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\proclaim Generalized Kac-Moody algebras.
Journal of Algebra Vol 115, No. 2, June 1988
Richard E. Borcherds,
Department of Pure Mathematics and Mathematical Statistics,
University of Cambridge,
16 Mill Lane,
Cambridge
CB2 1SB,
England.
We study a class of Lie algebras which have a contravariant bilinear
form which is almost positive definite. These algebras generalize
Kac-Moody algebras, and can be thought of as Kac-Moody algebras with
imaginary simple roots. Most facts about Kac-Moody algebras generalize
to these new algebras; for example, we prove a version of the
Kac-Weyl character formula, which is like the usual one except that it
has an extra correction term for the imaginary simple roots.
There are several ways in which these new algebras turn up. The fixed
point algebra of any Kac-Moody algebra under a diagram automorphism is
not usually a Kac-Moody algebra, but is one of these more general
algebras. There is also a generalized Kac-Moody algebra associated to
any even Lorentzian lattice of dimension at most 26 or any Lorentzian
lattice of dimension at most 10, and we give a simple formula for the
multiplicities of the roots of these algebras (but unfortunately I do
not know what the Cartan matrices are)! The numbers 10 and 26 come
from the ``no ghost''theorem.
We assume that most of Kac [3] is known, and only give proofs when
they differ significantly from the ones given there.
\proclaim 1.~Definitions.
This section consists mainly of definitions. We do not prove any of
the results stated here because they can all be proved by making
trivial changes to the usual proofs for Kac-Moody algebras.
A generalized Kac-Moody algebra $G$ (GKM algebra for short) will be
constructed from the following objects:
(1) A real vector space $H$ with a symmetric bilinear inner product (,).
(2) A set of elements $h_i$ of $H$ indexed by a countable set $I$,
such that $(h_i,h_j)\le 0$ if $i\ne j$ and $2(h_i,h_j)/(h_i,h_i)$
is an integer if $(h_i,h_i)$ is positive.
We write $a_{ij}$ for $(h_i,h_j)$ and call the matrix $a_{ij}$ the
symmetrized Cartan matrix of $G$ (SCM for short). The GKM algebra
$G$ associated to this is defined to be the Lie algebra generated by $H$
and elements $e_i$ and $f_i$ for $i$ in $I$ with the following relations:
(1) The image of $H$ in $G$ is commutative. (In fact the natural map from
$H$ to $G$ is injective so we can consider $H$ to be an abelian subalgebra
of $G$.)
(2) If $h$ is in $H$ then $[h,e_i]=(h,h_i)e_i$ and $[h,f_i]=-(h,h_i)f_i$.
(3) $[e_i,f_j]=h_i$ if $i=j$, 0 if $i\ne j$.
(4) If $a_{ii}$ is positive then $({\rm ad} e_i)^{1-2a_{ij}/a_{ii}}e_j=0$,
and similarly $({\rm ad} f_i)^{1-2a_{ij}/a_{ii}}f_j=0$.
(5) If $a_{ij}=0$ then $[e_i,e_j]=[f_i,f_j]=0$. (If $a_{ii}$ or $a_{jj}$
is positive this follows from (4).)
\medskip
The root lattice $Q$ is defined to be the free abelian group generated
by elements $r_i$ for $i$ in $I$, and $Q$ has a real-valued bilinear
form defined by $(r_i,r_j)=a_{ij}$. The Lie algebra $G$ is graded by
$Q$ by letting $H$ have degree 0, $e_i$ have degree $r_i$, and $f_i$
have degree $-r_i$. A root is a nonzero element $r$ of $Q$ such that
there are elements of $G$ of degree $r$. $r$ is called real if $(r,r)$
is positive and imaginary otherwise. It is called simple if it is one
of the $r_i$'s, positive if it is a sum of simple roots, and negative
if $-r$ is positive. Every root is either positive or negative. If $r$
and $s$ are in $Q$ we write $r\ge s$ if $r-s$ is a sum of simple roots.
{\it Warning.} For nonsymmetric Cartan matrices the elements
$e_i$, $f_i$, and $h_i$ above are not quite the same as the ones in Kac [3],
but are multiples of them.
