This paper defines a remarkable Lie algebra of infinite dimension and rank, and conjectures that it may be related to the Fischer-Griess Monster group.
The idea was discussed in [3] that there might be an infinite-dimensional Lie algebra (or superalgebra) L that in some sense "explains" the Fischer-Griess "Monster" group M. The present note produces some candidates for L based on recent discoveries about the Leech lattice. These candidates are described in terms of a particular Lie algebra L, of infinite rank.
We first review some of our present knowledge about these matters. It was proved by character calculations in [3, p. 3 171 that the centralizer C of an involution of class 2A in M has a natural sequence of modules affording the head characters (restricted to C). In [ 121, Kac has explicitly constructed these as C-modules. Now that Atkin, Fong, and Smith [ 1, 91 have verified the relevant numerical conjectures of [3] for M, we know that these modules can be given the structure of M-modules, but we have no idea how to do this explicitly.
Some of the conjectures of [3] have analogues in which M is replaced by a compact simple Lie group, and in particular by the Lie group E,. Most of the resulting statements have now been established by Kac. However, it seems that this analogy with Lie groups may not be as close as one would wish, since two of the four conjugacy classes of elements of order 3 in Es were shown in [ 151 (see [ 161) to yield modular functions that are not Hauptmoduls for any modular group. This disproves the conjecture made on p. 267 of [ 111, and is particularly distressing since it was the Hauptmodul property that prompted the discovery of the conjectures in [3], and it is this property that gives those conjectures almost all their predictive power.