\magnification=\magstep1
\vbadness=10000
\hbadness=10000
\tolerance=10000
\def\cite#1{[#1]}
\def\bora{1}
\def\borb{2}
\def\borc{3}
\def\kaca{4}
\def\bibitem#1#2{[#1]. #2 \par}
A characterization of generalized Kac-Moody algebras.
J. Algebra 174, 1073-1079 (1995).
Richard E. Borcherds,
D.P.M.M.S.,
16 Mill Lane,
Cambridge CB2 1SB,
England.
\bigskip
Generalized Kac-Moody algebras can be described in two ways:
either using generators and relations, or as Lie algebras with
an almost positive definite symmetric contravariant bilinear form.
Unfortunately it is usually hard to check either of these
conditions for any naturally occurring Lie algebra. In this paper we
give a third characterization of generalized Kac-Moody algebras
which is easier to check, which says roughly that any
Lie algebra with a root system similar to that of a generalized
Kac-Moody algebra is a generalized Kac-Moody algebra. We use this to
show that some Lie algebras constructed
from even Lorentzian lattices are generalized Kac-Moody algebras.
I thank the referee for suggesting several improvements and corrections.
Section 1 states the theorems of this paper, section 2 describes some
examples, and section 3 gives the proof of the theorems.
%\vfill\eject
{\bf 1. Statement of result.}
All Lie algebras are Lie algebras over the real numbers $R$. We will assume
the basic theory of generalized Kac-Moody algebras given
in \cite{\borb,\borc,\kaca}.
We first recall the definition of a generalized Kac-Moody algebra.
Suppose that $a_{ij}$ is a real square matrix indexed by $i$ and $j$
in some countable set $I$ with the following properties.
\item{1} $a_{ij}=a_{ji}$.
\item{2} If $i\ne j$ then $a_{ij}\le 0$.
\item{3} If $a_{ii}>0$ then $2a_{ij}/a_{ii}$ is an integer for all $j$.
Then we define the universal generalized Kac-Moody algebra
of $a_{ij}$ to be the Lie algebra generated by elements $e_i$, $f_i$,
and $h_{ij}$ for $i,j\in I$, with 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 ${\rm Ad}(e_i)^{1-2a_{ij}/a_{ii}}e_j=
{\rm Ad}(f_i)^{1-2a_{ij}/a_{ii}}f_j=0$.
\item{4} If $a_{ij}=0$ then $[e_i,e_j]=[f_i,f_j]=0$.
\item{} (The relations $[h_{ij},h_{kl}]$ are usually also included,
but these follow from the other relations.)
We define a generalized Kac-Moody algebra to be
a Lie algebra $G$ such that $G$ is a semidirect product
$A.B$, where $A$ is an ideal of $G$ which is the quotient of
a universal generalized Kac-Moody algebra by a subspace of its center,
and $B$ is an abelian subalgebra such that the elements $e_i$ and $f_i$
are all eigenvalues of $B$. In other words, $G$ can be obtained from
a universal generalized Kac-Moody algebra by throwing away
some of the center and adding some commuting outer derivations.
This is slightly more restrictive than some previous definitions
because we insist that $I$ is a countable set, but I do not know of
any interesting examples where $I$ is uncountable. Kac \cite{\kaca}
extended the definition by allowing the matrix $a_{ij}$ to be non
symmetrizable. In this case the generalized Kac-Moody algebra does not
have an invariant bilinear form, and the existence of such a form is
an essential condition in the theorems of this paper. I do not know
of any useful characterizations of the Lie algebras associated to non
symmetrizable matrices.
There is a characterization of generalized Kac-Moody algebras
given in \cite{\borc} as follows. A Lie algebra $G$ is a generalized
Kac-Moody algebra if it satisfies the following conditions.
\item{1} $G$ can be graded as $G=\oplus_{n\in Z}G_n$,
with $G_n$ finite dimensional for $n\ne 0$.
\item{2} $G$ has an involution $\omega$ which maps $G_n$ to
$G_{-n}$ and acts as $-1$ on $G_0$.
\item {3} $G$ has a symmetric invariant bilinear form $(,)$
which is preserved by $\omega$ and such that
$G_m$ and $G_n$ are orthogonal unless $m=-n$.
\item{4} If $g\in G_n$, $g\ne 0$, and $n\ne 0$, then $(g,\omega(g))>0$.
