0$.
Corollary 3.3 now follows from corollary 3.2.
Corollary 3.3 will allow us to lift elements in the root spaces of
norm 2 vectors. In the rest of this section we show how to
lift elements in the root spaces of norm 0 vectors.
\proclaim Lemma 3.4.
Suppose that the the elements $\Gamma_i$ for $i\in \Z$ are independent
formal variables. If $I=(i_1,\ldots, i_m)$ is a finite sequence of
integers then define $\Gamma_I$ to be $\Gamma_{i_1}\cdots
\Gamma_{i_m}$ and define $\Sigma(I)$ to be $i_1+\cdots +i_m$ and
define $l(I)$ to be $m$. Then all coefficient of the power series
$$E=\exp\Big( \sum_{m,n>0}
\sum_{I\in \Z^m}
\sum_{J\in \Z^n\atop \Sigma(J)=-\Sigma(I)>0}
{\Sigma(I)\over mn}\Gamma_{I}\Gamma_J\Big)$$
are integers.
Proof. If $I$ is any finite sequence of integers then $I^k$ means the
obvious concatenation of $k$ copies of the sequence $I$. We will call
a pair of finite sequences primitive if it is not of the form
$(I^k,J^k)$ for some $k\ge 2$. Any pair of finite sequences can be
written uniquely as $(I^k,J^k)$ for some primitive pair $(I,J)$,
which we call the primitive core of the pair $(I^k,J^k)$. Consider
the group $\Z\times \Z$ acting on pairs $(I,J)$ by the first $\Z$
acting as cyclic permutations of the elements of $I$ and the second
$\Z$ acting as cyclic permutations of the elements of $J$. We group
the pairs $(I,J)$ indexing the terms of the sum in the exponent of $E$
into equivalence classes, where we say two pairs of sequences are
equivalent if their primitive cores are conjugate under $\Z\times \Z$.
For any primitive element $(I,J)$ with $\Sigma(I)+\Sigma(J)=0$ we will
show that all coefficients of the exponential of the sum of all terms
in the equivalence class of $(I,J)$ are integral. This will show that
the coefficients of $E$ are integral because $E$ is an infinite
product over the set of orbits of primitive elements of expressions
like this. Let $m'$ and $n'$ be the number of orbits of $I$ and $J$
under the cyclic action of $\Z$. Then the exponential of the sum of
the terms equivalent to $(I,J)$ is
$$\eqalign{
&\exp\Big(\sum_{k>0}{\Sigma(I^k)\over l(I^k)l(J^k)}
(\hbox{number of orbits of $(I,J)$ under $\Z\times\Z$})
\Gamma_{I^k}\Gamma_{J^k}\Big)\cr
=&\exp\Big(\sum_{k>0} {k\Sigma(I)m'n'\over kmkn}
\Gamma_{I^k}\Gamma_{J^k}\Big)\cr
=&\exp\Big({\Sigma(I)m'n'\over mn} \sum_{k>0} \Gamma_I^k\Gamma_J ^k/k\Big)\cr
=&\exp\Big(-{\Sigma(I)m'n'\over mn} \log(1- \Gamma_I\Gamma_J )\Big)\cr
=& (1- \Gamma_I\Gamma_J )^{-\Sigma(I)m'n'/mn}\cr
}$$
The number $\Sigma(I)$ is divisible by $m/m'$ as $I$ is the concatenation
of $m/m'$ identical sequences and $\Sigma(I)$
is the sum of the elements of $I$,
and similarly $\Sigma(I)=-\Sigma(J)$ is divisible
by $n/n'$. Moreover $m/m'$ and $n/n'$ are coprime as $(I,J)$ is primitive.
Hence $\Sigma(I)m'n'/mn$ is an integer because it is equal to $\Sigma(I)$
divided
by two coprime factors $m/m'$ and $n/n'$ of $\Sigma(I)$.
This implies that $(1- \Gamma_I\Gamma_J )^{-\Sigma(I)m'n'/mn}$ has
integral coefficients,
and so $E$ does as well as it is an infinite product of expressions like this.
This proves lemma 3.4.
\proclaim Theorem 3.5. Let $V$ be the vertex algebra
(over the integers) of the double cover of an even lattice $L$.
Define $D^*V$ to be the sum of the spaces $D^{(i)}V$ for $i\ge 1$.
We recall from [B86] that $V/D^*V$ has a natural Lie algebra structure,
with the bracket defined by $[u,v]=u_0v$. Moreover this Lie algebra
acts on $V$ preserving the vertex algebra structure of $V$.
Suppose that $\alpha, \gamma\in L$ with
$\alpha$ orthogonal to the norm 0 vector $\gamma$.
Then
$$\exp\Big(\sum_{i>0} {x^i\over i}(\alpha(1)e^{i\gamma})_0 \Big)$$
is a lifting of $(\alpha(1)e^\gamma)_0$ in the universal enveloping algebra
of $(V/D^*V)\otimes \Q$ all of whose
coefficients map $V$ to $V$.
Proof. It is obvious that the element is group-like as it is
the exponential of a primitive element, so the only problem
is to show that it preserves the integral form $V$.
This element is also an automorphism of the vertex algebra $(V\otimes
\Q)[[x]]$ as it is the exponential of a derivation of this vertex
algebra. Hence to show its coefficients
preserve the integral form of $V$
it is sufficient to show that it maps each of the generators
$e^\beta$, $\beta\in L$, of the vertex algebra $V$ into $V[[x]]$.
Define the operators $\Gamma_i$ by
$$e^\gamma(z)= \sum_i\Gamma_iz^i.$$
Recall the following formulas from [B86].
$$\alpha(1)(z) = \sum_j \alpha(1)_{-j}z^{j-1} = \sum_j \alpha(j)z^{j-1}$$
All coefficients of $e^\gamma$ commute with everything in sight
as $\gamma$ is orthogonal to $\gamma$ and $\alpha$.
