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\proclaim Vertex algebras, Kac-Moody algebras, and the Monster.
{\it Proc Natl. Acad. Sci. USA}
Vol. 83, pp 3068-3071
Richard E. Borcherds, %author
Trinity College,
Cambridge
CB2 1TQ,
England.
\bigskip
Communicated by Walter Feit, December 13, 1985
{\bf ABSTRACT It is known that the adjoint representation of any
Kac-Moody algebra $A$ can be identified with a subquotient of a certain
Fock space representation constructed from the root lattice of $A$. I
define a product on the whole of the Fock space the restricts to the
Lie algebra product on this subquotient. This product (together with
an infinite number of other products) is constructed using a
generalization of vertex operators. I also construct an integral form
for the universal enveloping algebra of any Kac-Moody algebra that can
be used to define Kac-Moody groups over finite fields, some new
irreducible integrable representations, and a sort of affinization of any
Kac-Moody algebra. The ``Moonshine representation of the Monster
constructed by Frenkel and others also has products such as the
ones constructed for Kac-Moody algebras, one of which extends the
Griess product on the 196884-dimensional piece to the whole
representation. }
\proclaim Section 1.~Introduction.
Let $A$ be any Kac-Moody algebra all of whose real roots have norm
2.(Everything here can be generalized to all Kac-Moody algebras but
becomes a lot more complicated, so for simplicity I will mostly just describe
this case.) $A$ is defined by certain generators and relations
depending on the Cartan matrix of $A$, and one of the most important
problems about Kac-Moody algebras is to find a more explicit
realization of $A$. This has been done only when $A$ is finite
dimensional or when $A$ is affine, in which case $A$ can be realized as a
central extension of a twisted ring of Laurent series in some finite
dimensional Lie algebra. Here we will construct a realization of an
algebra that is usually slightly larger than $A$ and is equal to $A$
if $A$ is finite dimensional or affine (in which case it is equivalent
to the usual realization of $A$). For any even integral lattice $R$
(for example, the root lattice of $A$), we will first construct a
(well known) Fock space $V=V(R)$. Physicists have defined ``vertex
operators'' for every element of $R$, which map $V$ to the space
$V\{z,z^{-1}\}$ of formal Laurent series in $V$, and the coefficients
of these operators map $V$ to $V$. I define a sort of generalized
vertex operator for every element of $V$ instead of just for elements
of $R$. This operator is written as $:Q(u,z):(v)$ for $u,v$ in $V$,
and its coefficients are written as $u_n(v)$ for $u,v$ in $V$ and
integers $n$. These products on $V$ are not associative, commutative
or skew commutative but satisfy several more complicated identities.
The product $u_0(v)$ is not a Lie algebra product on $V$, but it is a
Lie algebra product on $V/DV$, where $DV$ is the image of $V$ under a
certain derivation $D$. This Lie algebra $V/DV$ contains the Kac-Moody
algebra $A$ as a subalgebra but is always far larger than $A$. To
reduce $V/DV$ to a smaller subalgebra, we will use the Virasoro
algebra. This is spanned by the operators $c_i$ and 1, where $c$ is a
certain element of $V$. The commutator of this algebra in $V/DV$ also
contains $A$ and is not much larger than $A$; for example, we can
calculate bounds on the dimensions of the root spaces of $A$ from this
that are sometimes the best possible.
If $V$ is the infinite-dimensional representation of the monster
constructed by Frenkel {\it et al.} (1) then $V$ also has products $u_n(v)$
that satisfy several identities.
\proclaim Section 2.~Construction of the Fock space $V$.
In this section, we recall the construction of a certain Fock space $V$
from an even lattice $R$ and put several structures on $V$, such as a
product, a derivation, and an inner product (see ref. 2).