The Weyl group of $G$ is generated by elements $w_i$ for every
{\it real} simple root $r_i$ of $G$ with the relations
$$w_i^2=1\qquad\hbox{and}\qquad(w_iw_j)^{m_{ij}}=1,$$ where $m_{ij}$
is 2, 3, 4, 6, or $\infty$ depending on whether
$4(a_{ij})^2/a_{ii}a_{jj}$ is 0, 1, 2, 3, or greater than 3. It acts
on $Q$ and $H$ by letting $w_i$ act as reflection in the hyperplane
perpendicular to $r_i$ or $h_i$. Something is said to be in the Weyl
chamber if it has inner product at most 0 with all real simple roots.
The Cartan involution $\omega$ is the involution of $G$ that acts as
$-1$ on $H$ and exchanges $e_i$ and $-f_i$. There is a unique
invariant bilinear form $(,)$ on $G$ extending the given form on $H$,
and we define the contravariant bilinear form $(,)_0$ on $G$ by
$(x,y)_0=-(x,\omega(y))$. (This will turn out to be positive definite
on the root spaces of $G$ other than $H$.)
We use $(,)$ to indicate the bilinear pairing between any of $Q$,
$Q^*$, $H$, and $H^*$ when such a pairing can be sensibly defined,
possibly using the map from $Q$ to $H$ which maps $r_i$ to $h_i$. We
let $\rho$ be the element of $Q^*$ such that $(\rho, r_i)={1\over 2}
(r_i)^2$. We say that a vector $x$ in any of $H$, $H^*$, $Q$, or $Q^*$
is in the Weyl chamber if $(x,r_i)\le 0$ for all $i$.
\proclaim 2.~Geometry of the Root System.
In this section we prove or state some facts about the root system
and Weyl group of a GKM $G$ that generalize results about Kac-Moody
algebras. In particular we prove enough about the root system so that
the arguments in Kac [3] can be used to prove that $(,)_0$ is almost
positive definite, and that $G$ is simple provided that certain
conditions are satisfied.
\proclaim Proposition 2.1. Every positive
root $r$ in $Q$ is conjugate under the Weyl group to a simple real
root or a positive root in the Weyl chamber.
{\it Proof.} We can assume that any positive root which is a conjugate
of $r$ has height at least equal to that of $r$. If $r$ is not in the
Weyl chamber, then there is a simple real root $s$ with $(r,s)>0$ so
that the reflection $r'$ of $r$ in the hyperplane of $s$ has height
less than $r$. By the assumption on $r$, $r'$ must be a negative root,
so $r$ must be the simple root $s$. Q.E.D.
\proclaim Proposition 2.2.
A positive root $r$ in the Weyl chamber of $Q$ is isotropic (i.e., has norm 0)
if and only if its support is affine or a root of norm 0. (The support
of a positive root $r$ is the set of simple roots appearing in the
expression of $r$ as a sum of simple roots.)
{\it Proof.} We can write $r=\sum k_ir_i$ with $k_i>0$ and $r_i$ some
set of simple roots. As $(r_i,r)\le 0$ and $(r,r)=0$ we must have
$(r_i,r)=0$ for all $i$. If all the $r_i$'s are real the proposition
follows from Kac [3] Proposition 5.7, while if some $r_i$ is imaginary
then $(r_i,r_j)\le 0$ for all $j$, so $(r_i,r_j)=0$ for all $j$, which
implies that $r=r_i$ as the support of $r$ is connected. Q.E.D.
Now we prove an important inequality for the roots of $Q$.
\proclaim Proposition 2.3.
If $r=\sum k_ir_i$ is in the Weyl chamber of $Q$, where the $k_i$'s are
positive integers and the $r_i$'s are some simple roots, then
$(r,r)\le 2(\rho,r)$ with equality if and only if $(r_i,r_i)\le 0$,
$(r_i,r_j)=0$ for $i\ne j$, and $(r_i,r_i)=0$ when $k_i>1$.
