ev1 −vn+1 → ev2 −v1 = −e−v1 +v2 → −e−v2 +v3 · · ·
then it is an automorphism of L̂.
As it is well known, a Lie algebra G of type An is isomorphic to
sl(n + 1, C) = {F ∈ Mn+1 (C) | trF = 0}
and the set T of diagonal matrices with trace 0 is a Cartan subalgebra. Here, we have
an isomorphism φ : sl(n + 1, C) → (VL )1 by φ(Eii − Ejj ) = (vi (−1)1 − vj (−1)1) and
φ(Eij ) = evi −vj , where Eij denotes a matrix with 1 at (i, j)-entry and zero else. We define
a permutation matrix C = (bij ) by cij = 1 if j − i ≡ 1 (mod n + 1) and cij = 0 else, then
the action of Coxeter element ξAn on G is given by a conjugate by C, that is,
ξAn : P → C −1 P C for P ∈ sl(n + 1, C).
Then the space of fixed point of ξAn is C =< En+1 , C, C 2 , ..., C n > ∩sl(n + 1, C), which
is a commutative Lie subalgebra of rank n. Set B = (ω ij ), where ω = e2πi/(n+1) . By the
direct calculation, we have
X
ω −is+tj Ei,j ,
σAn (Es,t ) = (ω −is )Es,t (ω tj ) =
i,j
where Es,t denotes a matrix with 1 at (s, t)-entry and 0 else. Therefore σAn : P → B −1 P B
is an automorphism of sl(n + 1, C) such that σS (C) = T . Since B is a symmetric matrix
and B tB = (δi+j,1 (mod n+1) ) is a permutation matrix (an automorphism of Dynkin diagram An ), (σS )2 commutes with θ.
4
Outline of proof
We assign Niemeier lattices N (A38 ) and N (A64 ) to nodes 3C and 5A, respectively, and
we denote it by N . Let P denote a root lattice of N and P = ⊕Pi a decomposition
into a direct sum of simple root lattices of type Ani . We first explain our strategy in
a weight one space. Generally, let G be a Lie algebra of type An (or type D4 ) and
T a Cartan subalgebra. Then θ : A → −t A is an involution of G which acts on T
as −1 and a Coxeter element ξ is given by ξ : A → P −1 AP for some permutation
matrix P , which implies θξ = ξθ. We also have that the fixed point space C = G <ξ>
6
of the automorphism ξ (Coxeter elmenet) is another Cartan subalgebra and so there is
an inner automorphism σ : A → B −1 AB such that σ(T ) = C, where B is a symmetric
matrix and BB is a permutation matrix, which implies that σ 2 commutes with θ. We
also have P −1 (B −1 P B)P ∈ C(B −1 P B). Then if some element ẽ is invariant under the
permutation and fixed by θ, then ẽ ∈ G <ξ> and σθσ −1 (ẽ) = σ 2 θ(ẽ) = ẽ. If ρ = σ −1 ξσ,
−1
then ξ(ρ(ẽ) = ξσ −1 ξσ(ẽ) = ρ(ẽ) and σθσ −1 ρ(ẽ) = ρ(ẽ). So if we set H+ = (G)<ξ,σθσ > ,
then < ẽ, ρ(ẽ) >⊆ H+ . Since ρ acts on T trivially, it is diagonal, that is, there is an
element v ∈ T such that ρ = exp(adv).
Our next step is to extend the above automorphisms to those of the Q
whole VOA VN .
(1) We extend θ and ξ to N̂ ,Qthat is, we will define an involution θN = θPi ∈ Aut(N̂ )
θN ξN = ξN θN , (ξN is not unique).
and a Coxeter element ξN = ξPi ∈ Aut(N̂ ) satisfying
√
(2) We define a ξN -invariant sublattice E (∼
= 2E8 ) of N .
(3) Let e be a special conformal vector of VE and we show e ∈ VE<ξN > by taking suitable
ξN .
Q
(4) We show that there is a sublattice F without roots and an automorphism σ = σPi
such that σ 2 is an automorphism (a permutation) of N̂ and σ −1 (VN<ξN > ) = VF . Moreover,
F is isomorphic to a sublattice of the Leech lattice Λ.
(5) Let ρAn denote an automorphism of VA∗n induced from A∗n /An , then ρAn σ = σξAn by
taking suitable σ.
Q
(6) We will show that for suitable choice of ci ∈ Z, ρN = σ( ξPcii )σ −1 ∈ AutVE plays the
same role of ρ√2E8 in [LYY].
Thus VA(e, ρN (e)) is isomorphic to a subVOA given by [LYY]. We will finally show:
(6) e, ρN (e) ∈ (V <ξ> )σ
−1 θσ
.
Therefore, we have the desired result:
σ(VA(e, ρN (e))) ⊆ VF<θ> ⊆ VΛ+ ⊆ V \ .
References
[B] R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl.
Acad. Sci. USA 83 (1986), 3068-3071.
[C] J.H.Conway, A simple construction for the Fisher-Griess Monster group, Invent.
Math. 79 (1985), 513-540.
[Atlas] J.H.Conway, R.T.Curtis, S.P.Norton, R.A.Parker and R.A.Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985.
[CS] J.H.Conway and N.J.A.Sloane, Sphere Packings, Lattices and Groups, SpringerVerlag, Berlin-New York, 1988.
[DLMN] C.Dong, H.Li, G.Mason and S.P.Norton, Associative subalgebras of Griess algebra and related tooics, Proc. of the Conference on the Monster and Lie algebra at
7
the Ohio State University, 1996, ed. by J.Ferrar and K.Harada, Walter de Gruyter,
Berline-New York, 1998, 27-42.
[FLM] I. Frenkel, J. Lepowsky, and A. Meurman, ”Vertex Operator Algebras and the
Monster”, Pure and Applied Math., Vol. 134, Academic Press, 1988.
[GN] G.Glauberman and S.P.Norton, On McKay’s connection between the affine E8 diagram and the Monster, CRM Proceedings and Lecture Notes, Vol.30, Amer. Math.
Soc., Providence, RI, 2001, 37-42.
[Gr] R.Griess, The Friendly Giant, Invent. Math. 69 (1982), 1-102.
[LYY] C. H. Lam, H. Yamada and H. Yamauchi, Vertex operator algebras, extended E8
diagram and McKay’s observation on the monster simple group, preprint.
[Mc] J.McKay, Graphs, singularities, and finite groups, Proc. Symp. Pure Mathe., Vol.37,
Amer. Math. Soc., Providence, RI, 1980, 183-186
[Mi1] M. Miyamoto, Griess algebras and conformal vectors in vertex operator algebras,
J. Algebra 179 (1996), 523-548.
[Mi2] M. Miyamoto, Vertex operator algebras generated by two conformal vectors whose
τ -involutions generate S3 . J. Algebra 268 (2003), 653–671.
[Mi3] M. Miyamoto, A new construction of the moonshine vertex operator algebra over
the real number field, Annals of Mathematics, 159 (2004), 535-596.
[SY] S. Sakuma and H. Yamauchi, Vertex operator algebra with two Miyamoto involutions generating S3 , J. Algebra 267 (2003), 272–297.
[Y] 2A-orbifold construction and the baby-monster vertex operator superalgebra, J. Algebra 284 (2005), 645-668.
8