For any even lattice $R$ there is a central extension
$$0\longrightarrow Z_2\longrightarrow\hat R\longrightarrow
R\longrightarrow 0$$ where $Z_2$ is a group of order 2 generated by an
element $\epsilon$ and $\hat R$ has an element $e^r$ for every element
$r$ of $R$, such that $e^re^s=\epsilon^{(r,s)}e^se^r$ and $e^re^{-r} =
\epsilon^{(r,r)}/2$. $\hat R$ is uniquely defined up to isomorphism by
these conditions, and the automorphism group of $\hat R$ is an
extension $Z_2^{\dim R}\cdot {\rm Aut}(R)$ (usually nonsplit). If $R$ is the
root lattice of a Kac-Moody algebra $A$, then ${\rm Aut}(R)$ is not usually
a subgroup of ${\rm Aut}(A)$ in any natural way, but ${\rm Aut}(\hat R)$ is, as we
can prove by constructing $A$ from $\hat R$
The Fock space $V$ is a rational vector space given by the tensor product
$$Q(R)\otimes S(R(1))\otimes S(R(2))\ldots.$$ Here $Q(R)$ is the
rational group algebra of $\hat R$ quotiented out by $\epsilon+1$, so
it has a basis of $e^r$ for $r$ in $R$ and $e^re^s=(-1)^{(r,s)} e^s
e^r$. $R(i)$ is a copy of the rational vector space of $R$, and its
elements are written $r(i)$ for $r$ in $R$. $S(R(i))$ is the symmetric
algebra on $R(i)$. A typical element of $V$ might be $e^rs(1)^3t(4)$
for $r,s,t$ in $R$.
$V$ has the following structures.
(i) $V$ is an algebra as each of the pieces of the tensor product
defining $V$ is. $V$ would be commutative except that $e^r$ and $e^s$
do not always commute.
(ii) There are linear maps $D$ and deg from $V$ to $V$ such that
$De^r=r(1)e^r$, $Dr(i)=ir(i+1)$, and $D$ is a
derivation. $\deg(e^r)={1\over 2}(r,r)e^r$, $\deg r(i)v=r(i)(iv+\deg
v)$. If $\deg u=iu$ we can say that $u$ has degree $i$. We write
$D^{(i)}$ for the operator $D^i/i!$.
(iii) $V$ has a Cartan involution $\omega$. $\omega$ acts on $R$ by
$\omega(e^r)=e^{-r}$, and this becomes an automorphism of $V$ with
$\omega(e^r)=e^{-r}$, $\omega(r(i))=-r(i)$.
(iv) We define the operators $r(i)$ on $V$ for $r$ in $R$ and integers
$i$ as follows: If $i>0$ then $r(i)$ is multiplication by $r(i)$. If
$i=0$ then $r(i)e^s=(r,s)e^s$. If $i<0$ then
$r(i)e^s=0$. $[r(i),s(j)]=j(r,s)$ if $i=-j$, 0 otherwise. (These
properties characterize the operators $r(i)$.)
(v) $V$ has a unique inner product $(,)$ such that the operator $r(i)$
is the adjoint of $r(-i)$, and $(e^r,e^s) =1$ if $r=s$, 0 otherwise.
(vi) The integral form $V_Z$ of $V$ is defined to be the smallest
subring of $V$ containing all the $e^r$ and closed under $D^{(i)}$ for
$i\ge 0$. This integral form is compatible with all the structures
above; i.e., it is preserved by the Cartan involution $\omega$ and the
operators $r(i)$, and the inner product is integral on it. It is
generated as a ring by $e^r$, $r(1)$, $(r(2)+r(1)^2)/2$,
$(2r(3)+3r(2)r(1)+r(1)^3)/6\ldots$, which are Schur polynomials in
$r(1)$, $r(2)/2$, $r(3)/3\ldots$. If $W$ is the sublattice of $V_Z$ of
elements of ``$R$ grading'' 0 and degree $i$, then the determinant of
$W$ is an integral power of the determinant of $R$ and, in particular,
if $R$ is unimodular then so is $W$. ($V$ is graded by the lattice $R$
by letting $e^r$ have degree $r$ and letting $r(i)$ have degree $0$.)
\proclaim Section 3.~Vertex Operators.
For each $u$ in $V$ we will define a map $u$ from $V$ to the ring of
formal Laurent series $V\{z,z^{-1}\}$. If $u$ is of the form $e^r$
then these operators are just vertex operators, and if $u$ is
a product of $r(i)$s then these operators have been constructed by
Frenkel (3).
