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\noindent {\bf A Monster Lie Algebra?}
\bigskip
Chapter 30 of ``Sphere packing, lattices and groups'' by Conway and Sloane,
and Adv. in Math. 53 (1984), no. 1, 75--79.
R. E. Borcherds, J. H. Conway, L. Queen
and N. J. A. Sloane
\bigskip
We define a remarkable Lie algebra of infinite dimension, and
conjecture that it may be related to the Fischer-Griess Monster group.
The idea was mooted in [C-N] that there might be an
infinite-dimensional Lie algebra (or superalgebra) $L$ that in some
sense ``explains'' the Fischer-Griess `Monster'' group $M$. In this
chapter we produce some candidates for $L$ based on properties of the
Leech lattice described in [C-S]. These candidates
are described in terms of a particular Lie algebra $L_\infty$ of
infinite rank.
We first review some of our present knowledge about these matters. It
was proved by character calculations in [C-N, p. 317] the centralizer
$C$ of an involution of class 2A in the Monster group has a natural
sequence of modules affording the head characters (restricted to $C$).
In [K], V. Kac has explicitly constructed these as $C$-modules. Now
that Atkin, Fong and Smith [F], [S] have verified the relevant
numerical conjectures of [C-N] for $M$, we know that these modules can
be given the structure of $M$-modules. More recently, Frenkel,
Lepowsky, and Meurman [F-L-M] have given a simple construction for the
monster along these lines, but this sheds little light on the
conjectures.
Some of the conjectures of [C-N] have analogs in which $M$ is replaced
by a compact simple Lie group, and in particular by the Lie group
$E_8$. Most of the resulting statements have now been established by
Kac and others. However, it seems that this analogy with Lie groups may not be
as close as one would wish, since two of the four conjugacy classes of
elements of order 3 in $E_8$ were shown in [Q] to
yield examples of modular functions neither of which are the Hauptmodul
for any modular group. This disproves the conjecture made on p. 267
of [K], and is particularly distressing since it was the Hauptmodul
property that prompted the discovery of the conjectures in [C-N], and it
is this property that gives those conjectures almost all their
predictive power.
The properties of the Leech lattice that we shall use stem mostly from
the facts about ``deep holes'' in that lattice reported in [C-S
Chapter 23]. Let $w~=~(0,1,2,3,\ldots,24~|~70)$. The main result of
[C-S Chapter 26] is that the subset of vectors $r$ in $II_{25,1}$ for
which $r \cdot r~=~2$, $r \cdot w~=~-1$ (the ``Leech roots') is
isometric to the Leech lattice, under the metric defined by
$d(r,s)^2~=~{\rm norm} (r-s)$. The main result of [C-S Chapter 27] is
that $\rm Aut ( II_{25,1} )$ is obtained by extending the Coxeter
subgroup generated by the reflections in these Leech roots by its
group of graph automorphisms together with the central inversion $-
1$. It is remarkable that the walls of the fundamental region for
this Coxeter group (which correspond one-for-one with the Leech roots)
are transitively permuted by the graph automorphisms, which form an
infinite group abstractly isomorphic to the group of all automorphisms
of the Leech lattice, including translations.
Vinberg [V] shows that for the earlier analogs $II_{9,1}$ and $II_{17,1}$
of $II_{25,1}$ the fundamental regions for the reflection subgroups
have respectively 10 and 19 walls, and the graph automorphism groups
have orders 1 and 2. For the later analogs $II_{33,1} ,\ldots,$ there
is no ``Weyl vector'' like $w$, so it appears
that $II_{25,1}$ is very much a unique object.