So $(\alpha(1)e^{i\gamma})_0$ is equal to
the coefficient of $z^{-1}$ in
$$\alpha(1)(z)(e^\gamma(z))^i
=\sum_j \alpha(j)z^{j-1} \sum_{I\in \Z^i}\Gamma_Iz^{\Sigma(I)}$$
Therefore $\sum_{i>0}{x^i\over i}(\alpha(1)e^{i\gamma})_0 $ is equal to
$A^++A^0+A^-$ where
$$\eqalign{
A^+&= \sum_{m>0} \sum_{j>0} \alpha(j) \sum_{I\in \Z^m\atop \Sigma(I)=-j}
{x^m\over m} \Gamma_I \cr
A^0&= \sum_{m>0} \sum_{j=0} \alpha(j) \sum_{I\in \Z^m\atop \Sigma(I)=-j}
{x^m\over m} \Gamma_I \cr
A^-&= \sum_{m>0} \sum_{j<0} \alpha(j) \sum_{I\in \Z^m\atop \Sigma(I)=-j}
{x^m\over m} \Gamma_I \cr
}$$
We would like to pull out a factor of $\exp(A^-)$
from $\exp(A^++A^0+A^-)$ because $\exp(A^-)(e^\beta)
=e^\beta$, but we have to be careful when doing this because $\alpha(i)$
does not commute with $\alpha(-i)$ if $i\ne 0$.
We can easily evaluate $[A^+,A^-]/2$ using the fact that
$[\alpha(i),\alpha(j)]=j(\alpha,\alpha)$ if $i+j=0$ and 0 otherwise, and
we find that
$$[A^+,A^-]/2= {1\over 2} (\alpha,\alpha)
\sum_{m,n>0}\sum_{k<0}
\sum_{I\in \Z^m\atop \Sigma(I)=k}
\sum_{J\in \Z^n\atop \Sigma(J)=-k}
{k\over mn}\Gamma_I\Gamma_J.$$
In particular this verifies the fact used below that $[A^+,A^-]$ commutes with
$A^+$ and $A^-$.
The exponential of this is the expression in lemma 3.4 raised to the power
of $(\alpha,\alpha)/2$, and therefore has integral coefficients by
lemma 3.4 and because $(\alpha,\alpha)$ has even norm.
So $\exp([A^+,A^-]/2)$ maps $V$ into $V[[x]]$.
Recall the
formula
$$\exp(A^++A^-)=\exp([A^-,A^+]/2)\exp(A^+)\exp(A^-)
$$
which is valid because $[A^+,A^-]$ commutes with $A^+$ and $A^-$.
(This is essentially just the first few terms of the
Baker-Campbell-Hausdorff formula.) Now we look at
$$\exp(A^++A^0+A^-)e^\beta =
\exp([A^-,A^+]/2)\exp(A^+)\exp(A^0)\exp(A^-)e^\beta
$$
It is obvious that $\exp(A^-)(e^\beta) = e^\beta$,
and we have checked above that all coefficients of
$\exp([A^-,A^+]/2)$ map $V$ to $V$. Hence to complete the
proof of theorem 3.5 it is sufficient to prove that
all coefficients of $\exp(A^0)$ and $\exp(A^+)$ map $V$ into $V$.
We check that $\exp(A^0)$ maps $V$ to $V[[x]]$. The follows
because $A^0$ is an infinite sum of expressions
like
$$\sum_{n>0} \alpha(0) \Gamma_{J^n}x^n/n=-\alpha(0)\log(1-x\Gamma_J)$$
where $J$ is a primitive sequence (in other words a sequence
that cannot be written in the form $I^m$ for some $m>1$).
Hence $\exp(A^0)$ is an infinite product of
terms of the form $(1-x\Gamma_{J})^{-\alpha(0)}$, which
map $V[[x]]$ to itself because $\alpha(0)$ has integral eigenvalue
$(\alpha,\beta)$ on the subspace of $V$ of degree $\beta\in L$.
This shows that all coefficients of $\exp(A^0)$ map $V$ into $V$.
Finally we have to show that all coefficients of $\exp(A^+)$ map $V$
into $V$.
As usual we divide the sum over elements $I$ in $A^+$ into
classes consisting of powers of conjugates of primitive elements $I$.
We see that $A^+$ is a sum over all orbits of primitive elements $I$
with $\Sigma(I)>0$ of
expressions like
$$l(I)\sum_{k>0}\alpha(\Sigma(I^k)){x^{l(I^k)}\over l(I^k)}\Gamma_{I^k}
=\sum_{k>0}\alpha(k\Sigma(I)){x^{kl(I)}\over k} \Gamma_I^k.
$$
(The factor at the front is the number
of conjugates of $I$ under the cyclic action of $\Z$, which
is equal to $l(I)$ because $I$ is primitive.)
So it is sufficient to show that the exponential of this
expression has integral coefficients.
Let $y_1,y_2,\ldots, $ be a countable number of independent
variables,
and identify $\alpha({k})$ with the symmetric function $\sum y_i^k$
of the $y$'s for $k>0$. (See [M chapter 1].)
Then
$$\eqalign{
&\exp\Big(\sum_{k>0}\alpha(k\Sigma(I)){x^{kl(I)}\over k}
\Gamma_I^k\Big)\cr
=&\exp\Big(\sum_i\sum_{k>0}y_i^{k\Sigma(I)}{x^{kl(I)}\over k}
\Gamma_I^k\Big)\cr
=&\exp\Big(\sum_i -\log(1-y_i^{\Sigma(I)}{x^{l(I)}}
\Gamma_I)\Big)\cr
=&\prod_{ i}{1\over 1-y_i^{\Sigma(I)}{x^{l(I)}}
\Gamma_I }.\cr
}$$
So we see that
$$\exp(A^+) = \prod_I\prod_{ i}{1\over 1-y_i^{\Sigma(I)}{x^{l(I)}}
\Gamma_I }
$$
where the product over $I$ is a product over all orbits
of primitive sequences $I$ with $\Sigma(I)>0$ under the cyclic action
of $\Z$. The last line is a power series in the elements $\Gamma_I$
and $x$ whose coefficients are symmetric functions in the $y$'s, and
hence are polynomials with integral coefficients in the complete
symmetric functions of the $y$'s. So we have to show that each
complete symmetric function, considered as a polynomial in the
$\alpha$'s with rational (not necessarily integral!) coefficients maps
$V$ to $V$. The complete symmetric functions are the polynomials
$e^{-\alpha}D^{(n)}(e^\alpha)$ considered as elements of the ring $V$
underlying the vertex algebra $V$ (as follows from [M, Chapter 1,
2.10]). By definition of the integral form $V$ these polynomials map
$V$ to itself.