We can define $Q(r,z)$ to be the formal expression
$$\sum_{i\ne 0}r(i)z^i/i +r(0)\log(z)+r$$
and define $Q(r(i),z)$ for $i\ge 1$ to be $(d/dz)^iQ(r,z)/(i-1)!$. If
$u=e^r\prod_ir_i(n_i)$ is an element of $V$ then we define
$Q(u,z)$ to be the formal expression
$$e^{Q(r,z)}\prod_iQ(r_i(n_i),z).$$ This is not an operator from $V$
to $V\{z,z^{-1}\}$ as it does not converge, but we can make it into an
operator by ``normal ordering'' it. This means that in each term of
the formal expression $Q(u,z)$ we rearrange all terms $e^r$ and $r(i)$
so that the ``creation operators'' $e^r$ and $r(i)$ $(i\ge 1)$ occur
to the left of all ``annihilation operators'' $r(i)$ $(i\le 0)$. Note
that all annihilation operators commute with each other, and so do all
creation operators except for $e^r$ and $e^s$. The normal ordering of
$Q(u,z)$ is denoted by $:Q(u,z):$, and this is a well-defined operator
from $V$ to $V\{z,z^{-1}\}$. We define $u_n(v)$ for $u,v$ in $V$ and
integers $n$ by
$$ u_n(v)= \hbox{the coefficient of } z^{-n-1} \hbox{ in } :Q(u,z):(v).$$
If $u$ and $v$ are in the integral form of $V$ then so is $u_n(v)$. If
$u$ and $v$ have degrees $i,j$ then $u_n(v)$ has degree $i+j-n-1$.
The operator $r(i)$ is equal to $r(1)_{-i}$.
\proclaim Section 4.~Vertex Algebras.
We will list some identities satisfied by the operators $u_n$ and show how
to construct Lie algebras from them. $u$, $v$, and $w$ denote elements of $V$,
and 1 is the unit of $V$.
For any even lattice $R$ the operators $u_n$ on $V$ satisfy the
following relations.
(i) $u_n(w)=0$ for $n$ sufficiently large (depending on $u$, $w$).
This ensures convergence of the following formulae.
(ii) $1_n(w)=0$ if $n\ne -1$, $w$ if $n=-1$.
(iii) $u_n(1) = D^{(-n-1)}(u)$.
(iv) $u_n(v) = \sum_{i\ge 0} (-1)^{i+n+1}D^{(i)}(v_{n+i}(u))$.
(v) $(u_m(v))_n(w)=\sum_{i\ge 0}(-1)^i{m\choose i}(u_{m-i}(v_{n+i}(w))
-(-1)^mv_{m+n-i}(u_i(w))).$
(The binomial coefficient ${m\choose i}$ is equal to
$m(m-1)\ldots(m-i+1)/i!$ if $i\ge 0$ and 0 otherwise.)
We will call any module with linear operators $D^{(i)}(u)$ and
bilinear operators $u_n(v)$ satisfying relations $i$-$v$ above a vertex
algebra, so $V$ is a vertex algebra. (When we work with Kac-Moody
algebras that so not have all real roots of norm 2, we can also
construct a space $V$ and operators $u_n$; however $n$ is not always
integral, $u$ lies in a subspace of $V$ depending on $n$, and $u_n$
acts on a space that is different from $V$.)
Another example of a vertex algebra is given by taking any ring with a
derivation [i.e., maps $D^{(i)}$ with $D^{(i)}=0$ for $i<0$, 1 for
$i=0$,
$$\eqalign{
D^{(i)}D^{(j)} &= {i+j\choose i} D^{(i+j)}\cr
D^{(i)}(uv) &= \sum_j D^{(j)}(u)D^{(i-j)}(v)]
}$$
and defining $u_n(v)$ to be $D^{(-n-1)}(u)v$. This satisfies
conditions $i$-$iii$ and $v$ and satisfies condition $iv$ if and
only if the ring is commutative. It also satisfies $u_n(v)=0$ if $n\ge
0$, and conversely and vertex algebra satisfying this comes from a
unique ring with derivation. Hence vertex algebras are a
generalization of commutative rings with derivations.
A module over a vertex algebra $V$ is a module $W$ with operators
$u_n$ on $W$ satisfying relations $i$-$v$ above for $u,v$ in $V$, $w$ in
$W$. In particular $V$ is a $V$ module. (Warning---if $V$ comes from a
ring with derivation then vertex algebra modules over $V$ are not the
same as ring modules over $V$.)