{\it Proof.} $2(\rho,r)-(r,r)=\sum k_i(r_i,r_i-r)$. If $r_i$ is real
then $(r_i,r_i)>0$ and $(r_i,-r)\ge 0$ so $k_i(r_i,r_i-r)>0$. If $r_i$
is imaginary then $r-r_i$ is a sum of simple roots and so has inner
product at most 0 with $r_i$, hence $k_i(r_i,r_i-r)\ge 0$. Hence
$2(\rho,r)\ge (r,r)$ and if equality holds then all the $r_i$'s are
imaginary and $(r_i,r_i-r)=0$ for all $i$. $r-r_i$ is a sum of some
simple roots including all the $r_j$'s for $j\ne i$ and $r_i$ if
$k_i>1$, and $r_i$ has inner product at most 0 with all these simple
roots, so $(r_i,r_j)=0$ if $i\ne j$ or if $k_i>1$. Conversely if the
$r_i$'s and $k_i$'s satisfy these conditions it is obvious that
$(r,r)=2(\rho,r)$. Q.E.D.
\proclaim Corollary 2.4.
If $r$ is a positive root then $(r,r)\le 2(\rho,r)$ with equality if
and only if $r$ is simple.
{\it Proof.} If $r$ is imaginary then $r$ is in the Weyl chamber so
the result follows from 2.3 because the support of $r$ is
connected. If $r$ is real and positive then we can keep on strictly
reducing $(\rho,r)$ by reflections in the Weyl group while keeping $r$
positive, until $r$ is a simple root when $(r,r)=2(\rho,r)$, so that
$(r,r)<2(\rho,r)$ if $r$ is not simple. Q.E.D.
\proclaim Corollary 2.5.
The contravariant inner product $(,)_0$ is positive definite on the
weight space of $r$ if $r$ is nonzero.
{\it Proof.}\footnote{$^*$}{This proof is not complete; see the third
edition of Kac [3] for a complete proof.} The proof in Kac [3, 11.7]
carries over to $G$ because $2(\rho,r)>(r,r)$ if $r$ is a positive
root that is not simple. Q.E.D.
\proclaim Corollary 2.6.
Any nonzero graded ideal of $G$ has nonzero intersection with the
Cartan subalgebra.
{\it Proof.} We can assume that the graded ideal has a nonzero
homogeneous element in the weight space of some positive root of
minimum possible height; it is easy to check that such an element has
inner product 0 with all elements of $G$, which contradicts
2.5. Q.E.D.
{\it Remark.} This implies that the definition of GKM algebras by
means of generators and relations is essentially the same as the
definition given by Kac for arbitrary matrices. (The center and outer
derivations may be different.)
\proclaim Corollary 2.7.
Suppose that $H$ is spanned by the $h_i$'s, there is no element of $H$
perpendicular to all the $h_i$'s, and the $h_i$'s cannot be divided
into two nonempty orthogonal sets. Then $G$ is either simple or the
quotient of the derived algebra of an affine Kac-Moody algebra by its
center.
{\it Proof.} This follows from 2.6 and exercise 4.10 of Kac [3].
\proclaim 3.~ A Characterization of GKM Algebras.
In this section we show that GKM algebras are essentially the same as
graded algebras with an ``almost positive definite'' contravariant
bilinear form. We will later use this to construct examples of GKM
algebras.
Let $G$ be a GKM algebra generated by its Cartan subalgebra $H$ and
the elements $e_i$ and $f_i$, and let $s_i$ be a collection of
positive integers such that only a finite number of the $s_i$'s are
equal to any given positive integer. We grade $G$ by putting
$\deg(H)=0$, $\deg(e_i)=-\deg(f_i)=s_i$. Recall that $G$ has a Cartan
involution $\omega$ with $\omega(e_i)=-f_i$ and a contravariant
bilinear form $(,)_0$ defined by $(x,y)_0=-(x,\omega(y))$. $G$ has the
following properties:
(1) $G$ is graded as the direct sum of $G_m$ for $m$ integral. $G_0$
is abelian and $G_m$ is finite dimensional if $m$ is nonzero.
(2) $G$ has an invariant bilinear form $(,)$ such that $G_m$ and $G_n$
are orthogonal unless $m=-n$.
(3) $G$ has an involution $\omega$ which is $-1$ on $G_0$ and which
maps $G_m$ into $G_{-m}$.
(4) The contravariant bilinear form $(x,y)_0:=-(x,\omega(y))$ is
positive definite on $G_m$ if $m$ is nonzero.
We now prove the converse.
\proclaim Theorem 3.1.
If $G$ is a Lie algebra satisfying (1) to (4) above and $K$ is the
kernel of $(,)$, then $K$ is in the center of $G$ and $G/K$ is a GKM
algebra.
