\magnification=\magstep1
\vbadness=10000
\hbadness=10000
\tolerance=10000
\newcount\sectionnumber \sectionnumber
=0 \newcount\subsectionnumber
\def\section#1{\advance\sectionnumber by 1 \bigbreak
{\bf\number\sectionnumber \ #1}
\medskip\par\nobreak \subsectionnumber=0}
\def\s{\number\sectionnumber{$\cdot$}}
\def\leftdenom{\prod_{r\in \pos}(1-e^r)^{p_{24}(1-r^2/2)}}
\def\ii{\hbox{$II_{25,1}$}}
\def\pos{\hbox{$\Pi^+$}}
\def\simple{\Pi^S}
\def\gkm{generalized Kac-Moody algebra }
\def\gkms{generalized Kac-Moody algebras }
\def\mult{{\rm mult}}
\def\height{{\rm ht}}
\def\Ad{{\rm Ad}}
\def\Aut{{\rm Aut}}
\def\half{{1 \over 2}}
\def\cite#1{[#1]}
\def\bibitem#1#2#3{\item{[#2]} #3}
\def\bora{1}
\def\borb{2}
\def\borc{3}
\def\bord{4}
\def\bore{5}
\def\borf{6}
\def\borg{7}
\def\conb{8}
\def\conc{9}
\def\cond{10}
\def\goda{11}
\def\guna{12}
\def\kaca{13}
\def\frea{14}
\def\freb{15}
\def\rada{16}
\def\sera{17}
\proclaim The monster Lie algebra.
Adv. Math. Vol 83, No. 1, September 1990, p. 30-47.
Richard E. Borcherds,
Department of pure mathematics and mathematical statistics,
16 Mill Lane,
Cambridge CB2 1SB,
England.
We calculate the multiplicities of all the roots of the
``monster Lie algebra''. This gives an example of a
Lie algebra all of whose simple roots and root multiplicities are
known and which is not finite dimensional or an affine Kac-Moody algebra.
There seem to be several
similar infinite dimensional Lie algebras, which to a sympathetic eye
appear to correspond to some of the sporadic simple groups.
\item 1 Introduction and notation.
\item 2 Generalized Kac-Moody algebras.
\item 3 The monster Lie algebra.
\item 4 There are no other simple roots.
\item 5 Corollaries.
\item 6 Lie algebras for other simple groups.
\section{Introduction}
We will prove
\proclaim Theorem 1. Let \ii be the 26 dimensional even unimodular
Lorentzian lattice, and let $\rho$ be a Weyl vector for
some choice of simple roots of its reflection group $W$ (so
$\rho$ has norm 0). We define the monster Lie algebra $M$
to be the \gkm with root lattice \ii whose simple roots are the simple
roots of $W$, together with the positive multiples of $\rho$, each
with multiplicity 24. Then any nonzero root $r\in\ii$ of $M$ has multiplicity
$p_{24}(1-r^2/24)$, which is the number of partitions of $1-r^2/2$ into
parts of 24 colours.
This is equivalent to the following ``denominator formula''.
$$e^\rho\leftdenom = \sum_{w\in W\atop n\in Z}\det(w)\tau(n)e^{w(n\rho)}$$
where $\tau(n)$ is the Ramanujan tau function, and $\pos$ is the
set of vectors of \ii which are either positive multiples of $\rho$
or have negative inner product with $\rho$.
We give a brief outline of the proof of theorem 1. In section 3
we construct a Lie algebra $M$ from the Lorentzian lattice \ii using vertex algebras.
The ``no ghost'' theorem (\cite{\goda}) allows us to calculate
the multiplicities of the roots of $M$, which are as given in theorem 1,
and also allows us to prove that $M$ is a \gkm. We can find the real
simple roots of $M$ by using Conway's theorem (\cite\conb ) which
describes the reflection group of the lattice \ii, and we
show that the positive multiples of $\rho$ are simple roots of
multiplicity 24.
To complete the proof of theorem 1 we have to show that $M$ has
no other simple roots.
To do this we examine both sides of the ``denominator formula'' of $M$.
We can evaluate one side of this by using
Hecke operators, and it turns out to be a sort of modular form.
The other side of the denominator formula is a sum of terms of the
form $q^n$ times a modular form, where $n$ is proportional to
the norm of $r+\rho$ for some simple root $r$. By looking at the asymptotic behaviour
of both sides at the cusp 0 we prove that all the $n$'s must be 0,
which can be used to show that there are no other simple roots and proves theorem
1.
Section 5 contains a number of corollaries of theorem 1; for example
any 25 dimensional unimodular positive definite lattice must have a root.
In section 6 we give some not very convincing evidence for
the existence of some more Lie algebras similar to the monster Lie algebra $M$
associated to some of the other sporadic simple groups.
The results we use here can be found in the following places. The results
about vertex algebras we use are taken from Borcherds \cite{\bora}, and proofs
of these have recently been written up by Lepowsky, Frenkel and
Meurman in their book \cite{\freb}. The no ghost theorem is proved in
Goddard and Thorn \cite{\goda} and in Frenkel \cite{\frea}.
We use some standard results
about Kac-Moody algebras which can all be found in Kac's book
\cite{\kaca}, and the results about \gkms that we need are in
Borcherds \cite{\borb} and \cite{\borc} and are summarized in section 2.
The results about the geometry of the
Leech lattice $\Lambda$ and the Lorentzian lattice \ii that
we need can be found either in the original papers by Conway, Parker,
and Sloane,
most of which are reprinted in the book \cite{\conc} by Conway and Sloane,
or in \cite{\bore}.
Finally the results on modular forms, Hecke operators and theta functions
that we use can be found in any book on modular forms, for example
Serre \cite{\sera} or Gunning \cite{\guna}.