If $V$ is any vertex algebra then $V/DV$ is a Lie algebra, where $DV$
is the sum of all the spaces $D^{(i)}(V)$ for $i\ge 1$ and where the
Lie algebra product is $[u,v]=u_0(v)$. Note that $u_0(v)$ is not
antisymmetric on $V$. Any $V$ module $W$ becomes a module for the Lie
algebra $V/DV$ by letting $v$ in $V/DV$ act as $v_0$ on $W$. (If $v$
is in $DV$ then $v_0$ is 0.) In particular $V$ is a $V/DV$ module and
is usually a nonsplit extension of the adjoint representation of
$V/DV$. The operators $D^{(i)}$ and the products $u_n(v)$ on $V$ are
invariant under the action of $V/DV$. ($V/DV$ can be extended to a
larger Lie algebra $V[z,z^{-1}]/DV[z,z^{-1}]$ of operators on $V$ that
is spanned by all the operators $u_n$, but this algebra does not leave
the products $u_n(v)$ invariant; see Section 8.)
The free vertex algebra on some set of generators does not exist
because of relation $i$. However if for each pair of generators $u,v$
we fix an integer $n(u,v)$ and include the relations $u_i(v)=0$ for
$i\ge n(u,v)$ then there is a universal vertex algebra with these
generators and relations. It can be constructed as a subalgebra of the
vertex algebra $V(R)$ for a certain lattice $R$ depending on the
$n(u,v)$s, and in particular any relation between the operators $u_n$
that holds for all the vertex algebras constructed from lattices can
be deduced from relations $i$-$v$.
\proclaim Section 5.~The Virasoro Algebra.
We will construct a representation of the Virasoro algebra on $V$
using some operators $c_n$, which are the Segal operators, and use
this to reduce the space $V/DV$.
We assume that $R$ is nonsingular, and we let $c$ be the element
${1\over 2} \sum_ir_i(1)r'_i(1)$ of $V$, where $r_i$ runs over some
base of $R$ and $r'_i$ is the dual base. We write $L_i$ for the
operator $c_{i+1}$, and we find that the $L_i$ have the following
properties:
$$\eqalign{
&L_{-1}=D,\qquad L_0=\deg \cr
&[L_i,L_j]=(i-j)L_{i+j}+(i^3-i)\dim(R)\delta_{i,-j}/12\cr
&L_{-i}\hbox{ is the adjoint of } L_i.\cr
}$$
In particular the operators $L_i$ and 1 span a copy of the Virasoro
algebra. If $R$ is a (possibly singular) lattice contained in a
nonsingular lattice $S$, then the operators $L_i$ for $i\ge -1$ on the
vertex algebra of $S$ restrict to operators on the vertex algebra of
$R$ that do not depend on the lattice $S$ containing $R$. In
particular if $i\ge -1$ then the operator $L_i$ can be defined on the
vertex algebra $V$ of $R$ even when $R$ is singular. We define the
physical subspace $P^i$ to be the elements $v$ of $V$ with $L_n(v)=iv$
if $n=0$, $0$ if $n\ge 1$. If $v$ is in $P^1$ then the operator $v_0$
commutes with the Virasoro algebra so it preserves all the spaces
$P^i$. If $v$ in $P^1$ is equal to $Du$ for some $u$ in $V$ then
$u$ is in $P^0$, so $P^1/DP^0$ is a Lie algebra acting on $V$ and
commuting with the Virasoro algebra. More generally if $u$ is in $P^1$
then
$$[L_j,u_k]=((j+1)(i-1)-k)u_{j+k}.$$
If $R$ contains the root lattice of the Kac-Moody algebra $A'$
(possibly quotiented out by some null lattice) then $A'$ can be mapped
to $P^1/DP^0$ by
$$\eqalign{
e_i&=e^{r_i}\cr
f_i&=-e^{-r_i}\cr
%- sign added 2002-9-30
h_i&=r_i(1).\cr
}$$
Here $r_i$ are the simple roots of $A$, and $e_i, f_i$, and $h_i$ are
the usual generators for the derived algebra $A'$ of $A$. It is easy
to check that these elements are in $P^1$ and satisfy the relations
for $A'$, so we obtain a representation of $A'$. If the root lattice
of $A'$ is singular, we can either quotient out by the kernel of the
bilinear form on it, in which case we will not obtain a faithful
representation of $A'$, or embed it in a nonsingular lattice $R$, in
which case some of the elements $r(1)$ will be outer derivations of
$A'$.
If $r$ is any nonzero vector of $R$ the then dimension of the $r$
subspace of $P^1$ or $P^0$ is $p_{d-1}(1-(r,r)/2)$ or
$p_{d-1}(-(r,r)/2)$, where $d$ is the dimension of $R$ and $p_{d-1}$
is the number of partitions into $d-1$ colors. Hence the dimension of
the $r$ subspace of $P^1/DP^0$ is equal to
$p_{d-1}(1-(r,r)/2)-p_{d-1}(-(r,r)/2)$, and this is an upper bound for
the multiplicity of roots of $A$ (provided $A$ is connected and not
affine so that the kernel of the map from $A'$ to $P^1/DP^0$ is in the
center of $A'$).
