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\proclaim Lattices like the Leech lattice.
Journal of Algebra, Vol. 130, No. 1, April 1990, 219--234.
Richard E. Borcherds,
Department of mathematics, University of California at Berkeley, Berkeley,
California 94720.
\bigskip
\proclaim Introduction.
The Leech lattice has many strange properties, discovered by Conway,
Parker, and Sloane. For example, it has covering radius $\sqrt{2}$,
and the orbits of points at distance at least $\sqrt{2}$ from all
lattice points correspond to the Niemeier lattices other than the
Leech lattice. (See Conway and Sloane [6, Chaps. 22-28].) Most of the
properties of the Leech lattice follow from the fact that it is the
Dynkin diagram of the Lorentzian lattice $II_{25,1}$, as in Conway
[4]. In this paper we show that several other well-known lattices, in
particular the Barnes-Wall lattice and the Coxeter-Todd lattice (see
[6, Chap. 4]) are related to Dynkin diagrams of reflection groups of
Lorentzian lattices; all these lattices have properties similar to
(but more complicated than) those of the Leech lattice. Conway and
Norton [5] showed that there was a strange correspondence between some
automorphisms of the Leech lattice, some elements of the monster, some
sublattices of the Leech lattice and some of the sporadic simple
groups. Many of these things also correspond to some Lorentzian
lattices behaving like $II_{25,1}$ and to some infinite dimensional
Kac-Moody algebras.
Most of the notation and terminology is standard. For proofs of the
facts about the Leech lattice that we use, see the original papers of
Conway, Parker and Sloane in chapters 23, 26, and 27 of Conway and
Sloane [6], or see Borcherds [1]. Lattices are always integral, and
usually positive definite or Lorentzian, although they are
occasionally singular. A root of a lattice means a vector $r$ of
positive norm such that reflection in the hyperplane of $r$ is an
automorphism of the lattice and such that $r$ is primitive, i.e., $r$
is not a nontrivial multiple of some other lattice vector; by a strong
root we mean a root $r$ such that $(r,r)$ divides $(r,v)$ for all $v$
in the lattice. (For example, any vector of norm 1 is a strong root.)
The symbols $a_n,b_n,\ldots, e_8$ stand for the spherical Dynkin
diagrams, and their corresponding affine Dynkin diagrams are denoted
by $A_n,B_n,\ldots,E_8$. In some of the examples we give later $E_8$
also stand for the $E_8$ lattice. The symbols $I_{n,1}$ and $II_{n,1}$
stand for odd and even unimodular Lorentzian lattices of dimension
$n+1$, which are unique up to isomorphism. The automorphism group
$\Aut(R)$ of a Lorentzian lattice $R$ means the group of automorphisms
that fix each of the two cones of negative norm vectors (so $\Aut(R)$
has index 2 in the ``full'' automorphism group of $R$).
The reflection group of a Lorentzian lattice is the group generated by
the reflections of its roots. For any Lorentzian lattice, one of the
two components of the norm $-1$ vectors can be identified with
hyperbolic space, and all automorphisms of the lattice act as
isometries on this space. In particular, a reflection of the lattice
can be thought of as a reflection in hyperbolic space, so the
reflection group of a lattice is a hyperbolic reflection group.
Section 1 contains several results useful for practical calculation of
Dynkin diagrams of Lorentzian lattices, Section 2 contains some
results about the reflection group of a sublattice fixed by some
group, and Section 3 applies the results of Sections 1 and 2 to the
Leech lattice to produce several lattices whose reflection groups
either have finite index in the automorphism group of behave like the
reflection group of the Lorentzian lattice $II_{25,1}$.
I thank J. Lepowsky for suggesting many improvements to this paper.
\proclaim 1.~Reflection groups of Lorentzian lattices.
In this section we give some theorems which help in calculating the
Dynkin diagram of a reflection group of a Lorentzian lattice. Vinberg
[7] described an algorithm for finding the Dynkin diagram of any
hyperbolic reflection group. In the special case of hyperbolic
reflection groups of Lorentzian lattices, his conditions for a root to
be simple can be sharpened slightly, which reduces the amount of work
needed for practical calculations for some of the lattices in Section
3. Most of the results of this section are not used later in this
paper, but they are useful in checking the examples in Section 3. We
let $R$ be any Lorentzian lattice.
In a finite group acting on a lattice every positive root can be
written uniquely as a sum of simple roots. This is not usually true for
hyperbolic reflection groups as the simple roots are not always
linearly independent, but in the case of Lorentzian lattices there is
still a ``canonical'' way to write a root as a sum of simple roots as
follows.
\proclaim Theorem 1.1.
