Vertex operator algebras, extended $E_8$ diagram, and McKay's observation on the Monster simple group

Abstract

Abstract. We study McKay's observation on the Monster simple group, which relates the 2A-involutions of the Monster simple group to the extended E 8 diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices L of the E 8 lattice obtained by removing one node from the extended E 8 diagram at each time. We then construct a certain coset (or commutant) subalgebra U associated with L in the lattice VOA V √ 2E 8. There are two natural conformal vectors of central charge 1/2 in U such that their inner product is exactly the value predicted by Conway (1985). The Griess algebra of U coincides with the algebra described in his Table 3. There is a canonical automorphism of U of order |E 8 /L|. Such an automorphism can be extended to the Leech lattice VOA V Λ , and it is in fact a product of two Miyamoto involutions. In the sequel (2005) to this article, the properties of U will be discussed in detail. It is expected that if U is actually contained in the Moonshine VOA V , the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.

An explicit Majorana representation of the group 32:2 of 3C-pure type

“…Then by the same calculations as in [LYY1], it is straightforward to verify that K α 1 < V OA(e M , ρe M ), K α 2 < V OA(e N , ρe N ), K α 3 < V OA(eÑ , ρeÑ ),…”

“…This correspondence is very useful for studying some mysterious phenomena of the Monster group and many problems about 2A-involutions in the Monster group may also be translated into questions about Ising vectors. For example, the McKay's observation on the affine E 8 -diagram has been studied in [LYY1] using Miyamoto involutions and certain VOA generated by 2 Ising vectors were constructed. Nine different VOA were 2010 Mathematics Subject Classification.…”

“…1 constructed in [LYY1] and they are denoted by U 1A , U 2A , U 2B , U 3A , U 3C , U 4A , U 4B , U 5A , and U 6A because of their connection to the 6-transposition property of the Monster group (cf. [LYY1,Introduction]), where 1A, 2A, .…”

“…It is also not very difficult to determine the central charge. To construct the VOA explicitly, we combine the construction of the so-called dihedral subVOA from [LYY1] and the construction of EE 8 -pairs from [GL].…”

“…In Section 2, we recall some basic definitions and notations. We also review the structure of the so-called 3C-algebra from [LYY1,LYY2]. In Section 3, we study the Griess algebra generated 9 Ising vectors such that the subgroup generated by the corresponding Miyamoto involutions has the shape 3 2 : 2 and any two Ising vectors generate a 3C subVOA U 3C .…”