This proves theorem 3.5.
\proclaim 4.~The fake monster smooth Hopf algebra.
We recall the construction of the fake monster Lie algebra [B90].
It is the Lie algebra of physical states of the vertex algebra of
the double cover of the lattice $II_{25,1}$. This Lie algebra has
an integral form $\m$ consisting of the elements represented by
elements of the integral form of the vertex algebra.
We recall some properties of $\m$:
\item{1} $\m$ is graded by the lattice $II_{25,1}$,
and the piece $\m_\alpha$ of degree $\alpha\in II_{25,1}$ has dimension
$p_{24}(1-\alpha^2/2)$ if $\alpha\ne 0$ and 26 if $\alpha=0$,
where $p_{24}(n) $ is the number of partitions of $n$ into
parts of 24 colors.
\item{2} $\m$ has an involution $\omega$ lifting the
involution $-1$ of $II_{25,1}$.
\item{3} $\m$ has a symmetric invariant integer valued bilinear
form $(,)$, and the pairing between $\m_\alpha\otimes \Q$
and $\m_{-\alpha}\otimes \Q$ is nonsingular.
\item{4} $\m\otimes \Q$ is a generalized Kac-Moody algebra.
The simple roots are given by the norm 2 vectors $r$
with $(r,\rho)=-1$, together with all positive multiples of $\rho$
with multiplicity 24, where $\rho$ is a primitive norm 0 vector
of $II_{25,1}$ such that $\rho^\perp/\rho$ is isomorphic to the Leech lattice.
(This follows from [B90. theorem 1].)
We define $U^+(\m)$ to be the $\Z$-subalgebra of
the universal enveloping algebra $U(\m\otimes \Q)$ generated
by the coefficients of the liftings of elements in root spaces of
the simple roots and their negatives
constructed in corollary 3.3 and theorem 3.5. (Note that all simple roots
of $\m$
have norms 2 or 0 so we can always apply one of these two types of liftings.)
\proclaim Theorem 4.1. There is a $II_{25,1}$-graded
Hopf algebra $U^+(\m)$ over $\Z$ with the following
properties.
\item {1} $U^+(\m)$ has a structural basis over $\Z$.
\item {2} The primitive elements of $U^+(\m)$ are an integral form of the
fake monster Lie algebra $\m$.
\item {3} For every norm 2 vector of $II_{25,1}$, $U^+(\m)$ contains
the usual (Kostant) integral form of the universal enveloping algebra
of the corresponding $sl_2(\Z)$.
Proof. The algebra $U^+(\m)$ is a $\Z$-Hopf subalgebra of
$U(\m\otimes\Q)$ as it is generated by coalgebras. Also $U^+(\m)$ is
obviously torsion free as it is contained in a rational vector space.
It is easy to check directly that the degree zero primitive elements
of $U^+(\m)$ are just the degree 0 elements of $\m$ and therefore form
a free $\Z$ module (of rank 26). If $\alpha$ is any nonzero element
of $II_{25,1}$ having nonzero inner product with some element
$\beta\in II_{25,1}$, then $(\alpha,\beta)u\in
\m_\alpha$ for any primitive element $u\in U^+(\m)$ of degree
$\alpha$, because $U^+(\m)$ maps $\m$, and hence $\beta$, to
$\m$ by corollary 3.3 and theorem 3.5.
This shows that all root spaces of primitive elements of
$U^+(\m)$ are free $\Z$-modules. We can now apply theorem 2.15 to see
that $U^+(\m)$ is a Hopf algebra with a structural basis. The
primitive elements of $U^+(\m)$ form an integral form of the fake
monster Lie algebra, because the fake monster Lie algebra over the
rationals is generated by the root spaces of simple roots and their
negatives, and all simple roots have norms 2 or 0. This proves theorem 4.1.
\proclaim 5.~A smooth Hopf algebra for the Virasoro algebra.
In this section we show that there is a Hopf algebra over $\Z$ with a
structural basis whose primitive elements form the natural
integral form of the
Virasoro algebra (theorem 5.7). In other words, there is a formal
group law over $\Z$ corresponding to the Virasoro algebra. Moreover this
Hopf algebra acts on the integral form of the vertex algebra of any
even self dual lattice.
Let $R$ be a commutative ring. We write $Hom(R,R)$ for the ring of
homomorphisms of the abelian group $R$ to itself, and $Der(R)$ for the
Lie algebra of derivations of the ring $R$, and $U(Der(R))$ for the universal
enveloping algebra of $Der(R)$ over $\Z$. Consider the group of all
element $a=\sum_i a_i\epsilon^i\in Hom(R,R)[[\epsilon]]$ with $a_0=1$
that induce automorphisms of the $\Z[[\epsilon]]$ algebra
$R[[\epsilon]]$. We can think of the elements of this group
informally as ``infinitesimal curves in the group of automorphisms of
$Spec(R)$''. We will call a derivation of $R$ liftable if it is of
the form $a_1$ for some $a$ as above.