If $r$ is a positive root of some hyperbolic reflection group $W$
acting on the lattice $R$ then $r$ can be written uniquely as $r=\sum
n_is_i$ where the $s$'s are a finite number of distinct simple roots of
$W$, the $n$'s are positive integers, and the following condition
holds:
\item{}
Let $T_r$ be the (possibly singular) lattice generated by linearly
independent elements $t_i$ with $(t_i,t_j)=(s_i,s_j)$. Then $t=\sum
n_it_i$ is conjugate to some $t_i$ under the reflection group of $T_r$
generated by the reflections of the roots $t_i$. ($T_r$ is singular if
the $s$'s are linearly independent, but the quotient of $T_r$ by the
kernel of its quadratic form is Lorentzian or positive definite.)
{\sl Proof.} Let $T$ be the (possibly infinite dimensional and
singular) lattice generated by linearly independent elements $t_i$ for
every simple root $s_i$ of $W$, with the inner product on $T$ defined
by $(t_i,t_j)=(s_i,s_j)$, and let $W'$ be the reflection group of $T$
generated by the reflections of the roots $t_i$. ($T$ can also be
described as the root lattice of the Kac-Moody algebra of the Dynkin
diagram of $W$.) We let $c$ be a vector of $R$ which has negative
inner product with all $s_i$, and define the height on $R$ or $T$ by
$ht(x)=-(x,c)$. As $W'$ is a Weyl group with linearly independent
roots, every positive root of $W'$ can be written uniquely as a sum of
simple roots. (A root of $W'$ is a vector of $T$ conjugate to some
$t_i$ under $W'$, and is called positive if its height is positive.)
We now check that every root of $W$ is the image of a unique root of
$W'$ under the map from $T$ to $R$ taking $t_i$ to $s_i$. It is
sufficient to check this for simple roots of $W$, as any root of $W$
is conjugate to a simple root, and it will follow for simple roots if
we show that no simple root $s$ can be written as a nontrivial sum
$\sum m_is_i$ with $m_i$ positive integers. If it could be, then one
of the $s_i$'s, say $s_0$, must be $s$ because $(s,\sum
m_is_i)=(s,s)>0$ and all simple roots except $s$ have inner product at
most 0 with $s$. Now the fact that $ht(s)=\sum m_iht(s_i)\ge
ht(s_0)=ht(s)$ implies that $m_0=1$ and there are no other $m$'s,
because all simple roots have positive height. Hence every positive
root $r$ of $W$ can be written uniquely in the form $r=\sum n_is_i$
such that $\sum n_it_i$ is a root of $W'$. $t=\sum n_it_i$ is a root
of $W'$ if and only if $t$ is conjugate to one of the $t_i$'s under
$W'$, or equivalently under the subgroup of $W'$ generated by the
reflections of the $t_i$'s appearing in the sum for $t$. Q.E.D.
We use this to prove a strengthened form of Vinberg's condition
(Vinberg [7]) for a root to be simple. This corollary is not used
later in the paper, but is sometimes useful for practical
calculations.
\proclaim Corollary 1.2.
Let $c$ be an element of $R$ which has inner product at most 0 with
all simple roots of some fundamental domain of some hyperbolic
reflection group $W$ acting on $R$ and call $-(c,r)$ the height of
$r$. Let $r$ be a root of $W$ of positive height. Then the following
are equivalent:
\item{(1)}
$r$ is simple.
\item{(2)}
$r$ has inner product at most 0 with all simple roots $s$ such that
$ht(s)\le ht(r)\min(1,|s|/(|r|\sqrt{2}))$.
\item{(3)}
$r$ has inner product at most 0 with all simple roots $s$ such that
there is an integer $n$ satisfying the two conditions
\itemitem{(a)}
$ht(s)/ht(r)\le 1/n\le s^2/r^2$
\itemitem{(b)}
if $n=1$ then $r^2\le s^2/2$.
{\sl Proof.} It is obvious that (1) implies (2) and an easy argument
shows that any root $s$ satisfying the condition in (3) also satisfies
the condition in (2), so that (2) implies (3). Hence we have to show
that if $r$ is not simple there is a simple root $s$ satisfying the
condition in (3) and having positive inner product with $r$.
Assume $r$ is not simple. We can write $r=\sum n_is_i$ as in 1.1 with
$n_i$ positive and $s_i$ simple. Let $s$ be a simple root of smallest
possible height having positive inner product with $r$. If $s$ has
height 0 then it satisfies the conditions of (3), so we can assume
that $s$ has positive height. The simple root $s$ must be equal to
some $s_i$, so $ht(s)\le ht(r)$, so we can define a positive integer
$n$ by $n\le ht(r)/ht(s) ht(r)/2$, so
$r-2s(s,r)/(s,s)=r-s$ as it must have height at least 0 (so it cannot
be $r-ms$ for $m>1$), so $2(r,s)=(s,s)$. Also $(r,r-s)\le 0$ because
$r-s$ is a sum of simple roots of height less than that of $s$ (as
$ht(s)>ht(r)/2$), so by the choice of $s$ they all have inner product
at most 0 with $r$. Hence $(r,r)\le (r,s)=(s,s)/2$. Q.E.D.