{\it Proof.} If $k$ is in the kernel $K$ of $(,)$ then it is in $G_0$
and so commutes with all elements of $G_0$ as $G_0$ is abelian. If $g$
is in $G_m$ for $m$ nonzero then $([k,g],h)=(k,[g,h])=0$ for all $h$,
so $[k,g]=0$ as $(,)_0$ is non-degenerate on $G_m$. Hence $k$ is in
the center of $G$. From now on we can assume that $(,)$ is
nondegenerate and we have to prove that $G$ is a GKM algebra.
We take $G_0$ to be the Cartan subalgebra $H$ of $G$. For positive $m$ we
let $E_m$ be the subspace of $G_m$ perpendicular under $(,)_0$ to the
subalgebra of $G$ generated by $G_n$ for $00$ whenever $s$ is a weight of
$L(r)$ not equal to $r$. The proof that this implies (1) and (2) of
the proposition is almost exactly the same as the proof of the last
part of 3.1. Q.E.D.
Now we assume that $r$ and $L(r)$ satisfy (1) and (2) above, and show
that $(,)$ is positive definite on $L(r)$. (2) implies that the set of
weights of $L(r)$ (which is a subset of the affine space $r+Q$) is
invariant under the Weyl group. We now prove some inequalities for the
weights of $L(r)$ similar to the inequalities of Section 2 for the
roots of $G$. We let $s$, $s_1$, and $s_2$ denote weights of $L(r)$.
\proclaim Proposition 4.2.
$(r,r)-(s_1,s_2)\ge 0$, and equality implies that $s_1=s_2$ and $s_1$
is conjugate to $r$ under the Weyl group. (For Kac-Moody algebras this
is proved in Kac [3, 11.4(a)].)
{\it Proof.} By acting on $s_1$ and $s_2$ with the Weyl group we can
assume that $(s_1,h_i)$ is at most 0 for all real simple roots
$h_i$. Then $(r,s_1-r)$ and $(s_1,s_2-r)$ are both at most 0 because
$r$ and $s_1$ have inner product at most 0 with all simple roots, and
$s_i-r$ is a sum of simple roots. Their sum is 0 by assumption, so
they are both 0 which implies that $(r,h_i)=0$ for all simple roots
$h_i$ in the support of $s_1-r$. By condition (1) above this implies
that $s_1=r$, and the condition $(s_1,s_2-r)=0$ then implies that
$s_2=r$ in the same way. Q.E.D.
\proclaim Proposition 4.3.
$(r-\rho)^2-(s-\rho)^2$ is at least 0, and equal to 0 only if
$s=r$. (See Kac [3,11.4(b)].)
{\it Proof.} We can act on $s$ by elements of the Weyl group, each
time strictly increasing $(s-\rho)^2$, until $(s,h_i)\le 0$ for all
real simple roots $h_i$, so we can assume that $(s,h_i)\le 0$ for all
real simple roots $h_i$. If $s$ is not $r$ then let $h_i$ be one of
the simple roots in the support of $s-r$. We have
$(h_i,r+s-2\rho)=(h_i,r)+(h_i,s-h_i)\le 0$ as $s-h_i$ is a sum of $r$
and some simple roots. Hence $(s-r,s+r-2\rho)\le 0$ which is
equivalent to $(r-\rho)^2-(s-\rho)^2\ge 0$. Equality implies that
$(h_i,r)=0$ for any $h_i$ in the support of $s-r$, which implies that
$s=r$ by (1). Q.E.D.
\proclaim Corollary 4.4.
The contravariant bilinear form $(,)$ on $L(r)$ is positive definite.
{\it Proof.} This follows from 4.3 and the proof of 11.7 in Kac [3].
\proclaim Corollary 4.5.
If $r$ satisfies (1) and (2) and $L(r)$ is defined to be the lowest
weight module satisfying the relations (1) and (2) then $L(r)$ is
simple.
{\it Proof.} This follows from 4.4. Q.E.D.