\proclaim Lemma 5.1. Let $R$ be any commutative algebra
with no $\Z$-torsion. Then any $a\in Hom(R,R)[[\epsilon]]$
with $a(0)=1$ that is an automorphism of $R[[\epsilon]]$
is the image of a unique group-like element $G_a$ of
$U(Der(R)\otimes \Q)[[\epsilon]]$ with $G_a(0)=1$ under
the natural map from $U(Der(R)\otimes \Q)$ to $Hom(R,R)\otimes \Q$.
Proof. We note that $\log(a)$
is a well defined element of $(Hom(R,R)\otimes \Q)[[\epsilon]]$
as $a=1+O(\epsilon )\in Hom(R,R)[[\epsilon ]]$.
As $a$ is a ring homomorphism it follows that
that $\log(a)$ is a derivation of $(R\otimes \Q)[[\epsilon]]$
and is therefore an element of $(Der(R)\otimes \Q)[[\epsilon]]$.
Now we consider $\log(a)$
to be an element of the universal enveloping algebra
$U(Der(R)\otimes \Q)[[\epsilon ]]$
and we define $G_a\in U(Der(R)\otimes \Q)[[\epsilon ]]$ by
$$G_a = \exp(\log(a)),$$
where the log is computed in $(Hom(R,R)\otimes \Q)[[\epsilon]]$
and the exponential
is computed in $U(Der(R)\otimes \Q)[[\epsilon ]]$.
It is obvious that the action of $G_a$ on $R[[\epsilon]]$ is the same
as that of $a$.
Also $G_a$ is group-like because it is the exponential of a primitive
element. It is easy to check that $G_a$ is the unique group-like
lifting of $a$ with $G_a(0)=1$, because the log of a group-like element
must be primitive and must therefore be the same as $\log(a)$.
This proves lemma 5.1.
\proclaim Corollary 5.2. Suppose that $R$ is a commutative
ring with no $\Z$ torsion such that $Der(R)$ is a free
$\Z$-module. Define $U^+(Der(R))$ to be the subalgebra of
$U(Der(R\otimes\Q))$ generated by all the coefficients of all
group-like elements of $U(Der(R)\otimes \Q)[[\epsilon]]$ that have
constant coefficient 1 and map $R[[\epsilon]]$ to $R[[\epsilon]]$. Then
$U^+(Der(R))$ is a Hopf algebra over $\Z$ with a structural basis, and
its primitive elements are the liftable primitive elements of
$Der(R)$.
Proof. Applying theorem 2.15 shows that $U^+(Der(R))$
is a Hopf algebra over $\Z$ with
a structural basis. By lemma 5.1 the space of primitive elements
of $U^+(Der(R))$
is the same as the space of liftable primitive elements of
$Der(R)$. This proves corollary 5.2.
{\bf Example 5.3} Suppose we take $R$ to be the algebra
$\Z[x][x^{-1}]$ of Laurent polynomials. Then $Der(R)=Witt$ is the
Witt algebra over $\Z$, which is spanned by the elements $L_m=-x^{m+1}{d\over
dx}$ for $m\in \Z$. All elements $L_m$ are liftable; for example, we
can use the automorphism of $R[[\epsilon]]$ taking $x$ to $x-\epsilon
x^{m+1}$ to show that $L_m$ is liftable. Therefore the Hopf algebra
$U^+(Witt)$ is a Hopf algebra over $\Z$ with a structural basis, whose
primitive elements are exactly the elements of the Witt algebra.
\proclaim Lemma 5.4. Let $N$ be an integer. Let $R_N$ be the representation
of $Witt$ with a basis of elements $e_n$, $n\in \Z$, with the action
given by $L_m(e_n)= (Nm+n)e_{m+n}$. Then the action of
$Witt$ on $R_N$ can be extended to an action of $U^+(Witt)$ on $R_N$.
Proof. For $N=-1$ the module $R_N$ is the module of first order
differential operators on $R=\Z[x][x^{-1}]$, so the automorphisms $G_a$
extend to $R_N[[\epsilon]]$. For other negative values of $N$ the $R$-module
$R_N$
is a tensor product of $-N$ copies of the module $R_{-1}$, and
$R_N$ for $N$ positive is the dual of $R_N$ for $N$ negative.
Therefore $U^+(Witt)$ extends to these modules as well. This proves
lemma 5.4.
We define $Witt_{\ge n}$ for $n=-1,1$ to be the subalgebra of
$Witt$ spanned by $L_i$ for $i\ge n$.
Let $V$ be the vertex algebra of some even lattice. It contains elements
$e^\alpha$ for $\alpha\in L$, so there are operators
$e^\alpha_i$ on $V$ for $i\in \Z$. The algebra $Witt_{\ge -1}$ also
acts naturally on $V$.
\proclaim Lemma 5.5.
Put $N=\alpha^2/2-1$ and $e_j=e^\alpha_{\alpha^2/2-1-j}$ for $j\in \Z$
and let $R_N$
be the space with the elements $e_j$ as a basis. Define an action of
the algebra $U^+(Witt_{\ge -1})$ on $R_N\otimes U^+(Witt_{\ge -1})$
using the action on $R_N$ as in lemma 5.4 and the action on
$U^+(Witt_{\ge -1})$ by left multiplication and the coalgebra structure
of $U^+(Witt_{\ge -1})$. If $u\in U^+(Witt_{\ge -1})$ then
$ue^\alpha_m= \sum_i e^\alpha_i u_i$ as operators on $V$, where $\sum_i
e^\alpha_i\otimes u_i=u(e^\alpha_m\otimes 1)$ is the image of
$e^\alpha_m\otimes 1$ under the action of $u$ on $R_N\otimes
U^+(Witt_{\ge -1})$. In particular if $U^+(Witt_{\ge -1})$
maps some element $v\in V$ to $V$ then it also maps $e^\alpha_i(v)$ to $V$
for any $\alpha\in L$ and $i\in \Z$.
Proof.
If $u$ is of the form $L_i$ for $i\ge -1$ this can be proved as follows.