We can summarize these conditions by saying that $G$ has an almost
positive definite symmetric contravariant bilinear form. Conversely,
it is almost true that any generalized Kac-Moody algebra satisfies the
conditions above. It is incorrectly stated in some previous papers of
mine that this is true but several people pointed out to me the
following two minor reasons why this is not quite true: if the $i$'th
and $j$'th rows of the Cartan matrix are equal and $i\ne j$ then
$\omega$ need not act as $-1$ on the subalgebra spanned by
$h_{ij}$ and $h_{ji}$, and if the Cartan matrix has an infinite number
of identical rows then it is not always possible to grade $G$ so that
$G_n$ always has finite dimension (because of the elements $h_{ij}$
for $i\ne j$). The elements $h_{ij}$ for $i\ne j$ seem to be of no use
in practice and is tempting just to add the relations that they should
be 0 to the definition of a generalized Kac-Moody algebra; the main
reason for not doing this is that they are nonzero in some central extensions
of simple generalized Kac-Moody algebras and
most central extensions of
groups or Lie algebras seem to turn out to be useful sooner or later.
There are many examples of Lie algebras satisfying the conditions above
which can be constructed using vertex algebras, and which are
therefore generalized Kac-Moody algebras. Unfortunately there are
many examples of Lie algebras which are generalized Kac-Moody algebras
for which it is very hard to verify the positivity condition (4)
directly. This condition is sometimes not satisfied for the
``obvious'' involution $\omega$, but is for some other choices of
$\omega$. We will prove the following theorem which removes this
difficulty.
\proclaim Theorem 1. Any Lie algebra $G$ satisfying the following 5 conditions
is a generalized Kac-Moody algebra.
\item{1} $G$ has a nonsingular invariant symmetric bilinear form $(,)$.
\item{2} $G$ has a self centralizing subalgebra $H$ (called the Cartan
subalgebra) such that $G$ is the sum of the eigenspaces of $H$ and all
the eigenspaces are finite dimensional. The nonzero eigenvalues
of $H$ acting on $G$ (which are elements of the dual of $H$)
are called the roots of $G$. (Note that the simple roots need
not be linearly independent.)
\item{3} $H$ has a regular element $h$; this means that
the centralizer of $h$ is $H$ and that there are only a finite number
of roots $\alpha$ such that $|\alpha(h)|0$ and $i\ne j$ then $a_{ij}\le 0$, $2a_{ii}/a_{ij}$
is integral,
and ${\rm Ad}(e_i)^{1-2a_{ij}/a_{ii}}e_j=
{\rm Ad}(f_i)^{1-2a_{ij}/a_{ii}}f_j=0$. We use the fact that
the norms of the roots of $G$ are bounded above (condition 4 of
theorem 1). The norms of
the roots of $e_i$ and $f_i$ are positive, which implies that the
Lie algebra $G$ is the direct sum of finite dimensional
representations of the copy of $sl_2(R)$ spanned by $e_i, f_i$, and $h_i$.
The element $e_j$ is killed by $f_i$ and is an eigenvector of $h_i$
with eigenvalue $a_{ij}$, so the fact that it generates a finite
dimensional representation of $sl_2(R)$ implies that
$2a_{ij}/a_{ii}$ must be a nonpositive integer and that
${\rm Ad}(e_i)^{1-2a_{ij}/a_{ii}}e_j=0$.
Similarly ${\rm Ad}(f_i)^{1-2a_{ij}/a_{ii}}f_j=0$.
We check that if $a_{ii}\le 0$ and $a_{jj}\le 0$ then
$a_{ij}\le 0$, and if $a_{ij}=0$ then $[e_i,e_j]=[f_i,f_j]=0$.
This follows from the condition (5) that
any two imaginary roots which are both positive or both negative have
inner product at most 0, and if they have inner product 0 their root
spaces commute.
We have checked all the conditions and relations for a generalized
Kac-Moody algebra, so that $G_{\le M} $ is a generalized Kac-Moody algebra,
so $G$ is a generalized Kac-Moody algebra. This proves
theorem 1.
\parindent = 0pt \everypar={\hangindent = .3in}
\medbreak {\bf References}\medskip\par\nobreak
\bibitem{\bora}{R. E. Borcherds, Vertex algebras, Kac-Moody algebras,
and the monster. Proc. Natl. Acad. Sci. USA. Vol. 83 (1986) 3068-3071.}
\bibitem{\borb}{R. E. Borcherds, Generalized Kac-Moody algebras. J. Algebra
115 (1988), 501--512.}
\bibitem{\borc}{R. E. Borcherds, Central extensions of generalized Kac-Moody
algebras. J. Algebra. 140, 330-335 (1991).}
\bibitem{\kaca}{V. G. Kac, ``Infinite dimensional Lie algebras'',
third edition, Cambridge
University Press, 1990. (The first and second editions (Birkhauser,
Basel, 1983, and C.U.P., 1985) do not contain the material
on generalized Kac-Moody algebras that we need.) }
\end