A Monster Lie Algebra ?

We define a remarkable Lie algebra of infinite dimension, and conjecture that it may be related to the Fischer-Griess Monster group. The idea was mooted in [C-N] that there might be an infinite-dimensional Lie algebra (or superalgebra) L that in some sense “explains” the Fischer-Griess ‘Monster” group M . In this chapter we produce some candidates for L based on properties of the Leech lattice described in [C-S]. 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 [C-N, p. 317] the centralizer C of an involution of class 2A in the Monster group has a natural sequence of modules affording the head characters (restricted to C). In [K], V. Kac has explicitly constructed these as C-modules. Now that Atkin, Fong and Smith [F], [S] have verified the relevant numerical conjectures of [C-N] for M , we know that these modules can be given the structure of M -modules. More recently, Frenkel, Lepowsky, and Meurman [F-L-M] have given a simple construction for the monster along these lines, but this sheds little light on the conjectures. Some of the conjectures of [C-N] have analogs in which M is replaced by a compact simple Lie group, and in particular by the Lie group E8. Most of the resulting statements have now been established by Kac and others. 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 E8 were shown in [Q] to yield examples of modular functions neither of which are the Hauptmodul for any modular group. This disproves the conjecture made on p. 267 of [K], and is particularly distressing since it was the Hauptmodul property that prompted the discovery of the conjectures in [C-N], and it is this property that gives those conjectures almost all their predictive power. The properties of the Leech lattice that we shall use stem mostly from the facts about “deep holes” in that lattice reported in [C-S Chapter 23]. Let w = (0, 1, 2, 3, . . . , 24 | 70). The main result of [C-S Chapter 26] is that the subset of vectors r in II25,1 for which r · r = 2, r · w = −1 (the “Leech roots’) is isometric to the Leech lattice, under the metric defined by d(r, s) = norm(r − s). The main result of [C-S Chapter 27] is that Aut(II25,1) is obtained by extending the Coxeter subgroup generated by the reflections in these Leech roots by its group of graph automorphisms together with the central inversion −1. It is remarkable that the walls of the fundamental region for this Coxeter group (which correspond one-for-one with the Leech roots) are transitively permuted by the graph automorphisms, which form an infinite group abstractly isomorphic to the group of all automorphisms of the Leech lattice, including translations. Vinberg [V] shows that for the earlier analogs II9,1 and II17,1 of II25,1 the fundamental regions for the reflection subgroups have respectively 10 and 19 walls, and the graph automorphism groups have orders 1 and 2. For the later analogs II33,1, . . . , there is no “Weyl vector” like w, so it appears that II25,1 is very much a unique object.