We can use the vector $w$ to define a root system in $II_{25,1}$. If
$v~\in~II_{25,1}$ then we define the height of $v$ by $- v \cdot
w$, and we say that $v$ is positive or negative according as its
height is positive or negative. We now define a Kac-Moody Lie algebra
$L_\infty$, of infinite dimension and rank, as follows: $L_\infty$ has
three generators $e(r)$, $f(r)$, $h(r)$ for each Leech root $r$, and
is presented by the following relations:
$$
\eqalign{
[e(r),h(s)]~ &=~r \cdot s~e(r)~,
\cr
[f(r),h(s)]~ &=~-r \cdot s~f(r),
\cr
[e(r),f(r)]~ &=~h(r),
\cr
[e(r),f(s)]~ &=~0,
\cr
[h(r),h(r)]~ &=~0~=~
[h(r),h(s)],
\cr
e(r)~\{ {\rm ad}~e(s) \} ^
{1-r \cdot s}~ &=~0~=~f(r)~
\{ {\rm ad}~f(s) \}^{1-r \cdot s} ,
\cr
}
$$
where $r$ and $s$ are distinct Leech roots. (We have quoted these
relations from Moody's excellent survey article [M]. Moody supposes
that the number of fundamental roots is finite, but since no argument
ever refers to infinitely many fundamental roots at once, this clearly
does not matter. See also [K89].)
Then we conjecture that $L_\infty$ provides a natural setting for the
Monster, and more specifically that the Monster can be regarded as a
subquotient of the automorphism group of some naturally determined
subquotient algebra of $L_\infty$.
The main problem is to ``cut $L_\infty$ down to size'. Here are some
suggestions. A rather trivial remark is that we can replace the
Cartan subalgebra $H$ of $L_\infty$ by the homomorphic image obtained
by adding the relations
$$
c_1~h(r_1 )~+~c_2~h(r_2 )~+~\ldots
~=~0
$$
for Leech roots $r_1 , r_2 , \ldots$, whenever
$c_1 , c_2 , \ldots$ are integers for which
$$
c_1 r_1~+~c_2 r_2~+~\ldots~=~0~.
$$
A more significant idea is to replace $L_\infty$ by some kind of
completion allowing us to form infinite linear combinations of the
generators, and then restrict to the subalgebra fixed by all the graph
automorphisms. The resulting algebra, supposing it can be defined,
would almost certainly not have any notion of root system.
Other subalgebras of $L_\infty$ are associated with the holes in the
Leech lattice, which are either ``deep'' holes or ``shallow'' holes (see
[C-S Chapter 23]).
(i) By [C-S Chapter 23], any deep hole corresponds to a Niemeier
lattice $N$, which has a Witt part which is a direct sum of root
lattices chosen from the list $A_n$ $(n~=~1,2,\ldots)$, $D_n$
$(n~=~4,5,\ldots)$, $E_6$, $E_7$ and $E_8$. Only 23 particular
combinations arise, and we shall take $A_{11} D_7 E_6$ as our standard
example. The graph of Leech roots contains a finite subgraph which is
the disjoint union of extended Dynkin diagrams corresponding to these
Witt components $W$ of $N$, and so our algebra $L_\infty$ has a
subalgebra $L[N]$ which is a direct sum of the Euclidean Lie algebras
$E(W)$ corresponding to those components (see [K89]). For
example, $L_\infty$ has a subalgebra
$$
E(A_{11} )~+~E (D_7 )~+~E(E_6 )~.
$$
Each such subalgebra of $L_\infty$ can be extended to a larger
subalgebra $L^*(N)$ having one more fundamental root, corresponding to
a ``glue vector' of the appropriate hole (see [C-S Chapter 25]). In
the corresponding graph, the new node is joined to a single special
node in each component. The graph for $L^*[A_{11} D_7 E_6 ]$ is shown in
Figure 30.1 of [C-S]. (A special node of a connected extended Dynkin
diagram is one whose deletion would result in the corresponding
ordinary diagram.) These hyperbolic algebras $L^*[N]$, having finite
rank, are certainly more manageable than $L_\infty$ itself. Since the
23 Niemeier lattices yield 23 constructions for the Leech lattice
([C-S Chapter 25]), it is natural to ask if we can obtain 23 different
constructions for the Monster using the Lie algebras $L[N]$ or $L^*[N]$.
(ii) Each shallow hole in the Leech lattice
(see [C-S Chapter 24])
corresponds to a maximal subalgebra of
$L_\infty$ of finite rank.