{\sl Remarks.} The condition in (3) is in some sense the best
possible. Vinberg's condition was that $r$ is simple if it has inner
product at most 0 with all simple roots $s$ such that
$ht(s)/|s|0.$$ If $r_i\ne r$ then $(r_i,r)\le 0$
and is a multiple of $(r,r)/2$, so at most one of the terms $(r,r_i)$
for $r\ne r_i$ is nonzero, and such a nonzero term must be
$-(r,r)/2$. $G$ acts transitively on the set of $r_i$'s, so either
they are all perpendicular in which case they form a Dynkin diagram
$a_1^n$, or each has nonzero inner product with exactly one other
$r_i$ and this inner product is $-(r,r)/2$, in which case they form a
Dynkin diagram $a_2^{n/2}$. This proves (1).
Finally if $\sigma$ is the element of the Weyl group of $a_1^n$ or
$a_2^n$ mapping $\rho$ to $-\rho$ (where $\rho$ is the Weyl vector of
$a_1^n$ or $a_2^n$) then the restriction of $\sigma$ to $R'$ maps $r'$
to $-r'$ and fixes the orthogonal complement of $r'$ in $R'$. Hence
the reflection of $r'$ is an automorphism of $R'$ lifting to the
element $\sigma$ of $W$, and this implies that the smallest positive
(real) multiple of $r'$ that is in $R'$ is a root of $R'$. (If $r$ is
a strong root then the conjugates of $r$ form a Dynkin diagram
$a_1^n$, and the corresponding root of $R'$ is the sum of these
conjugates and therefore also a strong root.) Q.E.D.
If $r$ is strong or the conjugates of $r$ form a Dynkin diagram of
type $a_2^n$ then the root of $R'$ that is a multiple of $r'$ is the
sum of the conjugates of $r$; otherwise it may be half the sum of the
conjugates of $r$.
Now we show that several ways of construction a reflection group of
$R'$ from $R$, $W$, and $G$ all give the same group.
\proclaim Theorem 2.2.
The following groups of automorphisms of the sublattice $R'$ of $R$
are the same.
\item{(1)} The elements of $W$ commuting with $G$.
\item{(2)}
The elements of $W$ fixing the subspace $R'$.
\item{(3)}
The reflection group $W'$ generated by the reflections of the vectors
$r'$ as $r$ runs through the simple roots of $W'$ whose projections
$r'$ have positive norm.
\item{(4)} Same as (3), with ``simple roots'' replaced by ``roots''.
\medskip
\item{} Moreover the subgroups of $W$ in (1) and (2) act faithfully on $R'$.
{\sl Proof.} It is obvious that the subgroup (1) of $W$ is contained
in the subgroup (2), and that the groups (3) and (4) are the same. We
will complete the proof by constructing an injective map from the
subgroup (2) of $W$ to the group (4), and then checking that the
restriction of this map from the group (1) to the group (4) is onto.
The nonzero intersections of hyperplanes of $W$ not containing $R'$
with $R'$ are hyperplanes of $R'$, and by 2.1 the reflections of these
hyperplanes are restrictions of elements of $W$ to $R'$. Hence the
intersection $D'$ of $D$ with $R'$ is a Weyl chamber for the
reflection group $W'$. If $w$ is any element of the subgroup (2) of
$W$ then there is an element $w'$ of $W'$ such that $ww'$ fixes the
Weyl chamber $D'$ of $W'$, and by Lemma 2.1 $w'$ can be lifted to an
element of $W'$, which we also denote by $w'$. Then $ww'$ is an
element of $W$ fixing $D$ and is therefore 1. This implies that the
restriction map from the subgroup (2) of $W$ to the group of
automorphisms of $R'$ is injective and maps into the group (4).
Finally we have to check that the composed map from (1) to (4) is
surjective; to do this it is sufficient to check that any reflection
of $W'$ is the restriction of some element of (1), but this follows
from 2.1. Q.E.D.
We now consider the special case where there is a nonzero vector $c$
having bounded inner product with all simple roots of $W$ (e.g., if
$W$ has a finite number of roots). This is a very strong restriction on
$W$. The existence of such vectors of negative norm is equivalent to
$W$ having only a finite number of simple roots, and if $c$ has norm 0
then the simple roots of $W$ look a bit like the union of several
cosets of some lattice together with a finite number of extra
roots. (See Section 3.) Two examples of this case are $II_{25,1}$ or
$I_{9,1}$ with $W$ the group generated by reflections of vectors of
norms 2 or 1, respectively. The simple roots can then be identified
with the points of the Leech lattice (as in Conway [4]) or with the
points of the $E_8$ lattice and in general there is a similar
description of the Dynkin diagram of the lattice, as in Section 3.