Remark. The multiplicities $p_{24}(1+n)$ of the roots of the monster
Lie algebra are given by Rademacher's formula
(\cite{\rada, Chapter 15})
$$p_{24}(1+n) = 2\pi n^{-13/2}\sum_{k>0}{I_{13}(4\pi \sqrt{n}/k)\over k}\sum_{0\leq h,h'0}(1-q^n)^{-24} =
\Delta(q)^{-1} = q^{-1} + 24 + 324q + 3200q^2+ 25650q^3 + 176256q^4
+ 1073720q^5 +\ldots.$ These are the multiplicities of roots of the
monster Lie algebra $M$.
\item{$\tau(n)$} is the Ramanujan tau function, whose generating
function is
\item{$\Delta(q)$}$ = \sum_n\tau(n)q^n =
q\prod_{n>0}(1-q^n)^{24} = q-24q^2+252q^3+\ldots.$
\item{$\Theta_\Lambda(q)$} = $\sum_{\lambda\in \Lambda}q^{\lambda^2/2} =
1 + 196560q^2 +\ldots$ is the theta function of the Leech lattice.
\item{$c(n)$} are the coefficients of the elliptic modular function $j(q)-720$.
\item{$j(q)$} is the elliptic modular function with $j(q) -720 = \sum_nc(n)q^n
= q^{-1} + 24 + 196884q
+\ldots = \Theta_\Lambda(q)/\Delta(q)$
\item {$q,\tau$} are complex variables, with $q=e^{2\pi i\tau}$, $|q|<1$,
and ${\rm Im}(\tau) >0$.
}
\section{Generalized Kac-Moody algebras}
We recall the results about \gkms that we need.
We will modify the original definition of \gkms in [\borb] slightly so that these algebras
are closed under taking universal central extensions, as in [\borc].
Suppose that $a_{ij}, i,j\in I$ is a symmetric (possibly infinite) real matrix
such that $a_{ij}\leq 0$ if $i\neq j$ and such that if $a_{ii}>0$
then $2a_{ij}/a_{ii}$ is an integer for any $j$. Then the universal
\gkm of this matrix is defined to be the Lie algebra
generated by elements $e_i, f_i, h_{ij}$ for $i,j\in I$
satisfying the following relations:
\item {1.} $[e_i,f_j] = h_{ij}$
\item{2.}$[h_{ij},e_k] = \delta_i^ja_{ik}e_k, [h_{ij},f_k] = -\delta_i^ja_{ik}f_k$
\item{3.}If $a_{ii}>0$ then $\Ad(e_i)^ne_j = \Ad(f_i)^nf_j = 0$, where $n =
1-2a_{ij}/a_{ii}$.
\item{4.}If $a_{ii}\leq 0, a_{jj}\leq 0$ and $a_{ij}=0$ then $[e_i,e_j] = [f_i,f_j]=0$.
If $a_{ii}>0$ for all $i\in I$ then
this is the same as the Kac-Moody algebra with symmetrized Cartan
matrix $a_{ij}$.
In general these algebras
have almost all the good properties that Kac-Moody algebras have,
and the only major difference is that \gkms are allowed to have
imaginary simple roots.
We list
some of their properties from [\borb] and [\borc].
(1) The element $h_{ij}$ is 0 unless the $i$'th and $j$'th columns of $a$
are equal. The elements $h_{ij}$ for which the $i$'th and $j$'th columns
of $a$ are equal form a basis for an abelian subalgebra $H$ of $G$, called
its Cartan subalgebra. (In the case of Kac-Moody algebras, the $i$'th and
$j$'th columns of $a$ cannot be equal unless $i=j$, so the only nonzero
elements $h_{ij}$ are those of the form $h_{ii}$, which are usually denoted
by $h_i$.) Every nonzero ideal of $G$ has nonzero intersection with $H$.
The centre of $G$ is contained in $H$ and contains all
the elements $h_{ij}$ for $i\neq j$. If $a$ is not the direct sum
of two smaller matrices, not the $1\times 1$ zero matrix, and not the matrix
of an affine Kac-Moody algebra, then $G$ modulo its centre is simple.
(2) If $a$ has no zero columns then $G$ is perfect and equal to its own
universal central extension. (This is why we need the elements $h_{ij}$
for $i\neq j$.)
(3) Suppose we choose a positive integer $n_i$ for each $i\in I$. Then we
can grade $G$ by putting $\deg(e_i) = -\deg(f_i) = n_i$. The degree 0
piece of $G$ is the Cartan subalgebra $H$.
(4) $G$ has an involution $\omega$ with $\omega(e_i) =-f_i, \omega(f_i) =-e_i$,
called the Cartan involution.
There is an invariant inner product (,) on $G$ such that
$(e_i,f_i) = 1$ for all $i$, and it also has the property that $-(g,\omega(g))>0$
whenever $g$ is a homogeneous element of nonzero degree.
(5) There is a character formula for simple highest weight modules $M_\lambda$
of $G$ with highest weight $\lambda$, which states that
$$ Ch(M_\lambda)e^\rho\prod_{\alpha\in \Pi^+}(1-e^{\alpha})^{\mult(\alpha)} =
\sum_{w\in W}\det(w)w(e^\rho
\sum_\alpha \epsilon(\alpha)e^{\alpha+\lambda})$$
The only case of this we need is for the one dimensional module, when
it becomes the denominator formula
$$ e^\rho\prod_{\alpha\in \Pi^+}(1-e^{\alpha})^{\mult(\alpha)} =
\sum_{w\in W}\det(w)w(e^\rho
\sum_\alpha \epsilon(\alpha)e^\alpha)$$
Here $\rho$ is the Weyl vector (i.e. a vector with $(\rho,r) = -r^2/2$
for all simple roots $r$), $\Pi$ is the set of positive roots
$\alpha$,
$W$ is the Weyl group, and $\epsilon(\alpha)$ is $(-1)^n$ if $\alpha$ is
the sum of $n$ distinct pairwise perpendicular imaginary simple roots that
are all perpendicular to $\lambda$,
and 0 otherwise.
