Introduction to the Monster Lie Algebra

for each element g of the monster, so that our problem is to work out what these Thompson series are. For example, if 1 is the identity element of the monster then Tr(1|Vn) = dim(Vn) = c(n), so that the Thompson series T1(q) = j(τ) − 744 is the elliptic modular function. McKay, Thompson, Conway and Norton conjectured [Con] that the Thompson series Tg(q) are always Hauptmoduls for certain modular groups of genus 0 (I will explain what this means in a moment.) In this paper we describe the proof of this in [Bor] using an infinite dimensional Lie algebra acted on by the monster called the monster Lie algebra. These Hauptmoduls are known explicitly, so this gives a complete description of V as a representation of the monster. We now recall the definition of a Hauptmodul. The group SL2(Z) acts on the upper half plane H = {τ ∈ C|Im(τ) > 0} by ( a b c d ) (τ) = aτ+b cτ+d . The elliptic modular function j(τ) is more or less the simplest function defined on H that is invariant under SL2(Z) in much the same way that the function e is the simplest function invariant under τ 7→ τ + 1. The element ( 1 1 0 1 ) of SL2(Z) takes τ to τ + 1, so in particular j(τ) is periodic and can be written as a Laurent series in q = e . The exact expression for j is