(Note that in the second case we are not using the full reflection
group, which has only a finite number (10) of simple roots.) These
facts are closely related to the facts that the covering radii of the
Leech lattice and $E_8$ are $\sqrt{2}$ and 1. We now show that if $R$
has such a vector $c$ then we can say a lot more about $R'$, and in
particular it also has such a vector.
\proclaim Lemma 2.3.
Let $c$ be a nonzero vector in the fundamental domain $D$ having
bounded inner products with all simple roots of the group $W$, and
assume that these simple roots span the vector space of $R$. If $z$ is
a norm 0 vector of $D$ not proportional to $c$ then the simple roots
of $W$ perpendicular to $z$ form an affine Dynkin diagram of rank
$\dim(R)-2$. (Recall that the rank of an affine Dynkin diagram is the
number of points minus the number of components.)
{\sl Proof.} For any vector $v$ of $R$ perpendicular to $z$, the
function mapping $r$ in $R$ to $r+(r,z)v-((r,v)+(v,v)(r,z)/2)z$ is
easily checked to be an automorphism of $R$ fixing all vectors
perpendicular to $z$ and $v$, and in particular fixing $z$. If the
simple roots of $W$ perpendicular to $z$ do not have rank $\dim(R)-2$
we can find a vector $v$ having inner product 0 with $z$ and all the
simple roots, and which is not a multiple of $z$. The automorphism of
$nv$ fixes $z$ and all the simple roots perpendicular to $z$, and
therefore fixes the Weyl chamber $D$ and hence acts on the set of
simple roots. We let $r$ be any simple root of $W$ not perpendicular
to $z$ and consider its images under the automorphism of $nv$ for
large $n$. The inner product of such an image with $c$ is
$-n^2(v,v)(r,z)(z,c)+$ terms in $n^1$ and $n^0$, and as the inner
product of simple roots with $c$ is bounded we must have
$(v,v)(r,z)(z,c)=0$. However, $(v,v)$ is nonzero because $(v,z)=0$ and
$v$ is not a multiple of $z$, $(r,z)$ is nonzero by assumption on $r$,
so $(z,c)=0$ and therefore $z$ is a multiple of $c$ as $(z,z)=0$ and
$z$ and $c$ are both in $D$. Q.E.D.
\proclaim Theorem 2.4.
Let $c$ be a nonzero vector of the fundamental domain $D$ of $W$
having bounded inner product with all simple roots of $W$. Then: either
\item{(1)}
The smallest normal subgroup of $\Aut(R')$ containing $W'$ is a
reflection subgroup of finite index in $\Aut(R')$, or
\item{(2)}
$c$ is in $R'$ and has norm 0, $W'$ is a reflection subgroup of
$\Aut(R')$ of infinite index and all simple roots of $W'$ have bounded
inner product with $c$. Any two conjugates of $c$ under $\Aut(R')$ are
conjugate under $W'$, and the subgroup of $W'$ fixing $c$ is an affine
reflection group that has a simple root for every orbit (under $G$) of
simple roots of $W$ perpendicular to $c$. (There may be no such roots,
in which case $W'$ is simply transitive on the conjugates of $c$ under
$\Aut(R')$.)
\item{}
Also, if there are no roots of $W'$ perpendicular to $c$ then $W'$ is
normal in $\Aut(R')$.
{\sl Proof.} First note that the subgroup of elements of $\Aut(R')$
that can be lifted to $\Aut(R)$ has finite index in $\Aut(R')$. If $c$
has nonzero norm then $W$ has a finite number of simple roots and
hence has finite index in $\Aut(R)$ so $W'$ has finite index in
$\Aut(R')$ and we are in case (1), so we can assume that $c$ has zero
norm. If $c$ is not fixed by $G$ then the sum of two conjugates of $c$
is a vector of nonzero norm with the same properties as $c$, so we can
assume that $c$ is fixed by $G$ and hence is in $R'$.
The group of automorphisms of $R'$ that can be lifted to $\Aut(R)$ has
finite index in $\Aut(R')$, so there are only a finite number of
conjugates of $c$ in $D'$, because any two conjugates of $c$ under
$\Aut(R)$ are conjugate under $W$. Suppose first that there is more
than one conjugate of $c$ under $\Aut(R')$ in $D'$. By Lemma 2.3 any
conjugate of $c$ in $D'$ other than $c$ has simple roots of $W'$
perpendicular to it forming an affine Dynkin diagram of rank
$\dim(R)-2$, so the same is true for $c$. Hence if $W''$ is any
reflection group of $R'$ containing $W'$ with Weyl chamber $D''$, the
index of $W''$ in its normalizer is the number of conjugates of $c$ in
$D''$ times the order of the group of automorphisms of $R'$ fixing $c$
and $D''$, which is finite. In particular if $W''$ is the smallest
normal subgroup of $\Aut(R')$ containing $W'$ then $W''$ is a normal
reflection subgroup of finite index in $\Aut(R')$, so we are in case
(1) again.