A standard vertex algebra calculation shows that
$$[L_i, e^\alpha_j] = ((i+1)(\alpha^2/2-1)-j) e^\alpha_{i+j}.$$
This
shows that lemma 5.5 is true when $u\in Witt_{\ge -1}$. If lemma 5.5
is true for two elements $u$, $u'$ of $U^+(Witt_{\ge -1})$ then it is
true for their product. If it is true for some nonzero integral
multiple of $u\in U^+(Witt_{\ge -1})$ then it is true for $u$ because
$V$ is torsion-free. To finish the proof we observe that the algebra
$U^+(Witt_{\ge -1})$ is generated up to torsion by the elements $L_m$
for $m\ge -1$. This proves lemma 5.5.
\proclaim Lemma 5.6. Suppose that $V$ is the vertex algebra
of the double cover of some even lattice, with the standard action of
$Witt_{\ge-1}$ on $V\otimes \Q$ ([B86]). This action extends to an action of
$U^+(Witt_{\ge -1})$ on $V$.
Proof. The vertex algebra $V$ is generated from the element $1$ by the
actions of the operators $e^\alpha_n$ for $n\in \Z$ and $\alpha\in L$.
Define $F^n(V)$ by defining $F^0(V)$ to be the space spanned by 1, and
defining $F^{n+1}(V)$ to be the space spanned by the actions of
operators of the form $e^\alpha_m$ on $F^n(V)$. Then $V$ is the union
of the spaces $F^n(V)$, so it is sufficient to prove that each space
$F^n(V)$ is preserved by the action of $U^+(Witt_{\ge -1})$. We will
prove this by induction on $n$. For $n=0$ is is trivial because
$L_n(1)=0$ for $n\ge -1$. If it is true for $n$, then it follows
immediately from lemma 5.5 that it is true for $n+1$.
This proves lemma 5.6.
The algebra $U^+(Witt_{\ge -1})$ can be $\Z$-graded in such a way that
$L_m$ has degree $m$. This follows easily from the fact that
we can find graded liftings of the elements $L_m$. We define
$U^+(Witt_{\ge 1})$ to be the subalgebra generated by the coefficients
of graded liftings of the elements $L_m$ for $m\ge 1$. It is easy to check
that this is a Hopf algebra with a structural basis whose Lie algebra of
primitive vectors has a basis consisting of the elements $L_m$ for $m\ge 1$.
It is $\Z_{\ge 0}$-graded, with all graded pieces being finite dimensional;
in fact the piece of degree $n\in \Z$ has dimension $p(n)$ where
$p$ is the partition function.
We let $U(Witt_{\ge 1})$ be the
universal enveloping algebra of the Lie algebra $Witt_{\ge 1}$.
It is $\Z_{\ge 0}$-graded in the obvious way and is a subalgebra
of $U^+(Witt_{\ge 1})$.
In the rest of this section we construct an integral form with a
structural basis for the universal enveloping algebra of the Virasoro
algebra over $\Z$. This result is not used elsewhere in this paper.
We recall that the Virasoro algebra $Vir$ is a central extension
of $Witt$ and is spanned by elements $L_i$ for $i\in \Z$ and
an element $c/2$ in the center, with
$$[L_m,L_n]=(m-n)L_{m+n} +{m+1\choose 3}{c\over 2}.$$
We identify $Witt_{\ge -1}$ with the subalgebra of $Vir$
spanned by $L_m$ for $m\ge -1$.
The Virasoro algebra $Vir$ has an automorphism
$\omega$ of order 2 defined by $\omega(L_m)=-L_{-m}$, $\omega(c)=-c$,
and $\omega$ extends to an automorphism of the universal enveloping algebra
$U(Vir\otimes\Q)$. We define $U^+(Vir)$ to be the
subalgebra of $U(Vir\otimes\Q)$ generated by
$U^+(Witt_{\ge -1})$ and $\omega(U^+(Witt_{\ge -1}))$.
For any even integral lattice $L$ there is a double cover
$\hat L$, unique up to non-unique isomorphism,
such that $e^ae^b=(-1)^{(a,b)}e^be^a$.
We let $V_{\hat L}$ be the (integral form of the)
vertex algebra of $\hat L$. This is $L$-graded, and
has a self dual bilinear form on it (more precisely,
each piece of given $L$-degree $\alpha$ and given eigenvalue under $L_0$
is finite dimensional and dual to the piece of degree $-\alpha$),
and if $L$ is self dual then
$V_{\hat L}$
has a conformal vector generating an action of the Virasoro algebra.
\proclaim Theorem 5.7. The subalgebra $U^+(Vir)$ of
$U(Vir\otimes \Q)$ has the following properties:
\item 1 $U^+(Vir)$ is a $\Z$-Hopf algebra with a structural basis.
\item 2 The Lie algebra of
primitive elements of $U^+(Vir)$ has a basis consisting of
the elements $L_n$ for $n\in \Z$ and the element $c/2$.
\item 3 $U^+(Vir)$ maps the vertex algebra (over $\Z$) of any even
self dual lattice to itself.
Proof. We first construct the action on the vertex algebra $V$
of an even self dual lattice. We have an action of $U^+(Witt_{\ge -1})$
on $V$ by lemma 5.6. The vertex algebra of an even self dual lattice
is also self dual under its natural bilinear form, so the adjoint
of any linear operator on $V$ is also a linear operator on $V$.
The adjoint of $L_m$ is $L_{-m}$, so the adjoint $U^+(Witt_{\le 1})=
\omega(U^+(Witt_{\ge -1}))$ also maps $V$ to itself. As these two
algebras generate $U^+(Vir)$ this proves that $U^+(Vir)$ acts on $V$.
Next we find the Lie algebra $P$ of primitive elements of $U^+(Vir)$.