This formula is still correct if $\rho$ is replaced by a vector
which has inner product $-r^2/2$ with all {\sl real} simple roots $r$,
because $e^{w(\alpha+\rho)-\rho}$ depends only on the inner product
of $\rho$ with the real simple roots. If $G$ is a Kac-Moody algebra then
there are no imaginary simple roots so the sum over $\alpha$ is 1 and
we recover the usual character and denominator formulas.
We also need the following characterization of \gkms from \cite{\borb}.
\proclaim Theorem \s.
A Z-graded Lie algebra $G = \bigoplus_{i\in Z}G_i$ is the quotient
of a generalized Kac-Moody algebra graded as in (3) above by
a subspace of its centre if and only if it has the following four properties.
\item {1} $G_0\subset [G,G]$
\item{2} $G$ has an involution $\omega$ which acts as -1 on $G_0$ and maps
$G_i$ to $G_{-i}$
\item{3} $G$ has an invariant bilinear form (,) such that $G_i$ and $G_j$
are orthogonal if $i\neq -j$, and such that $-(g,\omega(g))>0$ if $g$
is a nonzero homogeneous element of $G$ of nonzero degree.
\item{4} As a module over $G_0$, $G$ is a sum of finite dimensional submodules.
(If condition (1) is omitted it is still easy to describe $G$,
by adding some outer derivations to a \gkm.)
\section{The monster Lie algebra.}
We construct a \gkm algebra $M$, called the monster Lie algebra,
with root lattice \ii whose root multiplicities
are given by the number of partitions of an integer into parts of
24 colours. We find the real simple roots of this algebra by using
Conway's description of the reflection group of \ii, and we
show that all positive multiples of $\rho$ are simple roots of
multiplicity 24.
In this section we will prove
\proclaim Theorem 3. There is a \gkm $M$ with root lattice \ii
such that the multiplicity of the nonzero vector $r\in \ii$ is
$p_{24}(1-r^2/2)$. The real simple roots of $M$ are the norm 2
vectors $r$ with $(r,\rho)$ = -1, and the
positive multiples of $\rho$ are simple roots of multiplicity 24.
We will later show that $M$ has no other simple roots.
\proclaim Lemma 1. There is a Lie algebra $M$ with the following properties.
\item{1.} $M$ is graded by \ii. The piece of degree $r\in \ii, r\neq 0$ has
dimension $p_{24}(1-r^2/2)$.
\item{2.} $M$ has an involution $\omega$, which acts as -1 on $\ii$ and on
the piece of $M$ of degree $0\in \ii$.
\item {3.} $M$ has a contravariant bilinear form (,) such that the pieces of $M$
of degrees $i,j\in \ii$ are orthogonal if $i\neq j$ and such that (,)
is positive definite on the piece of $M$ of degree $i\in \ii$ if $i\neq 0$.
Proof: This Lie algebra was constructed in Borcherds [\bora]; we briefly recall
its construction. Given any nonsingular even lattice $L$ we can can construct
its vertex algebra $V$, which is an $L$-graded vector space with a large number
of operations. In particular there is an action of the Virasoro algebra
on it, where the Virasoro algebra is spanned by the operators 1 and $L_i,i\in Z$
with the relations
$$[L_i,L_j]=(i-j)L_{i+j}+ {i^3-i\over 12}\dim(L)\delta^i_{-j}$$
We let $P^i$ be the space of vectors $v\in V$ satisfying $L_0(v)=iv$,
$L_n(v)=0$ if $n<0$. Then one of the results of [\bora] is that $P^1/L_1(P^0)$
can be made into a Lie algebra. The space $V$ has an involution $\omega$
with the property (2) above and an inner product (,) which when
restricted to $P^1$ is singular on $L_1(P^0)$ and defines a contravariant
inner product on the Lie algebra $P^1/L_1(P_0)$. The pieces of this Lie
algebra of degrees $i,j\in L$ are orthogonal unless $i=j$.
So far this construction can be done for any even lattice. In the special case
of a 26 dimensional Lorentzian lattice, the ``no ghost'' theorem
of Goddard and Thorn [\goda] implies
that if $r\in L,r\neq 0$ then the restriction of the inner product (,)
to the degree $r$ piece of $P^1/L_1(P^0)$ is positive semidefinite
and the quotient by its kernel has dimension $p_{24}(1-r^2/2)$. The fact that
(,) is contravariant implies that its kernel is an ideal of $P^1/L_1(P^0)$,
so if we define $M$ to be the quotient of $P^1/L_1(P^0)$ by the kernel
of (,), for $L=\ii$, then $M$ has all the properties stated in the lemma. Q.E.D.
\proclaim Lemma 2. The Lie algebra $M$ is a \gkm.
Proof.
If we fix any negative norm vector $r$ of \ii not perpendicular to any
norm 2 vectors, then we can make $M$ into a $Z$-graded Lie algebra
by using the inner product with $r$ as the degree. All the
conditions of theorem 2 are satisfied for $M$,
so $M$ is a \gkm.
We now fix a primitive norm 0 vector $\rho$ of \ii which is not perpendicular
to any norm 2 vectors. (All such vectors $\rho$ are conjugate under
$\Aut(\ii)$.) We choose a fundamental domain for the reflection group $W$
of \ii containing $\rho$.
\proclaim Lemma 3. The positive norm simple roots of $M$ are the norm
2 vectors $r\in \ii$ with $(r,\rho)=-1$.
Proof: All norm 2 vectors of \ii have multiplicity 1, so the positive norm simple roots of
$M$ are just the simple roots of the reflection group of \ii. This reflection
group was described by Conway \cite{\conb} who showed that its simple roots are the norm 2 vectors
$r$
of \ii with $(r,\rho) = -1$. Q.E.D.
