zbMATH — the first resource for mathematics

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.

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
Full Text: DOI EuDML
[1] Alexander, D., Cummins, C., McKay, J., Simons, C.: Completely replicable functions. (Preprint) · Zbl 0831.11032
[2] Atiyah, M.F.: K-theory. New York Amsterdam: Benjamin 1967 · Zbl 0159.53401
[3] Borcherds, R.E.: Vertex algebras, Kac-Moody algebras, and the monster. Proc. Natl. Acad. Sci. USA83, 3068-3071 (1986) · Zbl 0613.17012
[4] Borcherds, R.E.: Generalized Kac-Moody algebras. J. Algebra115, 501-512 (1988) · Zbl 0644.17010
[5] Borcherds, R.E.: Central extensions of generalized Kac-Moody algebras. J. Algebra140, 330-335 (1991) · Zbl 0776.17021
[6] Borcherds, R.E.: Lattices like the Leech lattice. J. Algebra130 (No. 1), 219-234 (1990) · Zbl 0701.11021
[7] Borcherds, R.E., Conway, J.H., Queen, L., Sloane, N.J.A.: A monster Lie algebra? Adv. Math.53, 75-79 (1984); this paper is reprinted as Chap. 30 of [12] · Zbl 0555.17004
[8] Borcherds, R.E.: The monster Lie algebra. Adv. Math.83, No. 1 (1990) · Zbl 0734.17010
[9] Borcherds, R.E.: Vertex algebras (to appear) · Zbl 0956.17019
[10] Cartan, H., Eilenberg, S.: Homological Algebra Princeton: Princeton University Press 1956
[11] Conway, J.H.: The automorphism group of the 26 dimensional even Lorentzian lattice. J. Algebra80, 159-163 (1983); this paper is reprinted as Chap. 27 of [12] · Zbl 0508.20023
[12] Conway, J.H., Sloane, N.J.A.: Sphere packings lattices and groups (Grundlehren de Math. Wiss., vol. 290) Berlin Heidelberg New York Springer 1988 · Zbl 0634.52002
[13] Conway, J.H., Norton, S.: Monstrous moonshine. Bull. Lond. Math. Soc.11, 308-339 (1979) · Zbl 0424.20010
[14] Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of finite groups. Oxford: Clarendon Press 1985 · Zbl 0568.20001
[15] Frenkel, I.B.: Representations of Kac-Moody algebras and dual resonance models. In: Flato, et al. (eds.) Applications of group theory in theoretical physics. (Lect. Appl. Math., vol. 21, pp. 325-353) Providence, RI: Am. Math. Soc. 1985
[16] Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex operator algebras and the monster. Boston, MA Academic Press 1988 · Zbl 0674.17001
[17] Frenkel, I.B., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess monster with the modular functionJ as character. Proc. Natl. Acad. Sci. USA81, 3256-3260 (1984) · Zbl 0543.20016
[18] Frenkel, I.B., Huang, Y-Z., Lepowsky, J.: On axiomatic formulations of vertex operator algebras and modules. (Preprint) · Zbl 0789.17022
[19] Frenkel, I.B., Garland, H., Zuckerman, G.: Semi-infinite cohomology and string theory. Proc. Natl. Acad. Sci. USA83, 8442-8446 (1986) · Zbl 0607.17007
[20] Garland, H., Lepowsky, J.: Lie algebra homology and the Macdonald-Kac formulas. Invent. Math.34, 37-76 (1976) · Zbl 0358.17015
[21] Goddard, P., Thorn, C.B., Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model, Phys. Lett. B40 (No. 2), 235-238 (1972)
[22] Gunning, R.C.: Lectures on modular forms. (Ann. Math. Stud) Princeton: Princeton University Press 1962 · Zbl 0178.42901
[23] Kac, V.G.: Infinite dimensional Lie algebras, third ed. Cambridge: Cambridge University Press 1990; (the first and second editions (Basel: Birkh?user 1983, and C.U.P. 1985) do not contain the material on generalized Kac-Moody algebras that we need.)
[24] Kac, V.G., Moody, R.V., Wakimoto, M.: OnE 10. (Preprint) · Zbl 0674.17007
[25] Koike, M.: On Replication Formula and Hecke Operators. Nagoya University (Preprint)
[26] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math.74, 329-387 (1961) · Zbl 0134.03501
[27] Mahler, K.: On a class of non-linear functional equations connected with modular functions. J. Aust. Math. Soc.22A, 65-118 (1976) · Zbl 0345.39002
[28] Norton, S.P.: More on moonshine, Computational group theory, pp. 185-195. London: Academic Press 1984
[29] Norton, S.P.: Generalized Moonshine. (Proc. Symp. Pure Math., vol. 47 pp. 208-209) Providence, RI: Am. Math. Soc. 1987
[30] Serre, J.P.: A course in arithmetic. (Grad. Texts Math., vol. 7) Berlin Heidelberg New York: Springer 1973 · Zbl 0256.12001
[31] Thompson, J.G.: A finiteness theorem for subgroups of PSL(2,R) which are commensurable with PSL(2,Z). (Proc. Symp. Pure Math., vol. 37, pp. 533-555) Providence, RI: Am. Math. Soc. 1979
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.