Now we find a formula for the character of $L(r)$ when $r$ satisfies
(1) and (2). By following the argument in Kac [3,10.4] we find that
$$e^{-\rho} Ch(L(r))\prod (1-e^h)^{{\rm mult} (h)} =
\sum c_se^{s-\rho},\eqno{(3)}$$ where both sides are antisymmetric
under the Weyl group, the $c_s$'s are integers, the product is over
all positive roots $h$ of $G$, and the sum on the right is over some
weights $s$ such that $(r-\rho)^2-(s-\rho)^2=0$ and $s\ge r$ (i.e.,
$s$ is equal to $r$ plus a sum of simple roots). We let $S$ be the sum
of the terms on the right for which $s-\rho$ is in the Weyl chamber
(i.e., $(s-\rho,h_i)\le 0$ for all real simple roots $h_i$).
If $s-\rho $ is in the Weyl chamber then we write $s=r+\sum k_ir_i$
for some simple roots $r_i$ and positive integers $k_i$. We have
$(r-\rho)^2-(s-\rho)^2=0$, so
$$\eqalign{
\sum k_i(r_i,r)+\sum k_i(r_i,s-2\rho) &=
\sum k_i(r_i,r+s-2\rho)\cr
&= (s-r,r+s-2\rho)\cr &= (s-\rho)^2-(r-\rho)^2\cr &=0\cr }$$ For all
$i$, $(r_i,r)\le 0$ by the assumption on $r$. If $r_i$ is real then
$(r_i,s-2\rho)<(r_i,s-\rho)\le 0$ as $s-\rho$ is in the Weyl
chamber. If $r_i$ is imaginary then $(r_i, s-2\rho)=(r_i,s-r_i)\le 0$
as $s-r_i$ is equal to $r$ plus some simple roots.
Hence none of the terms $k_i(r_i,r)$ and $k_i(r_i,s-2\rho)$ is
positive, so they must all be 0 as their sum is 0. In particular all
the $r_i$'s must be imaginary and must have inner product 0 with
$r$. We also have $(r_i,s-r_i-r)= (r_i,s-2\rho)=0$ and $s-r_i-r$ is
equal to $\sum k_jr_j+(k_i-1)r_i$, so $k_jr_j$ and $(k_i-1)r_i$ have
inner product 0 with $r_i$, so $(r_i, r_j)=0$ unless $r_i=r_j$ and
$k_i=1$.
Hence for any term $s_se^{s-\rho}$ in $S$, $s$ is of the form $r+\sum
k_ir_i$, where the $k_i$'s are positive integers, all the $r_i$'s are
imaginary and have inner product 0 with $r$, and $(r_i,r_j)=0$ if
$i\ne j$ or $k_i\ge 2$. Any such $s$ lies in the Weyl chamber, so
$s-\rho$ lies in the interior of the Weyl chamber, hence the
right-hand side of (3) is equal to $\sum \epsilon(w)w(S)$. To complete
the proof of the character formula we now evaluate $S$ by computing the
terms of the left-hand side of (3) that contribute to $S$.
If $e^{r+\sum k_i r_i}$ is a term of ${\rm Ch} L(r)$ then some $r_i$
has nonzero inner product with $r$, hence the only terms of the form
$e^{r+\sum k_ir_i-\rho}$ of the right-hand side of (3) in $S$ are
those coming from
$$ e^{-\rho} e^r\prod (1-e^h)^{{\rm mult}h}.\eqno{(4)}$$ If
$(r_i,r_j)=0$ for $i\ne j$ then the coefficient of $e^{\sum
k_ir_i+r-\rho}$ in (4) is easily seen to be 0 if some $k_i$ is greater
than 1, and $(-1)^n$ otherwise, where $n$ is the number of simple
roots $r_i$ in $\sum k_ir_i$. Hence we find that
$$S=e^{r-\rho}\sum\epsilon(s)e^s,$$ where the sum is over all sums of
simple roots $s$. Here the sign $\epsilon(s)$ is defined by
$\epsilon(s)=(-1)^n$ is $s$ is the sum of $n$ distinct pairwise
perpendicular imaginary simple roots perpendicular to $r$, and
$\epsilon(s)=0$ otherwise. Putting everything together shows that
$${\rm Ch}
L(r)=e^\rho\sum_w\epsilon(w)w(S)/\prod_\alpha(1-e^\alpha)^{{\rm
mult}\alpha},$$ where $S$ is given above. Note that if $r$ is not
perpendicular to any imaginary simple roots (e.g., if there are no
such roots) then $S=e^{r-\rho}$ and the formula for ${\rm Ch}L(r)$ is
identical to the usual Kac-Weyl formula.