{\it Example:} If $R$ is the 18-dimensional even unimodular Lorentzian
lattice $II_{17,1}$ and $A$ is the Kac-Moody algebra whose Dynkin
diagram is that of $R$, then $A$ has roots of norm 2, 0, $-2$, and
$-4$ whose multiplicity is equal to the upper bound given
above. However there are several Kac-Moody algebras for which
numerical evidence suggests the better upper bound
$p_{d-2}(1-(r,r)/2)$ for the multiplicities of roots (3). Frenkel used
the no-ghost theorem to prove this stronger upper bound when $R$ is 26
dimensional and Lorentzian. In this case $P^1/DP^0$ has a normal
subalgebra such that the quotient by this subalgebra is a simple
algebra with root spaces of dimension $p_{d-2}(1-(r,r)/2)$ for $r\ne
0$.
If the lattice $R$ is odd we can use it to construct a ``super vertex
algebra'' $V$ acted on by a super Virasoro algebra spanned by elements
$1$, $L_i$, and $G_{i+1/2}$. The space $G_{1/2}P^{1/2}/DP^0$ is then a
Lie algebra. (Not a proper superalgebra!) For example if $R$ is $I_{9,1}$ then
$G_{1/2}P^{1/2}/DP^0$ has a normal subalgebra such that the quotient by
this subalgebra is a simple Lie algebra with root spaces of dimensions
equal to the coefficient of $x^{(1-(r,r))/2}$ in $\prod_{i\ge
1}(1-x^{i-1/2})^8(1-x^i)^{-8}$. This simple algebra contains the
Kac-Moody algebra that has a simple root $r'$ for every vector $r$ of
the lattice $E_8$, with $(r',s')=1-{1\over 2}(r-s,r-s)$. (This is
similar to the ``monster Lie algebra'', which has a simple root $r'$
for every vector $r$ of the Leech lattice, with $(r',s')=2-{1\over
2}(r-s,r-s)$.)
The operator $L_1$ can be used to describe the adjoint of $u_n$: if
$u$ has degree $i$ then the adjoint of $u_n$ is
$$(-1)^i\sum_jL_1^j(\omega(u))_{2i-j-n-2}/j!$$ In particular if $u$ is
in $P^1$ then the adjoint of $u_0$ is $-\omega(u)_0$, so the adjoint
of $e_i$ is $f_i$ and the adjoint of $h_i$ is $-h_i$.
\proclaim Section 6.~The Representations $L(r)$.
If $r$ is any element of the weight lattice $R'$ of $R$ we construct
an irreducible $A$ module whose largest weight is $r$, and these
representations generalize the highest weight and adjoint
representations of $A$.
We first assume that $r$ is in $R$. We take the space $P^i$ with
$i=(r,r)/2$. This has a maximal graded submodule not containing $e^r$,
and if we quotient out by this we get an irreducible module that we
denote by $L(r)$. $L(r)$ has the following properties.
(i) $L(r)$ is irreducible.
(ii) The weight $r$ has multiplicity 1, and if $(s,s)>(r,r)$ then $s$
has multiplicity 0. This implies that $L(r)$ is integrable, so in
particular if $s$ and $t$ are conjugate under the Weyl group they have
the same multiplicity.
(iii) $L(r)$ has a nonzero contravariant bilinear form, which is unique
up to multiplication by a constant. (This is not necessarily positive
definite unless $r$ is a highest or lowest weight vector.)
(iv) All weights of $L(r)$ have finite multiplicities. (I do not know
of any formula for the multiplicities except in the cases below.)
(v) If $r$ is a highest or lowest weight vector then $L(r)$ is the
corresponding highest or lowest weight module, and if $r$ is a real
root of $A$ then $L(r)$ is a quotient of the adjoint representation
(and equal to $A'$ modulo its center if this is simple).
(vi) If $r$ and $s$ are conjugate under the Weyl group then
$L(r)=L(s)$. (The converse is not true; for example $r$ and $s$ could
be two real roots of $A$ in different orbits of ${\rm Aut}(R)$.)