Hence we can assume that there is only one conjugate of $c$ in $D'$
under $\Aut(R')$. This implies that any two conjugates of $c$ under
$\Aut(R')$ are conjugate under $W'$, and if there are no roots of $W'$
perpendicular to $c$ then $W'$ is simply transitive on the conjugates
of $c$, because $W'$ is simply transitive on the conjugates of $D'$.
Finally assume that there are no roots of $W$ perpendicular to $c$,
and let $W''$ be the smallest normal subgroup of $\Aut(R')$ containing
$W'$. If $r$ is any root of $W''$, then $(r'',c)$ is equal to $(r',c)$
for some root $r'$ of $W'$ because $W'$ is transitive on the
conjugates of $c$ under $\Aut(R')$, and in particular $(r'',c)$ is
nonzero. Hence $W''$ is simply transitive on the conjugates of $c$
under $W''$, and as the same is true for $W'$, $W''$ is the same as
$W'$, so $W'$ is normal in $\Aut(R')$. Q.E.D.
{\sl Remark.} Suppose that $c$ is in $R'$ and $(r,c)$ divides $(v,c)$
for every simple root $r$ of $W$ and every vector $v$ of $R$ (for
example $R$ could be $II_{25,1}$). Then the same is true for every
root $r$ of $W'$ and every vector $v$ of $R'$.
\proclaim 3.~Examples.
The lattice $II_{25,1}$ has a nonzero norm 0 vector which has bounded
inner product with all simple roots, and by applying the construction
of the last section we can find many other Lorentzian lattices with
the same property. These lattices have properties similar to those of
$II_{25,1}$. For example, the Dynkin diagram of $II_{25,1}$ can be
identified with the Leech lattice which is closely related to the fact
that balls of radius $\sqrt{2}$ just cover the vector space of the
Leech lattice, and similarly for other Lorentzian lattices we can
describe their Dynkin diagrams in terms of some positive lattice, and
can cover some vector space with balls and half planes.
We now describe the geometry of the simple roots when there is a
non-zero norm 0 vector $c$ of $R$ having bounded inner product with
all the simple roots of $W$, where $W$ is a normal reflection subgroup
of $\Aut(R)$. We let $T_0$ (respectively $T_1$) be the set of points
of the real vector space of $R$ having inner product 0 (respectively
1) with $c$, and we write $V_0$ and $V_1$ for the quotients of $T_0$
and $T_1$, where we identify two points of $T_0$ or $T_1$ if their
difference is a multiple of $c$. $V_1$ is an affine space over the
$\dim(R)-2$ dimensional vector space $V_0$, and $V_0$ inherits a
positive definite inner product from $R$. For each simple root $r$ of
$W$ we let $S_r$ be the subset of points of $V_1$ represented by norm
0 vectors of $T_1$ that have inner product at least 0 with $r$. (Note
that every point of $V_1$ is represented by a unique norm 0 vector of
$T_1$.) We write $R_0$ and $R_1$ for the lattice vectors in $V_0$ and
$V_1$.
The fact that the Leech lattice is the Dynkin diagram of $II_{25,1}$
implies that the vector space of the Leech lattice is covered by balls
of radius $\sqrt{2}$ about each lattice point, each ball corresponding
to a simple root. This can be generalized by replacing $II_{25,1}$
with the lattice $R$ above as follows.
\proclaim Theorem 3.1.
The balls and half-spaces $S_r$ of the affine space $V_1$ corresponding
to the simple roots $r$ have the following properties:
\item{(1)}
If $r$ has height 0 $S_r$ is a closed half-space; otherwise it is a closed
ball with center $r/(r,c)$ and radius $|r/(r,c)|.$ (Warning--$S_r$ does not contain 0, because 0 is not in $V_1$!)
\item{(2)}
The sets $S_r$ cover $V_1$ and there are only a finite number of them
intersecting any bounded subset of $V_1$.
\item{(3)}
If two of these balls have radii $r_1$ and $r_2$ and the distance
between their centers is $d$, then $d^2\ge r_1^2+r_2^2$. The center of
any ball is not contained in any other set $S_r$. In particular, if
any of the sets $S_r$ are removed, the remaining sets do not cover
$V_1$.