The elements $L_{m}$ for $m\ge -1$ are obviously in $U^+(Vir)$ because
they are in $U^+(Witt_{\ge -1})$, and similarly $L_m$ for $m\le 1$ is in
$P$. The element $c/2$ is in $P$ because $[L_2,L_{-2}]= 4L_0+c/2$.
So $P$ contains the basis described in theorem 5.7, and we have
to prove that $P$ contains no elements other than linear combinations of these.
As $U^+(Vir)$ and hence $P$ are both $\Z$-graded with $L_m$ having
degree $m$, it is sufficient to show that the degree $m$ piece of
$P$ is spanned by $L_m$ if $m\ne 0$ and by $L_0$ and $c/2$ if $m=0$.
For $m\ne 0$ this is easy to check as we just map the Virasoro algebra
to the Witt algebra and use example 5.3. The case $m=0$ is harder
and we will use the actions on vertex algebras of even self dual
lattices $L$ constructed above. If $L$ is such a lattice then
$c$ acts on $V$ as multiplication by $\dim(L)$, and $L_0$ has
eigenspaces with eigenvalue any given positive integer (at least
if $L$ has positive dimension). So if $xL_0+yc/2$ is in $P$ for some
$x,y\in \Q$ then $xm+yn/2$ is an integer whenever $m$ is a positive
integer and $n$ is the dimension of a nonzero even self dual lattice.
As we can find such lattices for any positive even integer $n$, this
implies that $x$ and $y$ are both integers. Hence
the degree 0 piece of $P$ is spanned by $L_0$ and $c/2$. This completes
the proof that $P$ is spanned by $L_m, m\in \Z$, and $c/2$.
Finally we have to show that $U^+(Vir)$ has a structural basis.
We know that all the elements $L_m$ for $m\ge -1$ are liftable in
$U^+(Witt_{\ge -1})$, so they are also liftable in $U^+(Vir)$.
Similarly the elements $L_m$ for $m\le 1$ are liftable in
$\omega(U^+(Witt_{\ge -1}))$ and therefore in $U^+(Vir)$.
It is trivial to check that $c$ is liftable as it acts
as multiplication by some integer, so every primitive element of
$U^+(Vir)$ is liftable by lemma 2.5 and part 2 of theorem 5.7.
Also $U^+(Vir)$ is generated
by the coefficients of group-like elements because this is true
for $U^+(Witt_{\ge -1})$ and its conjugate under $\omega$. The fact that
$U^+(Vir)$ has a structural basis now follows from theorem 2.15.
This completes the proof of theorem 5.7.
Example. If $L$ is an even lattice of odd dimension then $L_{-2}(1)$
is not in the vertex algebra of $L$, because $[L_2,L_{-2}] =
4L_0+\dim(L)/2$. So part 3 of theorem 5.7 is false without the
assumption that $L$ is self dual.
\proclaim 6.~The no-ghost theorem over $\Z$.
The no-ghost theorem of Goddard and Thorn [G-T] implies that over
the reals, the contravariant form restricted to the
degree $\beta\ne 0$ piece of the fake monster Lie algebra $\m$
is positive definite and in particular nonsingular. We give a refinement of
this to the integral form of the degree $\beta$ piece $\m_\beta$, showing
that the any prime dividing the discriminant of the quadratic
form on $\m_\beta$ also divides the vector $\beta$.
In particular if $\beta$ is a primitive vector then
$\m_\beta$ is a self dual positive definite integral lattice.
I do not know whether or not the discriminant can be divisible by $p$ when
$\beta$ is divisible by $p$.
If $\lambda=1^{i_1}2^{i_2}\cdots$ is a partition with $i_1$ 1's, $i_2$
2's, and so on, then we define $P(\lambda)$ to be the integer
$1^{i_1}2^{i_2}\cdots$ and $F(\lambda)$ to be $ i_1!i_2!\cdots$ and
$|\lambda|$ to be $1i_1+2i_2+\cdots$ and $l(\lambda)$ to be
$i_1+i_2+\cdots$. We also define $p(n)$ to be the number of partitions
of $n$.
\proclaim Lemma 6.1. Suppose that $n$ is an integer.
Then
$$\prod_{|\lambda|=n} P(\lambda)=\prod_{|\lambda|=n} F(\lambda).$$
Proof. We will show that both sides are equal to
$$\prod_{i>0} i^{\Sigma_{j>0}p(n-ij)}.$$
The left hand side $\prod_{|\lambda|=n} P(\lambda)$ is equal to
$\prod_{i>0} i^{n(i)}$ where $n(i)$ is the number of times $i$ occurs
in some partition of $n$, counted with multiplicities. This number
$n(i)$ is equal to $\sum_j n(i,j)$ where $n(i,j)$ is the number of
times that $i$ occurs at least $j$ times in a partition of $n$. But
$n(i,j)$ is equal to $p(n-ij)$ because any partition of $n$ in which
$i$ occurs at least $j$ times can be obtained uniquely from a
partition of $n-ij$ by adding $j$ copies of $i$. So the left hand
side $\prod_{|\lambda|=n} P(\lambda)$ is equal to $\prod_{i>0}
i^{\Sigma_{j>0}p(n-ij)}$.
On the other hand the right hand side $\prod_{|\lambda|=n} F(\lambda)$
is equal to $\prod_{i>0} i^{m(i)}$ where $m(i)$ is the number of times that
there is a partition of $n$ with some number occurring at least $i$ times
(counting a partition several times if it has more than
one number occurring at least $i$ times). But $m(i)$ is equal to
$\sum_{j}n(j,i)=\sum_{j}p(n-ji)$.
This shows that the right hand side
$\prod_{|\lambda|=n} F(\lambda)$ is also equal to
$\prod_{i>0} i^{\Sigma_{j>0}p(n-ij)}$. This proves lemma 6.1.
\proclaim Lemma 6.2.
The submodule $U(Witt_{\ge 1})_n$ has index $\prod_{|\lambda|=n} F(\lambda)$
inside $U^+(Witt_{\ge 1})_n$.