If $r$ is an element of $R'$ not in $R$ then the construction above
does not work because $e^r$ is not in $V$, so we construct the space
$V_r$ by replacing $Q(R)$ in the tensor product defining $V$ by
$Q(R+r)$. All the operators $u_n$ for $u$ in $V$ act on $V_r$, and we
can construct $L(r)$ as a subquotient of $V_r$ as above.
Problem: Is $L(r)$ the only $A$ module satisfying conditions $i$ and
$ii$ above? (It is if $r$ is a highest or lowest weight vector, and in
this case condition $ii$ implies condition $i$.)
\proclaim Section 7.~Integral forms for Kac-Moody algebras.
We constructed an integral form $V_Z$ for $V$ in Section 3. Here we
will use this to find an integral form for the universal enveloping
algebra for the Kac-Moody algebra $A$.
For each $r$ in the weight lattice $R'$ we define the integral form
$L_Z(r)$ of $L(r)$ to be the elements of $L(r)$ represented by
elements in the integral form of $V_r$. (If $r$ is not in $R$ then the
integral form of $V_r$ is $e^rV_Z$.) Similarly the integral form $A_Z$
of $A$ is the set of elements of $A$ represented by elements of
$V_Z$. This acts on all the $L_Z(r)$ because $u_n$ preserves the
integral form of $V_r$ Finally we define the integral form $U_Z$ of
the universal enveloping algebra $U$ of $A$ to be the subalgebra of
$U$ preserving all the $L_Z(r)$s. Calculation shows that $U_Z$
contains $(e_i)^n/n!$ and $(f_i)^n/n!$ for all integers $n\ge 0$ where
the $e$s and $f$s are the generators for $A$. We can therefore use
$U_Z$ to define Kac-Moody groups over finite fields (or over any
commutative ring) in the same way that Chevalley groups are defined,
by using the automorphisms $\exp(te_i)$ and $\exp(tf_i)$ of
$L_Z(r)\otimes F$ for $t$ in the finite field $F$.
The element $c$ of $V$ is not usually in $V_Z$, but $V_Z$ can be
extended to a larger integral form containing $2c$ and containing $c$
if $\dim(R)$ is even. In any case the operators $L_i$ for $i\ge -1$
and $L_{\pm 1}^n/n!$ preserve $V_Z$. (Warning---these operators do not
preserve the integral form of $V_r$ for $r$ not in $R$.)
Tits has also constructed an integral form for Kac-Moody algebras (4).
\proclaim Section 8.~Affinization.
If $A$ is a Kac-Moody algebra with root lattice $R$ we can construct a
sort of affinization of $A$, which when $A$ is finite dimensional is
just the affine algebra of $A$ When $A$ is finite dimensional the
affinization is also a Kac-Moody algebra, but this is not true in
general.
To construct the affinization we form the lattice $R_1$ that is the
sum of $R$ and a one-dimensional lattice generated by $s$ with
$(s,s)=0$ and let $V_1$, $P_1^i$ be the Fock space and physical spaces
of $R_1$. Then we define the affinization $\bar A$ of $A$ to be the
subalgebra of $P_1^1/DP_1^0$ generated by the elements $e^{ns\pm r_i}$
where $n$ runs through the integers, and the $r$s are the simple roots
of $A$.
$\bar A$ is an extension $N.A[z,z^{-1}]$ of an algebra $N$ with an
infinite descending central series by the algebra of Laurent
polynomials in $A$. When $A$ is finite dimensional and simple, $N$ is
one dimensional and we recover the usual affinization of $A$. If $R_2$
is the lattice that is the sum of $R$ and a lattice generated by $s,t$
with $(s,s)=(t,t)=0$, $(s,t)=1$, then $\bar A$ has many
representations that can be constructed as subquotients of the
subspaces $P_2^i$ of $V_2$.