{\it Remark.} In this formula for ${\rm Ch} L(r)$, the right hand side
does not change if $\rho$ is replaced by any vector having inner
products $(r_i,r_i)/2$ with all {\it real } simple roots $r_i$.
{\it Remark.} The Peterson recursion formula (Kac [3, Ex. 11.12]) for
the multiplicities of roots of $G$ and the Freudenthal recursion
formula for the multiplicities of roots of lowest weight modules (Kac
[3, Ex. 11.14]) can be proved in just the same way that they are
proved for Kac-Moody algebras.
\proclaim 5.~Examples.
We describe four ways of constructing GKM algebras: from SCM's, from
Lorentzian lattices of dimension at most 10, from even Lorentzian
lattices of dimension at most 26, and from diagram automorphisms of
Kac-Moody algebras.
The most obvious way of constructing GKM's is to find any matrix
satisfying the conditions of an SCM and write down then generators and
relations for the corresponding GKM algebra. I do not know of any
interesting algebras found like this except for a few Kac-Moody
algebras.
A second way to construct GKM algebras is as the fixed points of a
diagram automorphism of a Kac-Moody algebra. More generally we have:
\proclaim Theorem 5.1.
The subalgebra of a GKM algebra $G$ with nonsingular Cartan subalgebra
fixed by a finite group $A$ of diagram automorphisms is a GKM algebra
with nonsingular Cartan subalgebra.
(A diagram automorphism is one that preserves the Cartan subalgebra,
permutes the $e_i$'s and permutes the $f_i$'s in the same way.)
{\it Proof.} As the group $A$ is finite we can grade $G$ as in 3.1 in
such a way that the grading is preserved by $A$. Any diagram
automorphism commutes with the Cartan involution $\omega$, so the
fixed point subalgebra $G^A$ satisfies the conditions (1) to (4) of
Theorem 3.1. The fixed subspace of the Cartan subalgebra is
nonsingular as the fixed subspace of any nonsingular inner product
space under a finite group is nonsingular. Hence by Theorem 3.1, $G^A$
is a GKM algebra.
{\it Remark.} $G^A$ will usually have an infinite number of imaginary
simple roots even if $G$ has none. The real simple roots of $G^A$ are
easy to describe: they correspond to the orbits of $A$ on the Dynkin
diagram of $G$ which are Dynkin diagrams of the form $A_1^n$ or
$A_2^n$.
For any even lattice $R$ there is a Lie algebra $A$ with ``root
lattice'' $R$ such that the dimension of the weight space of any
nonzero element $r$ of $R$ is $p_{d-1}(1-{1\over
2}r^2)-p_{d-1}(-r^2)$, where $d$ is the dimension of $R$ and $p_{d-1}$
is the number of partitions with $d-1$ colors. See Borcherds [1] for
details. The real vector space $H$ spanned by $R$ is an abelian
subalgebra of $A$ which is self centralizing if $R$ is
nonsingular. $A$ has an involution $\omega$ acting as $-1$ on $R$ and
an invariant bilinear form $(,)$ extending that of $R$.
If $R$ is Lorentzian we chose an element $r$ in $R$ of negative norm
which is not perpendicular to any norm 2 vectors of $R$ and grade $A$
by letting the weight space of $A$ corresponding to $s$ in $R$ have
degree $(r,s)$. $H$ is then the subalgebra of $A$ of elements of
degree 0. The ``no ghost'' theorem (Goddard and Thorn [4]) implies
that the bilinear form $(,)_0$ is positive definite on any weight
space other than $H$ if the dimension of $R$ is at most 25, so that by
Theorem 3.1 $A$ is a GKM algebra. If $R$ has dimension exactly 26
then the no ghost theorem implies that $(,)_0$ is positive
semidefinite and its kernel in the weight space corresponding to $r$
has codimension $p_{24}(1-{1\over 2}r^2)$ if $r$ is nonzero. Hence $A$
has a graded ideal such that the quotient $B$ of $A$ by this ideal is
a GKM algebra such that the weight space corresponding to $r$ in $R$
has dimension $p_{24}(1-{1\over 2}r^2)$.