Proof. Choose a graded lifting $1+a_{i,1}x+a_{i,2}x^2+\ldots$
of $L_i$ for each $i>0$. Then
the elements $a_{1,i_1}a_{2,i_2}\cdots$ for $1i_1+2i_2+\cdots=n$
form a base for $U^+(Witt_{\ge 1})$,
and the elements $i_1!a_{1,i_1}i_2!a_{2,i_2}\cdots$ for
$1i_1+2i_2+\cdots=n$ form a basis for $U(Witt_{\ge 1})_n$.
Therefore the index of $U(Witt_{\ge 1})_n$ in $U^+(Witt_{\ge 1})_n$
is
$$\prod_{i_1+2i_2+\cdots=n}i_1!i_2!\cdots=\prod_{|\lambda|=n}F(\lambda).$$
This proves lemma 6.2.
\proclaim Lemma 6.3.
Let $\gamma$ be a norm 0 vector of $II_{25,1}$. Suppose that $W$ is
the graded space of all elements generated by the action of the
elements $e^{-\gamma}D^{(i)}(e^\gamma)$ on $e^\beta$, so that $W$ is
acted on by the smooth integral form $U=U^+(Witt_{>0})$. Then the
graded dual $W[1/(\beta,\gamma)]^*$ of $W[1/(\beta,\gamma)]$ is a free
$U[1/(\beta,\gamma)]$-module on one generator.
Proof. We define $U_n$ and $W_n$ to be the degree $n$ pieces of $U$
and $W$. Let $w_\mu$ be the basis of elements
$\gamma(1)^{j_1}\gamma_2^{j_2}\cdots e^\beta$ for $W_n\otimes \Q$
parameterized by partitions $\mu=1^{j_1}2^{j_2}\cdots$ of $n$. The
$\Z$ module $W_n'$ spanned by the $w_\lambda$'s is not $W_n$ but has
index $\prod_{|\mu|=n} F(\mu)$ in it. We let the elements
$L_\lambda=L_1^{i_1}L_2^{i_2}\cdots$ be the basis for the space
$U_n\otimes \Q$ indexed by partitions $\lambda
=1^{i_1}2^{i_2}\cdots$. The $\Z$ module $U_n'$ spanned by the
$L_\lambda$'s is not $U_n$ but has index $\prod_{|\lambda|=n}
F(\lambda)$ in it by lemma 6.2. We define $m_{\lambda,\mu}$ for
$|\lambda|=|\mu|$ by $L_\lambda(w_\mu)=m_{\lambda,\mu}e^\beta$. We
will show that the determinant of the $p(n)$ by $p(n)$ matrix
$(m_{\lambda,\mu})$ is
$$
\prod_{|\lambda|=n} (\beta,\gamma)^{l(\lambda)}P(\lambda)F(\lambda)
$$
where $l(\lambda)$ is the number of elements of the partition $\lambda$.
We order the partitions by $\lambda> \mu $ if
$\lambda$ is the partition $\lambda_1+\lambda_2+\cdots$ with
$\lambda_1\ge \lambda_2\ge\cdots$,
$\mu$ is the partition $\mu_1+\mu_2+\cdots$ with
$\mu_1\ge \mu_2\ge\cdots$,
and $\lambda_1=\mu_1,\ldots,\lambda_{k-1}=\mu_{k-1}$, $\lambda_k>\mu_k$
for some $k$.
Then the matrix entry $m_{\lambda,\mu}$ is 0 if $\lambda>\mu$,
and is equal to
$$P(\lambda)F(\Lambda)(\beta,\gamma)^{l(\lambda)}$$
if $\lambda=\mu$. We can see this by repeatedly using the relation
$$\eqalign{
&\cdots L_{i_{2}} L_{i_1}
\gamma(j_1)\gamma(j_{2})\cdots e^\beta\cr
=&
\cases{
(\hbox{number of $j$'s equal to $j_1$}) (\beta,\gamma) j_1
\cdots L_{i_{2}}
\gamma(i_{2})\cdots e^\beta
&if $i_1=j_1$\cr
0&if $i_1>j_1$\cr}
\cr
}$$
for $j_1\ge j_2\ge \cdots$, which in turn follows from the
identities $[L_i,\gamma(j)]=j\gamma(j-i)$,
$L_i(e^\beta)=0=\gamma(-i)(e^\beta)$ for $i>0$, and
$\gamma_0e^\beta=(\beta,\gamma)$ As the matrix $(m_{\lambda,\mu})$ is
triangular its determinant is given by the product of the diagonal
entries
$m_{\lambda,\lambda}=P(\lambda)F(\Lambda)(\beta,\gamma)^{l(\lambda)}$.
Now we work out the index of $U^+(Witt_{>0})(e^{\beta^*})_n$ in
$W^*_n$ where $e^{\beta*} $ is the basis element of $W_0^*$ dual to
$e^\beta\in W_0$. This index is equal to
$${
(\hbox{Index of $U'_n$ in $W'_n$})
\over
(\hbox{Index of $U'_n$ in $U_n$})(\hbox{Index of $W'_n$ in $W_n$})
}
$$
The numerator of this expression is equal to the determinant
of the matrix $(m_{\lambda,\mu})$, and we calculated this earlier.
Substituting in the known values we find the index
of $U^+(Witt_{>0})(e^{\beta*})_n$ in $W^*_n$ is
$${
\prod_{|\lambda|=n} P(\lambda)F(\lambda)(\beta,\gamma)^{l(\lambda)}
\over
(\prod_{|\lambda|=n} F(\lambda))(\prod_{|\mu|=n} F(\mu))
}$$
Applying lemma 6.1 we see that this is equal to
$\prod_{|\lambda|=n}(\beta,\gamma)^{l(\lambda)}$. This is a unit
in $\Z[1/(\beta,\gamma)]$, so over the ring $\Z[1/(\beta,\gamma)]$
the map from $U^+(Witt_{>0})$ to $W^*$ is an isomorphism.