\proclaim Lemma 4. The
positive multiples of $\rho$ are simple roots of multiplicity 24.
Proof.
As $(\rho,\rho) = 0$ and $(\rho,r)<0$ for all other simple roots $r$,
no multiple of $\rho$ can be written as a sum of simple roots, unless
all the roots in the sum are perpendicular to each other. Hence the simple
root $n\omega, n>0$ has multiplicity equal to the multiplicity of the root
$n\omega$, which is 24. Q.E.D.
\section{There are no other simple roots.}
We complete the proof of theorem 1 by showing that the Lie algebra
$M$ of the previous section has no other simple roots.
\proclaim Lemma 1.
$$e^\rho\leftdenom
= \sum_{w\in W}\det(w)w(\sum_{n\in Z}\tau(n)e^{n\rho} - \sum_{r\in
\simple,r^2<0}e^{r+\rho}) $$
where $\pos$ is the set of positive roots of the monster Lie algebra,
and $\simple$ is the collection of simple roots of the monster Lie algebra,
counted with multiplicities.
Proof. We show that this is just the denominator formula of section 2
for the \gkm $M$.
The vector $\rho$ has inner product $-1 = r^2/2$ with all real simple roots
$r$ by Conway's theorem (Lemma 3 above), so to prove this is the denominator
formula
we have to check that
$$\sum_{n\in Z}\tau(n)e^{n\rho} - \sum_{r\in \simple,r^2<0}e^{r+\rho}
= e^\rho\sum_\alpha\epsilon(\alpha)e^\alpha $$
where $\epsilon(\alpha)$ is the sum of a term $(-1)^n$ for each way
of writing $\alpha$ as a sum of $n$ distinct pairwise perpendicular
imaginary simple roots. Two imaginary roots can only be perpendicular
if they are both of norm 0 and proportional, so if
$\alpha$ is not a multiple of $\rho$ then $\epsilon(\alpha)$ is
just the multiplicity of $\alpha$ as a simple root. All positive
multiples of $\rho $ have multiplicity 24 and are perpendicular to each other, so $\epsilon(n\rho)$
is the coefficient of $q^n$ in $\prod_n(1-q^n)^{24} = q^{-1}\sum_n\tau(n)q^n$,
which proves lemma 1.
We recall that $\ii =\Lambda\oplus U$, where $U$ is a two dimensional
lattice spanned by $\rho, \rho'$ with $\rho^2 = \rho'^2 = 0, (\rho,\rho') =
-1$. We project both sides of the denominator formula from the
group ring of the lattice \ii onto the group ring of the lattice $U$,
which makes them easier to calculate. More precisely, we define the projection
$P$ from the group ring of \ii to the space of Laurent series in $p$ and $q$
by
$$ P(e^r) = p^{-(\rho,r)}q^{-(\rho',r)}$$
so that $P$ applied to each side of the denominator formula is a Laurent
series in $p$ and $q$.
\proclaim Lemma 2. If $m>0$ then
$$-\log(P(\leftdenom))
= \sum_{m>0}T_m(j(q)-720)p^m + 24\sum_{n>0}{\sigma(n)\over n}q^n
$$
where $\sigma(n)$ is the sum of the divisors of $n$, $j(q)$ is the
elliptic modular function, and $T_m$ is a Hecke operator (see Serre
\cite{\sera}, especially proposition 12 of chapter VII, section 5).
In particular the coefficient of $p^m$ for $m>0$
is a modular function of $q$ of level 1.
Proof. The number of vectors of norm $2a$ of \ii projecting onto
the vector $n\rho+m\rho'$ of $U$ is the number of vectors of $\Lambda$
of norm $2a+2mn$,
so the sum of the multiplicities of the roots
of $M$ that project onto $n\rho+m\rho'\in U$ is equal to the coefficient
$c(mn)$ of $\Theta_\Lambda(q)\sum_np_{24}(1+n)q^n = j(q)-720 = \sum_n c(n)q^n$.
The left hand side of the formula above is therefore equal to
$$\eqalign{
& -\log\prod_{m\geq 0}\prod_{n\in Z,n>0 \hbox{if }m=0}(1-p^mq^n)^{c(mn)}\cr
=& \sum_{m\geq 0}\sum_{n\in Z,n>0\hbox{ if }m=0}\sum_{k\geq 1} c(mn)p^{mk}q^{nk}/k \cr
=& \sum_{m>0}\sum_{n\in Z}\sum_{00}\sum_{00} T_m(\sum_{n\in Z}c(n)q^n)p^m + c(0)\sum_{n>0}{\sigma(n)\over n}q^n \cr
}$$
which is equal to the right hand side. Q.E.D.
\proclaim Lemma 3.
$$P(e^\rho\leftdenom) =\Delta(q)\Theta_\Lambda(p) - \Theta_\Lambda(q)
\Delta(p)$$
where
$$\Delta(q) = q\prod_{n>0}(1-q^n)^{24} = \sum_n\tau(n)q^n = q-24q^2+252q^3\ldots$$
is the generating function of Ramanujan's tau function, and
$$\Theta_\Lambda(q) = \sum_{\lambda\in\Lambda} q^{\lambda^2/2}
= 1 + 196560q^2+\ldots$$
is the theta function of the Leech lattice $\Lambda$.
In particular the coefficient of $p^m$ is a modular form in $q$ of weight
12 which is holomorphic at all cusps.
Proof. The left hand side is equal to
$$q\exp(\log(P(\leftdenom)))
= q\prod_{i}(1-q^i)^{24}\exp(-\sum_{m>0}p^mT_m(j(q)-720))$$
by lemma 2, so the coefficient of $p^m$ is a modular form of weight 12 and level 1
for any $m$, because $q\prod_i(1-q^i)^{24}$ is a modular form of weight
12 and level 1, and each of the functions $T_m(j(q)-720)$ is a modular
function of level 1.