\item{(4)}
The points $z$ not in the interiors of any of the sets $S_r$ are in
natural 1:1 correspondence with the primitive norm 0 vectors of $R$ in
the fundamental domain $D$ of $W$ that are not multiples of $c$. The
roots $r$ such that $z$ lies on the surface of $S_r$ form an affine
Dynkin diagram of rank $\dim(R)-2$. (Warning--it is possible for any
affine Dynkin diagram to occur; even ``twisted'' ones.)
\item{(5)}
Let $R_0$ be the lattice of points of $V_0$ represented by points of
$R$, and let $L$ be the sublattice of $R_0$ of vectors perpendicular
to all roots of $W$ of height 0. Then $L$ acts by translation on the
set of $S_r$'s and has only a finite number of orbits on this set.
The proof is routine and will be omitted.
The first example of this behavior was found by Conway [4] for the
lattice $II_{25,1}$. $R_0$ and $L$ are both isomorphic to the Leech
lattice, and $R_1$ is the affine Leech lattice. The sets $S_r$ are all
balls of radius $\sqrt{2}$ with centers the points of the affine Leech
lattice, and the points $z$ not in the interiors of any of these balls
are the so-called ``deep holes'' of the Leech lattice. The lattice
points nearest to a deep hole form the affine Dynkin diagram of the
Niemeier lattice of the norm 0 vector corresponding to a deep hole. A
similar example is when $R$ is $I_{9,1}$ and $W$ is generated by the
reflections of norm 1 vectors, when $R_1$ is the $E_8$ lattice and $V_1$ is
covered by balls of radius 1 about each lattice point. Theorem 3.1
states that the general case is rather like this, except that the
balls do not necessarily have the same radius (and may degenerate into
half-spaces), their centers may form more than 1 orbit under the
lattice $L$, and affine Dynkin diagrams with roots of different lengths
can occur. (In the case of $II_{25,1}$, there is a natural
correspondence between the orbits of primitive norm 0 vectors and the
Niemeier lattices. For arbitrary Lorentzian lattices there is also a
correspondence between the orbits of primitive norm 0 vectors and a
finite number of positive definite lattices, but these lattices do not
necessarily have the same determinant.) Roughly speaking the simple
roots of $W$ correspond to a finite number of cosets of the lattice
$L$, with a finite number of simple roots left over if $L$ has
dimension less than that of $R_0$.
If we know the Dynkin diagram of the reflection group of some lattice,
we can find all normal reflection subgroups whose simple roots have
bounded inner product with some nonzero vector using the following
lemma.
\proclaim Lemma 3.2.
Let $W$ be a hyperbolic reflection group with fundamental domain $D$,
and let $W'$ be a normal reflection subgroup with fundamental domain
$D'$ containing $D$. Then the group $W$ is a spilt extension of $W'$
by the reflection group $H$ whose simple roots are the simple roots of
$W$ that are not roots of $W'$.
{\sl Proof. } $W$ is a split extension of $W'$ by the group $G$ of
elements of $W$ fixing $D'$. This group certainly contains the
reflections of any simple root of $W$ that is not a root of $W'$, and
hence contains the group $H$ generated by these reflections. If $E$ is
the union of all conjugates of $D$ under $H$, then all faces of $E$
are conjugates of faces of $D'$ under $W$ and hence are hyperplanes of
$W'$ because $W'$ is a normal subgroup of $W$. Hence $E$ is a union
of fundamental domains of $W'$ and in particular contains $D'$, so $H$
contains $G$ and is therefore equal to $G$. Q.E.D.
This means that we can sometimes find interesting normal reflection
subgroups of $\Aut(R)$ by finding subdiagrams of the Dynkin diagram of
$R$ which are affine or spherical and such that any simple root
conjugate under $\Aut(R)$ to some root of this subdiagram is already
in the subdiagram. For example, suppose $R$ is $I_{9,1}$. Then the
Dynkin diagram of $R$ has 9 roots of norm 2 forming an extended $E_8$
Dynkin diagram and one root of norm 1. Hence the reflection group
generated by the norm 2 vectors of $R$ has 11 simple roots and index 2
in $W$, while the quotient of $W$ by the subgroup generated by the
norm 1 vectors is isomorphic to the affine $E_8$ Weyl group. In fact
the simple roots of the reflection group generated by the norm 1
vectors are isometric to the $E_8$ lattice in the same way that the
simple roots of $II_{25,1}$ are isometric to the Leech lattice.
We now put everything together to prove the following theorem, which
shows that several well-known lattices behave like the Leech lattice.
\proclaim Theorem 3.3.
Let $G$ be a group of automorphisms of the Leech lattice $\Lambda$
such that the sublattice $\Lambda'$ of vectors of $\Lambda$ fixed by
$G$ has no roots, and let $R$ be the Lorentzian lattice that is the
sum of $\Lambda'$ and the two dimensional even unimodular Lorentzian
lattice $U$. Then $R$ has a norm 0 vector $c$ such that the simple
roots of the reflection group of $R$ are exactly the roots $r$ of $R$
such that $(r,c)$ is negative and divides $(r,v)$ for all vectors $v$
of $R$. The group $\Aut(R)$ is isomorphic to a split extension of its
reflection group by the group of affine automorphisms of $\Lambda'$.
