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Zbl 0799.17014
Borcherds, Richard E.
Monstrous moonshine and monstrous Lie superalgebras. (English)
[J] Invent. Math. 109, No.2, 405-444 (1992). ISSN 0020-9910; ISSN 1432-1297

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 [{\it P. Goddard} and {\it C. Thorn}, Phys. Lett. B 40, 235-238 (1972)] to prove {\it J. H. Conway} and {\it 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 {\it I. B. Frenkel}, {\it J. Lepowsky} and {\it 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.\par The first part of the paper constructs a $\bbfZ\sp 2$-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 $T\sb g(q)=\sum\sb{n\in\bbfZ} \text{Tr}(g\vert V\sb n)q\sp n$. Here $V=\oplus\sb{n\in\bbfZ} V\sb n$ 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 $T\sb g(q)$ is a Hauptmodul for a genus 0 subgroup of $\text{SL}\sb 2(\bbfR)$, 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.\par 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 $\Aut(\widetilde {\Lambda})= 2\sp{24}.2.Co\sb 1$, where $\widetilde{\Lambda}$ is the standard double cover of the Leech lattice, $\Aut (\widetilde{\Lambda})$ is the group of automorphisms that preserve the inner product on $Co\sb 1= \Aut(\Lambda)/ \bbfZ\sb 2$, 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.
[J.F.Hurley (Storrs)]

MSC 2000:: *17B67 Kac-Moody algebras
20D08 Simple groups: sporadic finite groups
17B70 Graded Lie algebras
17B68 Virasoro and related algebras
17B81 Applications of Lie algebras to physics
81R10 Repres. of infinite-dim. groups and algebras from quantum theory
22E67 Loop groups and related constructions

Keywords: vertex algebras; homology groups; generalized Kac-Moody superalgebras; no-ghost theorem; string theory; moonshine conjectures; monster simple group; monster Lie algebra; denominator formulas; Thompson series; Hauptmoduls

Citations: Zbl 0424.20010; Zbl 0543.20016; Zbl 0674.17001

Cited in: Zbl 1163.11320 Zbl 1146.11026 Zbl 1100.81036 Zbl 1100.11016 Zbl 1144.58014 Zbl 1085.17009 Zbl 1038.17022 Zbl 1040.17025 Zbl 0990.17020 Zbl 0938.20012 Zbl 0907.11014 Zbl 0985.11017 Zbl 0929.17028 Zbl 0977.17030 Zbl 0903.20007 Zbl 0886.14015 Zbl 0870.16024 Zbl 0828.17029 Zbl 0839.11018

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