Therefore if we let $F(p,q)$ stand for the left hand side
of the expression above, it has the properties that
the coefficient of $p^m$ is a modular form of weight 12 and level 1
for each $m$ and is zero for $m<0$. Moreover $F(p,q) = -F(q,p)$
because it is antisymmetric under reflection in the
root $\rho-\rho'$ of \ii. We will show that there is only one
function $F(p,q) $ with these properties whose coefficient of
$q^1p^0$ is 1, which will prove lemma 3 because
$\Delta(q)\Theta_\Lambda(p) - \Theta_\Lambda(q)\Delta(p)$ also
has these properties.
The coefficients must all be modular forms holomorphic at the cusp
$\infty$ because the coefficient of $q^np^m$ is 0 if $n<0$ by
antisymmetry. We will now repeatedly use the fact that a holomorphic
modular form of weight 12 and level 1 is determined by its coefficients
of $q^0$ and $q^1$. The coefficient of $p^0q^0$ of $F$ is 0 by antisymmetry
and the coefficient of $p^0q^1$ is 1 by assumption, so the
coefficient of $p^0q^n$ is determined for all $n$ because the coefficient
of $p^0$ is a modular form of weight 12 and level 1. Similarly
the coefficients of $p^1q^0$ and $p^1q^1$ are -1 and 0 by
antisymmetry, so the coefficient of $p^1q^n$ is determined.
Finally for any $m$ the coefficients of $p^mq^0$ and $p^mq^1$ are determined
by antisymmetry, so the coefficient of $p^mq^n$ is determined.
This proves lemma 3.
This identifies the left hand side of the denominator formula. We now
examine the right hand side by splitting up the terms into
orbits under an action of the Leech lattice.
We define the height $\height(r)$ of a vector $r\in \ii$ to be $-(r,\rho)$.
We prove that there
are no simple roots of negative norm by induction on the height, so
we assume that we are given a positive integer $m$ such that there
are no simple roots of negative norm of height less than $m$, and we will prove
there are none of height $m$.
\proclaim Lemma 4.
$$p^{-m}P(\sum_{w\in W,n\in Z\atop\height(nw(\rho))=m}\det(w)\tau(n)e^{nw(\rho)}
- \sum_{r\in \simple,\height(r)=m}e^{\rho+r}) $$
is a modular form of weight 12 and level 1.
Proof.
If $r$ is any simple root of negative norm, then $\height(w(r))>\height(r)$
for any nontrivial element $w$ of $W$. By assumption there are no
simple roots of negative norm of height less than $m$, so
the only way that $w(e^{r+\rho})$ can have height $m$
for some imaginary simple root $r$ is that either $r$ is a multiple of
$\rho$ or $w=1$.
Therefore the sum in lemma 4 is the coefficient of $p^m$ of
$$\sum_{w\in W} \det(w)w(\sum_{n\in Z}\tau(n)e^{n\rho}-\sum_{r\in \Pi^S,r^2<0}
e^{r+\rho})$$
and lemma 4 now follows from lemmas 1 and 3.
\proclaim Lemma 5. Recall that $\ii = \Lambda\oplus U$. We write vectors
of \ii in the form $(v,m,n)$ with $v\in \Lambda$, $m\in Z$ and $n\in Z$,
and this vector has norm $\lambda^2-2mn$. For each
$\lambda\in \Lambda$ the map taking $(v,m,n)\in \ii$ to
$(v+m\lambda,m,n+(v,\lambda)+m\lambda^2/2) $ is an automorphism of \ii,
and this defines an action of $\Lambda$ on \ii.
Proof: An easy check.
(This gives a natural identification of $\rho^\perp/\rho = \Lambda$
with the group of automorphisms of \ii which fix $\rho$ and every vector
of $\rho^\perp/\rho$.)
\proclaim Lemma 6. For any nonzero integer $m$ there
are only a finite number of orbits of vectors of \ii of some fixed norm and height
$m$ under the action of $\Lambda$.
Proof: Any vector $(v,m,n)\in \ii$
of given norm and nonzero height is determined by $v$, so by lemma 5
the number of orbits of such vectors under the action of $\Lambda$ is at
most the order $m^{24}$ of $m\Lambda/\Lambda$. Q.E.D.
\proclaim Lemma 7. Suppose that $A$ is any orbit of vectors of \ii
of norm $-2n$ and height $m$ under the action of $\Lambda$. Then
$$p^{-m}P(\sum_{a\in A}e^a) = q^{n/m}\Theta_A(q)$$
where $\Theta_A(q)$ is a modular form of weight 12 with nonnegative coefficients
and ``constant term'' $m^{-12}$ at the cusp 0.
(The constant term of a modular form $\Theta(q)$ of weight $k$
at the cusp 0 is the limit of $\tau^k\Theta(e^{2\pi i\tau})$ as
$\tau$ tends to 0 along the positive imaginary axis.)
Proof. Let $(v,m,(v^2/2+n)/m)$ be any vector in the orbit $A$. Then
$$\eqalign{p^{-m}P(\sum_{a\in A}e^a)& = \sum_{a\in A}q^{-(a,\rho')}\cr
& = \sum_{\lambda\in
\Lambda}q^{-((v+m\lambda,m,(v^2/2+n)/m+(v,\lambda)+m\lambda^2/2),\rho')}\cr
& = \sum_{\lambda\in \Lambda}q^{ v^2/2m+n/m+(v,\lambda)+m\lambda^2/2}\cr
&= q^{n/m}\sum_{\lambda\in\Lambda}q^{m(\lambda-v/m)^2/2}\cr
& = q^{n/m}\Theta_A(q)\cr}$$
where $\Theta_A(q)$ is the theta function of a coset of the lattice
$\sqrt m\Lambda$ of determinant $m^{24}$ and is therefore a modular form
of weight $\dim(\Lambda)/2 = 12$ with nonnegative coefficients.
