Note that equation (25) mentioned in the quote above is gY(v, z)g−1 =Y(g·v, z), for g ∈M, where z can be interpreted as a nonzero complex variable, and Y(v, z) is a vertex operator. We are now prepared to unpack this quote. Triality is a relationship between three vector spaces. The authors had discussed triality in their previous paper (for more information on that paper see [5] in the References page). If we look at V\, we find it was built as a vertex operator algebra. This structure has a graded dimension of J(τ) and rank 24. To say that V\ acts irreducibly on itself ties into our discussion earlier of irreducible representations. Algebras are groups, and hence there is a way for them to act on themselves. We send an element of the algebra to an endomorphism of the algebra. Some of those actions are irreducible representations. The second part of the theorem says that the Monster acts on V\, and preserves the algebra structure. To say that the Monster acts faithfully on V\ means that the identity element of the Monster is the only element that leaves every element in V\ fixed [4]. That the Monster acts homogeneously means that the image of any homogeneous submodule is completely contained in another homogeneous submodule. Frenkel, Lepowsky, and Meurman also defined the Monster as “the group Mof linear automorphisms of the moonshine module V\ generated by C and σ” [6]. The group C was a group defined to act naturally onV\. It was “the centralizer of an involution in the Monster” [6]. The symbol σ represents an automorphism of V\. Finally, in 1992, Borcherds used Frenkel, Lepowsky, and Meurman’s vertex operator algebra and a result from string theory to construct an infinite dimensional Lie algebra, which he called the monster Lie algebra. By building a theory of mathematical objects called generalized Kac-Moody algebras and deriving some identities for those (the denominator and twisted denominator identities), he was able to show that these identities apply to the monster Lie algebra and the coefficients of the McKay-Thompson series. Because of that, the proof of Conway and Norton’s conjecture could be done using only a finite number of verifications. This was helped by the fact that Conway and Norton had already done some verifications. Hence, Borcherds proved the monstrous moonshine correspondence [11]. In this way, the connection between the J-function and the Monster was made clear. Vertex operator algebras now play a role in string theory, a type of theoretical physics [11]. For two fields that may seem so disjointed at first glance, namely, theoretical physics and group theory, it is fascinating that the Monster seems to be a connection between them. Page 48 Riley • An Overview of Monstrous Moonshine
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