This proves lemma 6.3.
Fix a norm 0 vector $\gamma\in L$.
We recall that the transverse space is the subspace
of elements $v\in V$ such that $L_i(v)=0$ for $i>0$,
$L_0(v)=v$, and $\gamma(i)(v)=0 $ for $i<0$.
It is easy to check that the transverse space $T_\beta$ of degree
$\beta$ with $(\beta,\gamma)\ne 0$ is positive definite, and the no-ghost
theorem [G-T] works by showing that the natural map from
$T_\beta$ to the space of physical states (modulo null vectors)
of degree $\beta$ is an isomorphism.
\proclaim Lemma 6.4. The transverse space $T_\beta$ of degree
$\beta\in II_{1,1}$ has determinant dividing a power
of $(\beta,\gamma)$.
Proof. For any vector $v$ of $V$ there is a $U^+(Witt_{>0})$ equivariant
map from $W^*$ to $V$ taking $1$ to $v$ over $\Z[1/(\beta,\gamma)]$ by
lemma 6.3. Dualizing we see that this means there is a vector in
$W\otimes V$ of the form $v\otimes 1+$ (terms involving some
$\gamma(i)$) which is fixed by $U^+(Witt_{>0})$ and hence also fixed
by $Witt_{>0}$. This gives a map from $V$ to the transverse space
which is an isomorphism over $\Z[1/(\beta,\gamma)]$. This isomorphism
preserves the bilinear form because $\gamma$ has norm 0 so all the
terms involving $\gamma(i)$ have zero inner product with all other
terms. Hence the transverse space is self dual over
$\Z[1/(\gamma,\beta)]$. This proves lemma 6.4.
The next theorem is an extension of the no-ghost theorem from rational
vector spaces to modules over $\Z$. Recall that the usual no-ghost
theorem [G-T] says that if $\alpha\in II_{25,1}$ is nonzero then the
space of physical states (over \Q) of degree $\alpha$ is positive
definite and spanned by the transverse space, which has dimension
$p_{24}(1-\alpha^2/2)$. This describes the space of physical states as
a rational vector space, but it also has a natural lattice inside it and
we can ask about the structure of this lattice.
\proclaim Theorem 6.5.~(The no-ghost theorem over \Z). Suppose $\alpha$
is a nonzero vector of $II_{25,1}$ which is
$n$ times a primitive vector. Then the discriminant of the space of
physical states of degree $\alpha$ divides a power of $n$.
In particular the space of physical states is a self dual lattice
if $\beta$ is primitive.
Proof. For every prime $p$ coprime to $n$ we can find a norm zero
vector $\gamma$ with $(\gamma,\beta)$ coprime to $p$, because the norm
0 vectors of Leech type span $II_{25,1}$. By lemma 6.4 this implies
that the discriminant of the space of physical states of degree
$\beta$ is coprime to $p$. Hence this discriminant divides a power of
$n$. This proves theorem 6.5.
The proof of this theorem implies that all spaces of physical states
of primitive vectors of the same norm are isomorphic over all $p$-adic
fields, in other words in the same genus. It is certainly not always
true that they are isomorphic over the integers. For example, for a
norm 0 vector the space of physical states is isomorphic to the
corresponding Niemeier lattice, and not all Niemeier lattices are
isomorphic.
\proclaim 7.~An application to modular moonshine.
In this section $\m$ will stand for the monster Lie algebra
over $\Z[1/2]$ (see [B-R], [B98]) rather than the fake monster Lie algebra.
We write $\Z_p$ for the ring of $p$-adic numbers.
The paper [B98] showed that Ryba's modular moonshine conjectures
[R] for primes $p\ge 13$ were true provided the following assumption
was true:
\proclaim Assumption. If $m0$ (V),
Bull. Soc. Math. France 84 1956 207--239
reprinted in
J. Dieudonn\'e, ``Choix d'\oe uvres math\'ematiques. Tome II.''
Hermann, Paris, 1981, ISBN:\quad 2-7056-5923-4, p. 600--632.
\item{[D72]} E. J. Ditters, Curves and formal (co)groups,
Inv. Math. 17 (1972) 1-20.
\item{[F-G-Z]} I. Frenkel, H. Garland, G. J. Zuckerman,
Semi-infinite cohomology and string theory. Proc.
Nat. Acad. Sci. U.S.A. 83 (1986), no. 22, 8442--8446.
\item{[G-T]} 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.
\item{[H]} M. Hazewinkel, ``Formal groups and applications''.
Pure and Applied Mathematics, 78.
Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers],
New York-London, 1978. ISBN: 0-12-335150-2
\item{[K]} B. Kostant, Groups over $Z$.
1966 Algebraic Groups and Discontinuous Subgroups (Proc.
Symposium. Pure Math., Boulder, Colorado., 1965) pp. 90--98
Amer. Math. Soc., Providence, R.I.
\item{[L-Z]} B. Lian, G. J. Zuckerman, Moonshine cohomology.
Moonshine and vertex operator
algebra (Kyoto, 1994).
$\hbox{S\=urikaisekikenky\=usho}$
$\hbox{K\=oky\=uroku}$ No. 904 (1995), 87--115.
\item{[M]} I. G. Macdonald, Symmetric functions and Hall polynomials.
Second edition. Oxford Mathematical Monographs. Oxford Science Publications.
The Clarendon Press, Oxford
University Press, New York, 1995. x+475 pp. ISBN: 0-19-853489-2
\item{[R]}{A. J. E. Ryba, Modular Moonshine?,
In ``Moonshine, the Monster, and related topics'',
edited by Chongying Dong and Geoffrey Mason.
Contemporary Mathematics, 193. American Mathematical Society,
Providence, RI, 1996. 307-336. }
\item{[Sh]} P. B. Shay, An obstruction theory for smooth formal group
structure,
Preprint, Hunter college, CUNY.
\bye