The most interesting example of such an algebra is got by taking $R$
to be the 26-dimensional even unimodular Lorentzian lattice
$II_{25,1}$. The real simple roots of the corresponding algebra $B$
generate the ``monster Lie algebra'' and are the simple roots of the
reflection group of $R$ which were described by Conway [2]. He showed
that $II_{25,1}$ has a norm 0 vector $\rho$ such that the simple roots
of the reflection group are just the norm 2 vectors of $R$ which have
inner product $-1$ with $\rho$, and these are the real simple roots of
$B$. (More precisely they are the images of the simple roots of $B$
under the natural map from the root lattice to $R$.) The norm 0 simple
roots of $B$ are not difficult to find: they are the positive
multiples of $\rho$, each with multiplicity $24$ (or more precisely
there are 24 simple roots mapping onto each positive multiple of
$\rho$). I have not been able to find any negative norm simple roots
of $B$; I have checked that there are no simple roots mapping onto $r$
in $R$ for all vectors $r$ of norm $-2$ or $-4$ and most vectors of
norm $-6$. ($R$ has 121 orbits of norm $-2$ vectors and 665 orbits of
norm $-4$ vectors, but any vector of $R$ can be the image of several
different roots of $B$.)
There is a similar construction using Lorentzian lattices $R$ of dimension
$d$ at most 10 instead of even Lorentzian lattices of dimension at
most 26. (Borcherds [1].) In this case the multiplicity of the root
$r$ of the GKM algebra constructed from $R$ is
$p'_{d-1}((1-r^2)/2)-p'_{d-1}(-r^2/2)$ if $d$ is at most 9, and
$p'_8((1-r^2)/2)$ if $d$ is 10. Here $p'_d(n)$ is the coefficient of
$x^n$ in
$$\prod_i(1-x^i)^{-d}(1+x^{i-1/2})^d.$$ The most interesting case of
this is got by taking $R$ to be the 10-dimensional odd unimodular
Lorentzian lattice $I_{9,1}$. $R$ has a norm 0 vector $2\rho$ such
that the real simple roots of the GKM algebra $A$ of $R$ are the norm 1
vectors of $R$ which have inner product $-1$ with $2\rho$, and the
norm 0 simple roots are the positive multiplicities of $2\rho$, each
with multiplicity 8. (The real simple roots are the simple roots of
the reflection group of $R$ generated by the reflections in
hyperplanes perpendicular to norm 1 vectors.)
The algebra $A$ has many simple roots of negative norm, unlike the
corresponding algebra for $II_{25,1}$ which appears to have no roots
of negative norm. To obtain an algebra which has no roots of negative
norm, we define the monster Lie superalgebra $B$ to be the following
GKM superalgebra:
(1) The root system of $B$ is the (nonintegral) lattice generated by
$R$ and $\rho$, with $R+\rho$ the roots of the ``super'' part of $B$.
(2) The real simple roots of $B$ are those of $A$. The imaginary
simple roots of $B$ all have norm 0 and are the positive multiples of
$\rho$ each with multiplicity 8, and they are ``superroots'' if the are
odd multiples of $\rho$.
Some calculations I have done, together with the analogy between the
monster Lie algebra and the monster Lie superalgebra, suggest the
conjecture that the root $r$ of $B$ has multiplicity
$p'_8((1-r^2)/2)$. Note that if $r^2$ is even, then by one of Jacobi's
identities this is equal to the coefficient of $x^{1-(1/2)r^2}$ in
$$8\prod_i(1-x^i)^{-8}(1+x^i)^8.$$ This conjecture implies that the
Lie algebra part of $B$ is isomorphic to $A$.
Nearly everything about GKM algebras can be generalized with little
difficulty to GKM superalgebras by copying the ways for generalizing
Kac-Moody algebras to Kac-Moody superalgebras.
\proclaim References.
\item{1.} {R. E. Borcherds, Vertex algebras, Kac-Moody algebras,
and the monster. Proc. Natl. Acad. Sci. USA. Vol. 83 (1986) 3068-3071.}
\item{2.}{J. H. Conway, The automorphism group of the
26-dimensional even Lorentzian lattice. J. Algebra 80 (1983) 159-163.
}
\item{3.}{V. G. Kac, ``Infinite dimensional Lie algebras'',
Birkh\"auser,
Basel, 1983.}
\item{4.}{P. Goddard and C. B. Thorn, Compatibility of the dual
Pomeron with unitarity and the absence of ghosts in the dual resonance
model, Phys. Lett., B 40, No. 2 (1972), 235-238.}
\bye