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Monstrous moonshine and monstrous Lie superalgebras. (English) Zbl 0799.17014
The author constructs two families of generalized Kac-Moody superalgebras. The first family is used, in conjunction with the no-ghost theorem of string theory [P. Goddard and C. Thorn, Phys. Lett. B 40, 235-238 (1972)] to prove J. H. Conway and S. P. Norton’s moonshine conjectures [Bull. Lond. Math. Soc. 11, 308-339 (1979; Zbl 0424.20010)] for the infinite-dimensional representation of the monster simple group constructed by I. B. Frenkel, J. Lepowsky and A. Meurman [Proc. Natl. Acad. Sci. USA 81, 3256-3260 (1984; Zbl 0543.20016) and ‘Vertex operator algebras and the monster’ (1988; Zbl 0674.17001)]. The second family is used to produce some new infinite-product identities by the same sort of process that produces the Macdonald identities from the denominator formulas of the affine Kac- Moody algebras. The paper closes with a list of eight open questions and conjectures about the Lie algebras and superalgebras the author constructs.
The first part of the paper constructs a \(\mathbb{Z}^ 2\)-graded Lie algebra acted on by the monster \(G\). This generalized Kac-Moody algebra, called the monster Lie algebra, provides the information the author uses to establish the main conjecture of Conway and Norton. He calculates the “twisted denominator formulas” of the monster Lie algebra, which provides enough information to determine the Thompson series \(T_ g(q)=\sum_{n\in\mathbb{Z}} \text{Tr}(g| V_ n)q^ n\). Here \(V=\oplus_{n\in\mathbb{Z}} V_ n\) is the infinite-dimensional graded representation of \(G\) of Frenkel-Meurman-Lepowsky, and \(g\in G\). The main result of the first part of the paper is that \(T_ g(q)\) is a Hauptmodul for a genus 0 subgroup of \(\text{SL}_ 2(\mathbb{R})\), so \(V\) satisfies a Conway-Norton conjecture. This also establishes the conjecture of McKay et al. that there is some graded module for \(G\) whose Thompson series are Hauptmoduls.
In the second part of the paper, the author constructs several Lie superalgebras similar to the monster Lie algebra. They comprise two classes: an algebra or superalgebra of rank 2 for many conjugacy classes \(g\) of \(G\), and a Lie superalgebra for many of the conjugacy classes of the group \(\operatorname{Aut}(\widetilde {\Lambda})= 2^{24}.2.Co_ 1\), where \(\widetilde{\Lambda}\) is the standard double cover of the Leech lattice, \(\operatorname{Aut} (\widetilde{\Lambda})\) is the group of automorphisms that preserve the inner product on \(Co_ 1= \operatorname{Aut}(\Lambda)/ \mathbb{Z}_ 2\), one of Conway’s sporadic groups, and the periods denote group extensions. Automorphisms of \(\Lambda\) of orders 1, 2, and 3 produce Lie algebras of ranks 26, 18, and 14, the first of which is called the fake monster Lie algebra.

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20D08 Simple groups: sporadic groups
17B70 Graded Lie (super)algebras
17B68 Virasoro and related algebras
17B81 Applications of Lie (super)algebras to physics, etc.
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
22E67 Loop groups and related constructions, group-theoretic treatment
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