Monstrous moonshine and monstrous Lie superalgebras 
Richard Borcherds
Invent. Math. 109, 405-444 (1992). 

We prove Conway and Norton's moonshine conjectures for the infinite 
dimensional representation of the monster simple group constructed by 
Frenkel, Lepowsky and Meurman. To do this we use the no-ghost theorem 
from string theory to construct a family of generalized Kac-Moody 
superalgebras of rank 2, which are closely related to the monster and 
several of the other sporadic simple groups. The denominator formulas 
of these superalgebras imply relations between the Thompson functions 
of elements of the monster (i.e. the traces of elements of the monster 
on Frenkel, Lepowsky, and Meurman's representation), which are the 
replication formulas conjectured by Conway and Norton. These replication 
formulas are strong enough to verify that the Thompson functions have 
most of the ``moonshine'' properties conjectured by Conway and Norton, 
and in particular they are modular functions of genus 0. We also construct 
a second family of Kac-Moody superalgebras related to elements of Conway's 
sporadic simple group Co_1. These superalgebras have even rank between 2 
and 26; for example two of the Lie algebras we get have ranks 26 and 18, 
and one of the superalgebras has rank 10. The denominator formulas of 
these algebras give some new infinite product identities, in the same 
way that the denominator formulas of the affine Kac-Moody algebras give 
the Macdonald identities.