There is a second way to construct the affinization of $A$. If $V$ is
any vertex algebra we make $W=V[z,z^{-1}]$ into a vertex algebra by
defining
$$uz^m_i(vz^n):=\sum_j{m\choose j} u_{i+j}(v)z^{m+n-j},$$
and we make $V$ into a $W$ module by defining
$$uz_i^m(v):=u_{i+m}(v).$$ (This is a special case of the tensor
product of two vertex algebras acting on the tensor product of two of
their modules; in this case the vertex algebras are $V$ and the vertex
algebra of the ring $Z[z,z^{-1}]$ with derivation
$D^{(i)}z^j={j\choose i}z^{j-i}$, and their modules are $V$ and a
one-dimensional module generated by an element 1 with $z_i^j(1)=1$ if
$i+j=-1$, 0 otherwise. Note that this one-dimensional module is not a
module for the ring $Z[z,z^{-1}]$.) The affinization of $A$ is then a
subalgebra of the Lie algebra $W/DW$, and this Lie algebra acts on
$V$. In particular we obtain a formula for the commutator of two
operators $u_m$ and $v_n$ on $V$:
$$[u_m,v_n]=\sum_j{m\choose j}u_j(v)_{m+n-j}.$$ If $V$ is constructed
from a lattice $R$ and $u$ is in $W=V[z,z^{-1}]$ then $Du=0$ if and
only if $u$ is a multiple of 1, and the operator $u_0$ on $V$ is 0 if
and only if $u=Dv$ for some $v$ in $W$. If $u$ is in $V$ then this
holds if and only if $u=Dv+a$ for some $v\in V$ and some constant $a$.
$V$ is usually irreducible under the action of $\bar A$.
\proclaim Section 9.~The monster.
Frenkel {\it et al.} (1) constructed an infinite-dimensional graded
representation $V$ of the monster $F$. This representation can be
given the structure of a vertex algebra that is invariant under $F$
and is similar to the vertex algebras constructed from positive
definite lattices (or more precisely to the subspace of the
complexification of such algebras fixed by the Cartan involution
$\omega$---i.e., their ``compact forms''). In particular it has an
element $c$ such that the operators $L_i=c_{i+1}$ give a
representation of the Virasoro algebra and it has a positive definite
inner product such that the adjoint of $u_n$ is given by the formula
in {\it Section 5} (with $\omega(u)=u$). One important difference
between this algebra and the ones coming from lattices is that the
piece of degree 1 is 0 dimensional. We will call this vertex algebra $V$
the Monster vertex algebra.
Any vertex algebra $V$ with these properties (except that it does not
have to have an action of the monster $F$ on it) also has the
following two properties for $u$ and $v$ in the degree 2 piece of $V$.
(i) $u_1(v)=v_1(u)$. When $V$ is the Monster vertex algebra this is
essentially the Griess product (5). Also $u_1$ is self adjoint.
(ii) Norton's inequality (see ref. 6):
$x=(u_1(u),v_1(v))-(u_1(v),u_1(v))$ is nonnegative and zero if and
only if the operators $u_1$ and $v_1$ commute. In fact $x$ is the norm
of $w=u_0(v)-{1\over 2}D(u_1(v))$, and the operator $w_2$ is the
commutator of $u_1$ and $v_1$.
There are a large number of other vertex algebras with these properties.
In particular the Griess product can be extended to the whole of $V$
in a natural way, and there are also an infinite number of other
products $u_n(v)$ on $V$ invariant under the action of the Monster on
$V$.
The product $u_1(v)$ is not symmetric on the whole of $V$. If we want
symmetric products we can define the product $\times_n$ for any
integer $n$ by
$$u\times_nv = \sum_{i\ge 0}{(-1)^i\over i+1}D^{(i)}(u_{n+1+i}(v))$$
and these are symmetric or antisymmetric depending on whether $n$ is
even or odd. If $n=0$ this is equal to the Griess product $u_1(v)$ on
the degree 2 piece of $V$, and $D(u\times_0v)=u_0(v)+v_0(u)$. These
products have these properties for any vertex algebra over the
rational numbers but do not seem to be as natural as the products
$u_n(v)$.
This research was supported by Trinity College, Cambridge, England and
by the Natural Sciences and Engineering Research Council of Canada.
\item{1.} Frenkel, I. B., Lepowsky, J. and Meurman, A. (1984) Proc. Natl. Acad. Sci. USA 81, 3256-3260.
\item{2.} Frenkel, I. B. and Kac, V. G. (1980) Invent. Math. 62, 23-66.
\item{3.} Frenkel, I. B. (1985) Lect. Appl. Math. 21, 325-353.
\item{4.} Tits, J. (1981) Th\'eorie des groupes, R\'esume de Cours,
Annuaire du Coll\`ege de France, pp. 75-87.
\item{5.} Griess, R. L., Jr. (1982) Invent. Math. 69, 1-102.
\item{6.} Conway, J. H. (1985) Invent. Math. 79, 513-540.
\bye