{\sl Proof.} The group of affine automorphisms of the Leech lattice
$\Lambda$ can be identified with the group of diagram automorphisms of
$II_{25,1}$, so $G$ can be considered to be a subgroup of
$\Aut(II_{25,1})$, and it is easy to check that that sublattice of
points of $II_{25,1}$ fixed by $G$ is isomorphic to $R$. The theorem
then follows from 2.4, except that we still have to check that the
group $W'$ of 2.4 is the full reflection subgroup of $R$. $II_{25,1}$
has a vector $c$ which has inner product 1 with all simple roots of
$II_{25,1}$, and from this it follows that $(r,c)$ divides $(r,v)$ for
all simple roots $r$ of $W'$ and all vectors $v$ of $R$, so $(r,c)\le
|(s,c)|$ for any conjugate $s$ of $r$. But then by 1.3 $r$ is a simple
root for the full reflection group of $R$, hence $W'$ is the full
reflection group of $R$. Q.E.D.
{\sl Remarks.} The fact that $(r,c)$ divides $(r,v)$ for all $v$
implies that $(r,c)$ is either $(r,r)$ or $(r,r)/2$; both cases
occur. Any root of $R$ has even norm dividing $2|G|$. There is a
similar theorem with $II_{25,1}$, $\Lambda$ and the full reflection
group of $II_{25,1}$ replaced by $I_{9,1}$, $E_8$, and the reflection
group of $I_{9,1}$ generated by the reflections of norm 1 vectors; of
course all roots of $W'$ will be strong roots.
{\sl Examples.} $\Aut(\Lambda)$ has elements of orders 2 and 3 whose
fixed lattices $\Lambda_{16}$ and $K_{12}$ have no roots and have
dimensions 16 and 12 respectively. ($\Lambda_{16}$ is the Barnes-Wall
lattice, and $K_{12}$ is the Coxeter-Todd lattice; see Conway and
Sloane [6, Chap. 4].) These lattices have the same relation to the
baby monster and $Fi_{24}$ that $\Lambda$ has to the monster. (See
Conway and Norton [5].) The theorem above shows that there are 18 and
14 dimensional Lorentzian lattices $R$ associated to them whose Dynkin
diagrams can be described in terms of the 16 and 12 dimensional
lattices. Note that they have roots of norms 4 and 6 as well as roots
of norm 2, so the geometry is rather more complicated than that of the
Leech lattice. The simple roots of $R$ correspond to some of the
vectors of the dual of $\Lambda'$ (but not necessarily all of them;
for example not the ones within $\sqrt{2}$ of a lattice vector.) The
vectors of $\Lambda'$ itself correspond to norm 2 simple roots in
$R$. For example, if we take $\Lambda'$ to be $\Lambda_{16}$ of
dimension 16 and determinant 256, then $R$ has a simple root of norm 2
for every vector of $\Lambda_{16}$ and a simple root of norm 4 for
every vector in one of 120 cosets of $\Lambda_{16}$. If we draw a
sphere of radius $\sqrt{2}$ about every point of $\Lambda_{16}$, and
of radius 1 about every point of these 120 cosets, then these spheres
just cover the vector space of $\Lambda_{16}$ in the same way that
spheres of radius $\sqrt{2}$ around points of $\Lambda$ just cover the
vector space of $\Lambda$. We also get large numbers of ``deep holes''
in $\Lambda_{16}$ (e.g., $F_4^4$) which behave like the deep holes of
$\Lambda$. (Likewise $K_{12}$ has deep holes of hype $G_2^6$ and so
on.)
$\Aut(\Lambda)$ also has an element of order 2 whose fixed sublattice
is $E_8(2)$ (i.e., the lattice $E_8$ with the norms of all vectors
doubled). In this case $\Lambda'$ has roots, and the reflection group
of $R$ has finite index in $\Aut(R)$. Similarly if $R$ is the sum of
$E_8(n)$ and the two dimensional even unimodular Lorentzian lattice
$U$ for $2\le n\le 6$ then the reflection group of $R$ has finite
index in $\Aut(R)$. (In fact, it follows from 2.4 that this is true
for any lattice $R$ fixed by some group of automorphisms of $\Lambda$
whose roots span its vector space.) For example when $n=6$ the Dynkin
diagram has 4 roots of norm 2, 2 or norm 4, 1 of norm 6, and 10 of
norm 12 and its automorphism group has order 4; the theorems of
Section 1 are useful for doing these calculations.