The generalized Jacobi inversion formula (Gunning \cite{\guna}, section 20)
states that
$$\Theta_A(e^{2\pi i\tau}) = {(-i\tau)^{-12}\det(\sqrt{m}\Lambda)}^{-1/2}
\sum_{\lambda\in \Lambda}e^{2\pi i((v,\lambda)-\lambda^2/2\tau)/m}$$
so $\Theta_A(q)$
has constant term $(\det(\sqrt
m\Lambda))^{-1/2} =
m^{-12}$ at the cusp 0. Q.E.D.
\proclaim Lemma 8. The sum $$\sum_Aq^{n(A)/m}\Theta_A(q)$$
where the sum is over all orbits $A$ of simple roots of height $m$,
is a modular form of weight 12. Each $n(A)$ is a positive integer
equal to $-(a+\rho)^2/2 = m-a^2/2$ for $a\in A$
and there are only a finite number of orbits $A$ with any given value of $n(A)$.
Each function $\Theta_A(q)$ is a modular form of weight 12 with nonnegative
coefficients which has constant term $m^{-12}$ at the cusp 0.
Proof. This follows from lemmas 4, 6, and 7 because if $a$ is a vector of
norm at most 0 with $(a,\rho)\leq 0$ which is not a multiple of $\rho$ then $(a+\rho)^2<0$.
We now complete the proof of theorem 1 by showing that the sum in
lemma 8 must be empty. We will do this by studying the behaviour
of the sum as $\tau$ tends to 0 through values on the positive imaginary axis.
The sum in lemma 8 is a sum of power series with positive coefficients whose
formal sum converges for $|q|<1$, so the sum of the power series also
converges for $|q|<1$. If $f(\tau)$ is a modular form of weight 12
with constant term $c$ at the cusp 0 then
$$f(\tau) = \tau^{-12}c + s(\tau),$$
where $s(\tau)$ tends to 0 faster than any power of $\tau$ as
$\tau$ tends to 0 along the imaginary axis.
Therefore
$$\tau^{-12}c + s(\tau) = \sum_Am^{-12}(e^{2\pi in(A)\tau/m}\tau^{-12} +s_A(\tau))$$
where each term in the sum on the right is nonnegative, and $s$ and $s_A$
are functions tending to zero faster than any power of $\tau$ as $\tau$
tends to zero along the imaginary axis.
In particular the right hand side is bounded as $\tau$ tends to 0,
so there are only a finite number of terms in the sum because
each term on the right tends to $m^{-12}$.
Therefore
$$c - \sum_{n>0}m^{-12}r(n)e^{2\pi in\tau/m}$$
tends to 0 faster than any power of $\tau$ as $\tau$ tends
to 0 along the positive imaginary axis,
where $r(n)$ is the number of orbits
$A$ with $n(A)=n$. All the $r(n)$'s are nonnegative integers, so this is impossible
unless they are all 0, so there are no simple roots of norm at most 0 and
height $m$. This proves theorem 1.
\section{Corollaries}
We deduce several results about the monster Lie algebra and
about lattices from our main result.
\proclaim Corollary 1. The universal central extension $\hat M$ of the monster
Lie algebra $M$ is a \ii-graded Lie algebra, and if $0\neq r\in \ii$ then
the subspace of $\hat M$ of degree $r$ is mapped isomorphically onto that of
$M$. The subspace of $\hat M$ of degree $0\in \ii$ (i.e. its ``Cartan
subalgebra'') can be represented naturally as the sum of a one dimensional space for
for each vector of the Leech lattice and a space of dimension $24^2=576$
for each positive integer.
Proof. This follows from the description of the Cartan subalgebra of the
universal central extension of a generalized Kac-Moody algebra in \cite{\borc}
or section 2
as the sum of a space of dimension $n^2$ for each simple root of multiplicity
$n$. The monster Lie algebra $M$ has a real simple root of multiplicity 1
for each vector of the Leech lattice, and a simple root of multiplicity 24
for each positive multiple of $\rho$. Q.E.D.
\proclaim Corollary 2.
$$e^\rho\leftdenom = \sum_{w\in W} \sum_{n\in Z}
\det(w)\tau(n)e^{\omega(n\rho)}$$
This is just the denominator formula for the \gkm M.
\proclaim Corollary 3. If $r\in \pos$ then
$$ (r+\rho)^2m(r) =
\sum_{\alpha,\beta\in\pos\atop\alpha+\beta=r}(\alpha,\beta)m(\alpha)m(\beta)$$
where $m(r)$ is defined to be $\sum_{n>0,n\in Z,r/n\in\ii}\mult(r/n)/n$
(which is equal to the multiplicity of $r$ if $r$ is primitive).
This is just the Peterson recursion formula for the multiplicities of the roots
of $M$, which is valid for \gkms. Notice that for the Peterson
recursion formula to hold $\rho$ must satisfy $(\rho,r) =
-r^2/2$ for all simple roots $r$ (not just the real ones), which would not be true if $M$ had any
simple roots $r$ of negative norm.
\proclaim Corollary 4. Let $N$ be the subalgebra of the monster
Lie algebra $M$ generated by the elements of $M$ whose degree has norm
2 (so that $N$ is the Kac-Moody subalgebra of $M$ generated by
the real simple roots of $N$). Then the multiplicity of the root
$r\in \ii$ of $N$ is at most $p_{24}(1-r^2/2)$, and equality
holds whenever $r$ is in the fundamental domain of $W$ and
$\height(r)>|r^2|$.
Proof. The inequality for the multiplicity of $r$ is clear because
$N$ is a subalgebra of $M$. If $r$ is in the fundamental
domain of $W$, then $r$ has inner product at most 0 with all
simple roots, so it is not possible for $r$ to be the sum of $\rho$
and some other simple roots if $|(r,\rho)| = \height(r) > r^2$.