We are making various calculations concerning $L_\infty$ (finding the
multiplicities of certain roots via the Weyl-MacDonald-Kac formula,
etc.). It is worth noting that these calculations are facilitated by
the remarkable recent discovery that the Mathieu group
$M_{12}$ is generated by the two permutations
$$
t~\mapsto ~|2t|~,~~~~~~t~\mapsto ~11-t~( {\rm mod} ~23)~,
$$
of the set $\{ 0,1,2,3,4,5,6,7,8,9,10,11 \}$, where $|x|$ denotes the
unique $y$ in this set for which $y~\equiv~\pm x$ (${\rm mod}~23$).
(See [C-S Chapter 11]. This discovery arose from the study of
properties of various standard playing-card shuffles. We have noticed
that there are many other elements of $M_{12}$ which have simple
formulae in this ``card numbering'', for example $t~\mapsto
~|~t^3~|$.) The simplest transformation (see [C-S Chapter 28])
between the usual Euclidean coordinates for the Leech lattice and its
Lorentzian coordinates uses this description of $M_{12} $.
\proclaim References.
\item{[C-N]} J. H. Conway, S. Norton, Monstrous moonshine,
Bull. London. Math. Soc. 11 (1979) 308-339.
\item{[C-S]}
J. H. Conway, N. J. A. Sloane, Sphere packings, lattices and
groups. Third edition. Grundlehren der Mathematischen
Wissenschaften
290. Springer-Verlag, New York, 1999. ISBN:
0-387-98585-9
\item{[F]} P. Fong, Characters arising in the
Monster--modular connection. P.S.P.M. 37 (1980), 557--559.
\item{[F-L-M]}
I. B. Frenkel, J. Lepowsky, A. Meurman,
Vertex operator algebras and the monster, Academic press 1988.
\item{[K]}
V. G. Kac, An elucidation of ``Infinite dimensional algebras $\ldots$ and
the very strange formula''. Adv. in Math. 35 (1980), 264--273.
\item{[K89]}
V. G. Kac, ``Infinite dimensional Lie algebras'',
third edition, Cambridge
University Press, 1990.
\item{[M]} R. V. Moody,
Root systems of hyperbolic type, Adv. in Math. 33 (1979), 144--160.
\item{[Q]} L. Queen,
Some relations between finite groups, Lie groups, and modular functions,
PhD. dissertation, Univ. of Cambridge, 1980.
Modular functions and finite simple groups, PSPM 37 (1980), 561--566.
Modular functions arising from some finite groups, MTAC 37 (1981), 547--580.
\item{[S]} R. F. Smith, On the head characters of the monster simple group,
In ``Finite groups--coming of age'', edited by J. McKay,
Contemp. Math. 45 (1985) p. 303--313.
\item{[V]}
\`E. B. Vinberg,
Some arithmetical discrete groups in \hbox{Loba\v cevski\u\i} spaces.
Discrete subgroups of Lie groups and applications to moduli
(Internat. Colloq., Bombay, 1973), pp. 323--348. Oxford Univ. Press,
Bombay, 1975.
\bigskip
Remark added 1998: The Lie algebra of this paper is indeed closely
related to the monster simple group. In order to get a well behaved
Lie algebra it turns out to be necessary to add some imaginary simple
roots to the ``Leech roots''. This gives the {\it fake} monster Lie
algebra, which contains the Lie algebra of this paper as a large
subalgebra. See ``The monster Lie algebra'', Adv. Math. Vol. 83,
No. 1, Sept. 1990, for details (but note that the fake monster Lie
algebra is called the monster Lie algebra in this paper). The term
``monster Lie algebra'' is now used to refer to a certain
``$Z/2Z$-twisted'' version of the fake monster Lie algebra. The
monster Lie algebra is acted on by the monster simple group, and can
be used to show that the monster module constructed by Frenkel,
Lepowsky, and Meurman satisfies the moonshine conjectures; see
``Monstrous moonshine and monstrous Lie superalgebras'', Invent. Math.
109, 405-444 (1992).
\bye