If $R$ is a Lorentzian lattice, there are 3 possibilities for the
``non-reflection group'' of $R$ which is the quotient of the
automorphism group of $R$ by the subgroup generated by
reflections. (These can be thought of as the ``elliptic'',
``parabolic'', and ``hyperbolic'' cases, although this terminology
should not be taken too seriously.)
\item{(1)}
This group is finite. This case includes many of the lattices whose
dimension and determinant are both small.
\item{(2)}
The non-reflection group if $R$ is infinite, but has a free abelian
subgroup of finite index. This case is the one mostly studied in this
paper, and seems to be rare. The existence of a free abelian subgroup
of finite index is equivalent to the existence of a nonzero vector of
norm at most 0 fixed by all automorphisms of a fundamental domain of
the reflection group, so $II_{25,1}$ is one example of this case, and
3.3 gives a few other examples.
\item{(3)}
The general case: everything else. This case seems to include most
Lorentzian lattices, possibly all of dimension more than 26. In
Borcherds [2] the non-reflection group is calculated for a few
unimodular lattices, and in these cases turns out to be a direct
limit of a finite number of finite groups. Many of the results of
Borcherds [2] still hold when $\Lambda$ is replaced by some lattice
$R$ in class (2) above, so it would be possible to find some more
lattices whose non-reflection groups could be presented as a direct
limit of finite groups.
The monster Lie algebra (Borcherds, Conway, Queen, and Sloane, [6,
Chap. 30], or Borcherds [3]) is a generalized Kac-Moody algebra with
root lattice $II_{25,1}$ whose positive simple roots are the simple
roots of $II_{25,1}$. (It also has simple roots of norm 0, so it is
not a Kac-Moody algebra.) (Note added 1998: this algebra is now called
the fake monster Lie algebra.) If $G$ is a finite group of diagram
automorphisms of $II_{25,1}$ then by Borcherds [3, theorem 3.1] the
subalgebra of the monster Lie algebra fixed by $G$ is still a
generalized Kac-Moody algebra. Note that the class of generalized
Kac-Moody algebras is invariant under the operation of taking the
subalgebra fixed by a finite group of diagram automorphisms, but the
class of Kac-Moody algebras is not. We can therefore define the baby
monster Lie algebra, the $Fi_{24}$ Lie algebra, and so on to be the
subalgebras of the monster Lie algebra fixed by the appropriate
group. There algebras are generalized Kac-Moody algebras whose
positive simple roots are the simple roots of the corresponding
Lorentzian lattice., and which have simple roots of norm 0 which are
multiples of the vector $c$. There is some numerical evidence that the
monster Lie algebra has no simple roots of negative norm (note added
in 1998: this has been proved), so it is natural to ask if this is
true for the new algebras (note added 1998: this is false).
{\sl Problems.} Find all Lorentzian lattices which have a nonzero
vector which has bounded inner product with all simple roots. (Note
that this includes as a special case the problem of finding all
Lorentzian lattices whose reflection group has finite index.) Are
there only essentially a finite number of such lattices of dimension
greater than 2? (Note added 1998: this has been proved by Nikulin if
we assume that $c$ has norm at most 0.) (Obviously any multiple of
such a lattice has the same property.) Is $II_{25,1}$ the only such
lattice of dimension at least 26? Is $I_{9,1}$ the only lattice of
dimension $\ge 10$ such that the simple roots of the reflection group
generated by strong roots have bounded inner product with some vector
$c$?
\proclaim References.
\item{1.} R. E. Borcherds, The Leech lattice. Proc. Roy. Soc. London Ser.
A 398 (1985), no. 1815, 365--376.
\item{2.} R. E. Borcherds,
Automorphism groups of Lorentzian lattices. J. Algebra 111 (1987), no. 1,
133--153.
\item{3.} R. E. Borcherds,
Generalized Kac-Moody algebras. J. Algebra 115 (1988), no. 2, 501--512.
\item{4.} J. H. Conway,
The automorphism group of the $26$-dimensional even unimodular Lorentzian
lattice. J. Algebra 80 (1983), no. 1, 159--163.
\item{5.} J. H. Conway, S. P. Norton,
Monstrous moonshine. Bull. London Math. Soc. 11 (1979), no. 3,
308--339.
\item{6.} J. H. Conway, N. J. A. Sloane,
Sphere packings, lattices and groups.
Grundlehren der
Mathematischen Wissenschaften 290. Springer-Verlag, New
York-Berlin, 1988. ISBN: 0-387-96617-X
\item{7.} \`E. B. Vinberg,
Some arithmetical discrete groups in Loba\v cevski\u\i ~spaces.
in ``Discrete subgroups
of Lie groups and applications to moduli''
(Internat. Colloq., Bombay, 1973), pp. 323--348. Oxford Univ. Press,
Bombay, 1975.
\bye