Remark. The algebra $N$ is the original ``monster Lie algebra'' defined
by Conway, Queen and Sloane in \cite{\borg}.
(This definition missed out the norm 0 simple roots of $M$,
possibly because algebras with imaginary simple roots had not been studied
at the time.)
Remark. The inequality of corollary 4 for the
multiplicities of the roots of $N$ was first proved by Frenkel \cite{\frea}
by showing that the space $P^1/L_1P^0$ of section 3 is a module
for $N$. The lattice \ii has 121 orbits of vectors of norm -2, all
but 2 of which have multiplicity 324 as roots of $N$,
and has 665 orbits of vectors of norm
-4, all but 3 of which have norm 3200 as roots of $N$.
Recall that in lemma 1 of section 3 we defined some subspaces
$P^i$ of the space $V$
consisting of lowest weight vectors of the Virasoro algebra.
The elements of the monster Lie algebra commute with all elements of
the Virasoro algebra, considered as endomorphisms of the space $V$
of section 3, so all the spaces $P^i$ are representations of the
monster Lie algebra.
\proclaim Corollary 5. If $i<0$ then the space $P^i$ is a sum
of highest weight and lowest weight representations of $M$.
Proof. The height of any weight of $P^i, i<0$ cannot be 0
as all its weights have negative norm. As $M$ is generated by
elements of height -1,0 or 1 this means that $P^i_+$ and $P^i_-$
are both representations of $M$, where $P^i_+$ and $P^i_-$
are the sums of the elements of $P^i$ whose degrees have positive or
negative height. The representation $P^i_+$ has the property
that the norms of its weights are bounded above (by $2i$) and
its weights all have negative inner product with all
imaginary simple roots of $M$, so by the results on representations
of \gkms in \cite{\borb} $P^i_+$ is a sum of lowest weight
representations of $M$. Similarly $P^i_-$ is a sum of highest weight
representations.
\proclaim Corollary 6. Suppose that $n$ is 1 or a prime. Then
the height of vectors $v$ of norm $-2n$ in the fundamental domain
of the reflection group of \ii is a linear function of the theta
function of the lattice $v^\perp$.
Proof. The Peterson recursion formula shows that for primitive vectors of
any fixed negative norm the height is a linear function of the theta
functions of the cosets $a+v^\perp$ for $a$ in the dual of the lattice
$v^\perp$. If $n$ is 1 or a prime then the theta function of any of these
cosets is a linear function of the theta function of the dual of
$v^\perp$, which is in turn a linear function of the theta function of
$v^\perp$.
\proclaim Corollary 7. Let $v$ be a vector of \ii of negative norm,
and suppose for simplicity that there is no norm 0 vector $z$
such that $|(z,v)|<|(v,v)/2|$. Let $n$ be the number of norm 0
vectors $z$ with $(v,z) = (v,v)/2$.
\item{1.} If $v$ has norm -2 then its height is $(r+18-4n)/12$.
\item{2.} If $v$ has
norm -4 then its height is $(r+20 -2n)/8$.
\item{3.}If $v$ has norm -6
then its height is $(r+20 -n )/6$.
Proof. These are just special cases of corollary 6.
Remark. If $v$ has inner product -1 with some norm 0 vector of height $h$
then $v^\perp$ is isomorphic to the sum of a Niemeier lattice and a 1
dimensional lattice, and $v$ has height $2h+1$.
Remark. The results of corollary 7 for vectors of norm -2 and -4
were originally proved by different methods in \cite{\bord}.
There is a list of the 121 orbits of norm -2 vectors and the 665 orbits of
norm -4 vectors of \ii in \cite{\bord}.
\proclaim Corollary 8. \item{1.} If $L$ is an even 25 dimensional positive definite
lattice
of determinant 2 then the number of roots of $L$ is 6 mod 12 unless $L$
is the sum of a Niemeier lattice and a 1 dimensional lattice.
\item{2.} If $L$ is
a 25 dimensional unimodular positive definite lattice which is not the
sum of a Niemeier lattice and a one dimensional lattice then the number of
norm 2 vectors plus twice the number of norm 1 vectors is 4 mod 8 (and in
particular any such lattice has roots).
\item{3.} If $L$ is a 26 dimensional even, positive definite lattice
of determinant 3 with no roots of
norm 6 then the number of roots of $L$ is 0 mod 6.
Proof. This follows from corollary 7 by using the following facts.
There is a 1:1 correspondence between orbits of norm -2 vectors of \ii
and even 25 dimensional positive definite lattices of determinant 2, such that
$v^\perp$ is isomorphic to $L$. There is a 1:1 correspondence between orbits of
norm -4 vectors of \ii and 25 dimensional positive definite unimodular lattices
$L$ such that $v^\perp$ is isomorphic to the sublattice of even vectors of
$L$, and the number of norm 1 vectors of $L$ is equal to the number of norm 0
vectors which have inner product -2 with $v$. Finally there is a 1:1
correspondence between norm -6 vectors $v$ of \ii and roots $r$ of
26 dimensional even positive definite lattices of determinant 3 such that
$v^\perp$ is isomorphic to $r^\perp$. In each case we can work out the theta
function of $L$ from that of $v^\perp$, and this gives the results of
corollary 8.
Remark. In \cite{\bord} there is an algorithm for classifying the even 26
dimensional positive definite lattices of determinant 3. In particular there is
a unique such lattice with no
roots, and its automorphism group is $2\times ^3\!\!D_4(2)$. There are also unique
such lattices with root systems $a_1^3$, $a_2$, and $g_2$, so the congruence
above is best possible.
Finally we have the following completely useless curiosity. (There are
presumably even more curious and useless such formulae for vectors
of norms -4, -6,... .)
