0} (1-e^{i\rho})^{24/(p+1)}(1-e^{pi\rho})^{24/(p+1)}) \cr}$$ where $\sum_{i>0} p_g(1+i)q^i = 1/\eta_g(q)$. For $p$= 2, 3, 5, 7, and 11 these Lie algebras seem to correspond to the baby monster, the Fischer group $Fi_{24}$, the Harada Norton group, the Held group and the Mathieu group $M_{12}$ in the same way that the fake monster Lie algebra corresponds to the monster simple group. Example 3. The fake Conway Lie superalgebra of rank 10. This is the algebra described at the end of section 2, and seems to correspond to Conway's simple group $Co_1$. We let $g\in \Aut(\hat\Lambda)$ be an element of order 2 which is the lift of an element of order 2 of $\Aut(\Lambda)$ which fixes a lattice $\Lambda^g$ of $\Lambda$ of dimension 8. This lattice $\Lambda^g$ is isomorphic to the $E_8$ lattice with all norms doubled, so if we halve all the norms of the lattice $\Lambda^g\oplus II_{1,1}$ we get the nonintegral lattice of determinant 1/4 which is the dual of the sublattice of even vectors of $I_{9,1}$. A calculation similar to that in example 1 but using lemma 11.2 shows that the denominator formula of this Lie superalgebra is $$e^{\rho}\prod_{r\in \Pi^+}(1-e^r)^{\mult(r)} = \sum_{w\in W} \det(w)w(e^{\rho}\prod_{n>0}(1-e^{n\rho})^{(-1)^n8})$$ where the multiplicity of the root $r=(v,m,n)\in L$ is equal to $$\mult(r) = (-1)^{(m-1)(n-1)}p_g((1-r^2)/2)= (-1)^{m+n}|p_g((1-r^2)/2)|,$$ and $p_g(n)$ is defined by $$\sum p_g(n)q^n = q^{-\half}\prod_{n>0}(1-q^{n/2})^{-(-1)^n8}.$$ \section{Open questions} We list a few conjectures and open questions about the Lie algebras and superalgebras we have constructed. (1) Prove the assumptions about the modular forms used in example 2 in section 14. In principle this should be easy (at least when they are true) because they just involve checking a finite number of identities between modular forms. (2) Investigate the Lie algebras and superalgebras coming from other elements of the monster or $\Aut(\Lambda)$ and write down their denominator formulas explicitly in some nice form. (3) Find a natural construction for these Lie algebras and superalgebras (i.e. other than by generators and relations). We used natural constructions for the monster Lie algebra and the fake monster Lie algebra from vertex algebras, but I do not know of any similar constructions for most of the other Lie algebras. The easiest case is the superalgebra of rank 10 which can be constructed from superstrings on a 10-dimensional torus. (The even part of the rank 10 superalgebra was constructed in \citex{\bora, \borb}; the odd part is more difficult to construct.) A natural construction should give actions of various finite groups on these Lie algebras; for example the double cover of the baby monster should act on the baby monster Lie algebra. (4) Describe the Lie bracket from $V_{ab}\otimes V_{cd}$ to $V_{(a+c)(b+d)}$ of the monster Lie algebra explicitly in terms of the vertex algebra operations on $V$. (5) Is the baby monster Lie algebra a subalgebra of the monster Lie algebra in a way that preserves the action of the double cover of the baby monster? Similarly for the other monstrous Lie algebras. (6) Are there any generalized Kac-Moody algebras, other than the finite dimensional, affine, monstrous or fake monstrous ones, whose simple roots and root multiplicities can both be described explicitly? The monstrous and fake monstrous algebras are both finite families, each with a few hundred members, corresponding roughly to the conjugacy classes in the monster and in $\Aut(\hat\Lambda)$. One example of a denominator formula for a Lie superalgebra of rank 3 is the identity $$\sum_{i+j+k=0}(-p)^{jk}(-q)^{ik}(-r)^{ij} =\prod_{i+j+k>0}((1-p^iq^jr^k)/(1+p^iq^jr^k))^{c(ij+jk+ki)}$$ where $c(i)$ is defined by $\sum_nc(n)q^n=\prod_{n>0}(1+q^n)/(1-q^n) = 1+2q+4q^2+8q^3+14q^4+\ldots$. (7) Find all completely replicable functions. (The ones with integral coefficients were found by computer in \citex{\alea}.) Are nontrivial completely replicable functions always modular functions of genus 0? A proof of this conjecture of Norton's \citex{\nora} which was not a case by case verification would be much neater than the argument in section 9. (8) Is it possible to say anything interesting from Lie algebras constructed from the vertex algebras of lattices (other than the Leech lattice) as in section 6? The two obvious candidates are the lattices $E_8$ and $E_8\oplus E_8$, so that the corresponding Lie algebras have root lattices $II_{9,1}$ and $II_{17,1}$. 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