\proclaim Corollary 9. Suppose $v$ is a norm -2 vector of \ii not of height 1
that is in the fundamental chamber of the reflection group of \ii. Then
$$\eqalign{
&d(d-51)/2 -\hbox{\sl number of components of the Dynkin diagram of $v^\perp$}
\cr
&+\hbox{\sl number of orbits of $R$ under $\sigma$} = 324.\cr
}$$
Here $d$ is the rank of the Dynkin diagram of $v^\perp$, $R$ is the set
of simple roots of the reflection group of \ii which have inner product
$-1$ with $v$, and $\sigma $ is minus the opposition involution of $v^\perp$,
which acts on the set $R$.
Proof. In \cite{\bord} it is shown that the left hand side of this is
the multiplicity of the root $v$ of $M$, so corollary 9 follows from
theorem 1.
\section{Lie algebras for other simple groups.}
There appear to be many other non affine Kac-Moody algebras similar to
the one in this paper, many of which seem to be associated in a rather
obscure way to many of the other sporadic simple groups in the monster.
In \cite{\borf} it is shown that if $G$ is any finite subgroup of
Conway's group $\Aut(\Lambda)$, then the lattice $\Lambda^G$ of vectors
fixed by $G$ has many of the properties of $\Lambda$. In particular
if it has no roots then the reflection group of the Lorentzian lattice $\Lambda^G\oplus U$
has a norm 0 Weyl vector $\rho$. This suggests the following question.
Problem. Let $G$ be a finite subgroup of $\Aut(\ii)$ fixing $\rho\in\ii$,
and let $L$ be the Lorentzian lattice of vectors fixed by $G$. Is there a
``nice'' Lie algebra with root lattice $L$? In particular, some
experiments suggest the following conjecture.
Conjecture. Let $g$ be an element of $M_{24}\subset \Aut(\Lambda)$
of order $h$
with cycle shape $1^{24}$, $1^82^8,$ $1^63^6,$ $1^45^4,$
$1^22^23^26^2,$ $1^37^3,$ $1^211^2,$ $1^12^17^114^1,$ $1^13^15^115^1,$ or
$1^123^1$. Let $\Lambda^g$ be the lattice of vectors of $\Lambda$ fixed by $g$
and let $L$ be the Lorentzian lattice $\Lambda^g\oplus U$. If $g$ has cycle
shape $1^{n_1}2^{n_2}\ldots$ let $\Delta_g(q)$ be the modular form
$\eta(q)^{n_1}\eta(2q)^{n_2}\ldots = \sum_n\tau_g(n)q^n$.
Then we conjecture that
\item 1 $\Lambda^g$ has no roots. By the results of \cite{\borf} this implies
that the reflection group of $L$ has a norm 0 Weyl vector $\rho$ (i.e.,
$\rho$ has inner product $-(r^2/2)$ with every simple root $r$ of the
reflection group).
\item 2 The modular form $\Delta_g(q)$ has multiplicative coefficients
and is the cusp form of smallest weight for the normalizer $\Gamma_0(h)+$
of the subgroup $\Gamma_0(h)$ of
$PGL^+_2(Q)$.
\item 3 Let $M_g$ be the generalized Kac-Moody algebra whose real simple roots
are those of the reflection group of $L$, and whose imaginary simple roots
are the multiples $n\rho$ of $\rho$, with multiplicity equal to the number of
fixed points of $g^n$, so that $\Delta_g(q) = q\prod_n(1-q^n)^{\mult(n\rho)}$.
Define $p_g$ by $\sum p_g(n)q^n = 1/\Delta_g(q)$.
Then the root $r$ of $L$ has a multiplicity given by some expression
involving $p_g$, possibly
$$\sum_{d|h,(r,L)} p_g(-r^2/d)$$
where the sum is over all positive integers $d$ dividing both $h$ and
all the numbers $(r,s), s\in L$.
This seems to be the simplest expression that gives the correct multiplicity
for positive roots, multiples of $\rho$, and \ii. It does not work
if $g$ is one of the elements of $M_{24}$ of cycle shape $1^42^24^4$ or
$1^22^14^18^2$.
Conway and Norton \cite{\cond} showed that there was a natural correspondence
between automorphisms of $\Lambda$ and certain conjugacy classes in the monster simple group, and these elements of the monster often have centralizers
which are almost sporadic simple groups. For example, the elements
of $M_{24}$ of cycle shapes $1^{24}$, $1^82^8$, $1^63^6$, $1^45^4$ and $1^37^3$
correspond in this way to the monster, the baby monster, the Fischer group
$Fi'_{24}$, the Harada Norton group, and the Held group. This suggests
that the Lie algebras associated to these elements of $M_{24}$ might
be connected in some way to these simple groups. However, there is
no known direct connection even between the monster Lie algebra and
the monster simple group.
There also seems to be an interesting Lie superalgebra.
Let $U$ be the nonintegral lattice spanned by two vectors
$\rho,\rho'$ with $\rho^2=\rho'^2 =0, (\rho,\rho')=-1/2$,
and let $L$ be the direct sum of $U$ and the $E_8$ lattice.
We define a generalized Kac-Moody superalgebra whose root lattice
is $L$ by letting its real simple roots be the
norm 1 vectors $r$ of $L$ with $(\rho,r) = -1/2$, and letting its
imaginary simple roots be the positive multiples of $\rho$,
each with multiplicity 8, where the even multiples of $\rho$
are even roots, and the odd multiples of $\rho$ are odd (or ``super'')
roots. Then we conjecture that the multiplicity of a nonzero
root $r\in L$ is the coefficient of $q^{(1-r^2)/2}$ in
the modular form
$$q^{1/2}\prod_{i>0}(1-q^i)^{-8}(1+q^{i-1/2})^8.$$
of level 2. This modular form has the same relation to the
theta functions of odd unimodular lattices as $\Delta(q)$ has
to the theta functions of even unimodular lattices.
If we could prove that there was some generalized Kac-Moody
superalgebra with these root multiplicities, then it would follow
from an argument similar to the proof of theorem 1 that
its simple roots are just the ones above.
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\bye