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Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 75–100 | Cite as

\(\varvec{{\mathbb {Z}}_3}\)-orbifold construction of the Moonshine vertex operator algebra and some maximal 3-local subgroups of the Monster

  • Hsian-Yang Chen
  • Ching Hung Lam
  • Hiroki Shimakura
Article

Abstract

In this article, we describe some maximal 3-local subgroups of the Monster simple group using vertex operator algebras (VOA). We first study the holomorphic vertex operator algebra obtained by applying the orbifold construction to the Leech lattice vertex operator algebra and a lift of a fixed-point free isometry of order 3 of the Leech lattice. We also consider some of its special subVOAs and study their stabilizer subgroups using the symmetries of the subVOAs. It turns out that these stabilizer subgroups are 3-local subgroups of its full automorphism group. As one of our main results, we show that its full automorphism group is isomorphic to the Monster simple group by using a 3-local characterization and that the holomorphic VOA is isomorphic to the Moonshine VOA. This approach allows us to obtain relatively explicit descriptions of two maximal 3-local subgroups of the shape \(3^{1+12}.2.{{\mathrm{Suz}}}{:}2\) and \(3^8.\Omega ^-(8,3).2\) in the Monster simple group.

Mathematics Subject Classification

Primary 17B69 Secondary 20B25 

Notations

\(\langle \cdot ,\cdot \rangle \)

The (normalized) invariant bilinear form on a VOA.

\(a_{n}\)

The n-th mode of an element \(a\in V\).

\({{\mathrm{Aut}}}V\)

The automorphism group of a VOA V.

\(C_G(H)\)

The centralizer of a subgroup H in a group G.

G.H

A group extension with normal subgroup G such that the quotient by G is H.

\(K_{12}\)

The Coxeter–Todd lattice of rank 12.

\({\mathbb {M}}\)

The Monster simple group.

\(N_G(H)\)

The normalizer of a subgroup H in a group G.

\(\Omega ^-_8(3)\)

The commutator subgroup of the orthogonal group of 8-dimensional quadratic space over \({\mathbb {F}}_3\) of minus type.

\(O_3(G)\)

The maximal normal 3-subgroup of a finite group G.

O(L)

The automorphism group of L preserving the inner product \(\langle \cdot |\cdot \rangle \).

O(Rq)

The orthogonal group of the quadratic space R with quadratic form q.

\(\tau \)

A fixed-point free automorphism of a lattice of order 3 or its lift to an automorphism of the lattice VOA of order 3.

\(\Lambda \)

The Leech lattice.

R(V)

The set of isomorphism classes of irreducible V-modules.

\(V_L\)

The lattice VOA associated to an even lattice L.

\(V_\Lambda ^\tau \)

The fixed-point subspace of \(\tau \) in \(V_\Lambda \), a subVOA of \(V_\Lambda \).

\(V^\natural \)

The Moonshine VOA.

\(V^\sharp \)

The holomorphic VOA of central charge 24 constructed as in (3.1).

\({{\mathrm{wt}}}(M)\)

The lowest L(0)-weight of an irreducible (g-twisted) module M.

\(\omega \)

The conformal vector of a VOA.

W

The \(\tau \)-fixed-point subspace \(V_{K_{12}}^\tau \) of the Lattice VOA \(V_{K_{12}}\) associated to \(K_{12}\)

\(\xi \)

\(\exp (2\pi \sqrt{-1}/3)\).

Z(G)

The center of a finite group G.

\({\mathbb {Z}}_n\)

The cyclic group of order n.

Notes

Acknowledgements

The authors wish to thank Xingjun Lin for helpful comments on unitary vertex algebras.

References

  1. 1.
    Borcherds, R.E.: Vertex algebras, Kac–Moody algebras, and the Monster. Proc. Nat’l. Acad. Sci. USA 8(3), 3068–3071 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carpi, S., Kawahigashi, Y., Longo, R., Weiner, M.: From vertex operator algebras to conformal nets and back. Mem. Am. Math. Soc. arXiv:1503.01260
  3. 3.
    Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Eynsham. Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G, Thackray (1985)Google Scholar
  4. 4.
    Chen, H., Lam, C.H.: Quantum dimensions and fusion rules of the VOA \( V^\tau _{{L_{{\varvec {{\cal{C}}}} \times {\varvec {{\cal{D}}}}}}}\). J. Algebra 459, 309–349 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Conway, J.H., Sloane, N.J.A.: The Coxeter–Todd lattice, the Mitchell group, and related sphere packings. Math. Proc. Camb. Philos. Soc. 9(3), 421–440 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dong, C., Griess, R.L., Lam, C.H.: Uniqueness results for the moonshine vertex operator algebra. Am. J. Math. 129, 583–609 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dong, C., Lepowsky, J.: The algebraic structure of relative twisted vertex operators. J. Pure Appl. Algebra 110, 259–295 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dong, C., Li, H., Mason, G.: Modular-invariance of trace functions in orbifold theory and generalized Moonshine. Comm. Math. Phys. 214, 1–56 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dong, C., Lin, X.J.: Unitary vertex operator algebras. J. Algebra 397, 252–277 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dong, C., Mason, G.: The Construction of the Moonshine Module as a \(Z_p\)-Orbifold, Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups (South Hadley, MA, 1992), Contemporay Mathematics, vol. 175. American Mathematical Society, Providence (1994)Google Scholar
  11. 11.
    Dong, C., Griess Jr., R.L.: Automorphism groups and derivation algebras of finitely generated vertex operator algebras. Mich. Math. J. 50, 227–239 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Frenkel, I.B., Huang, Y., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Am. Math. Soc. 104, viii+64 (1993)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Frenkel, I.B., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, vol. 134. Academic Press, Boston (1988)zbMATHGoogle Scholar
  14. 14.
    Huang, Y.Z., Kirillov Jr., A., Lepowsky, J.: Braided tensor categories and extensions of vertex operator algebras. Commun. Math. Phys. 337, 1143–1159 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kac, V.G., Raina, A.K., Rozhkovskaya, N.: Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, 2nd edn. Advanced Series in Mathematical Physics, 29. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2013)Google Scholar
  16. 16.
    Kitazume, M., Lam, C.H., Yamada, H.: 3-State Potts model, Moonshine vertex operator algebra, and \(3A\)-elements of the Monster group. Int. Math. Res. Not 23, 1269–1303 (2003)Google Scholar
  17. 17.
    Lam, C.H., Sakuma, S., Yamauchi, H.: Ising vectors and automorphism groups of commutant subalgebras related to root systems. Math. Z 255(3), 597–626 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lam, C.H., Yamauchi, H.: On 3-transposition groups generated by \(\sigma \)-involutions associated to \(c=4/5\) Virasoro vectors. J. Algebra 416, 84–121 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lepowsky, J.: Calculus of twisted vertex operators. Proc. Natl. Acad. Sci. USA 8(2), 8295–8299 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li, H.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure Appl. Algebra 9(6), 279–297 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Miyamoto, M.: Griess algebras and conformal vectors in vertex operator algebras. J. Algebra 179, 523–548 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Miyamoto, M.: A \({\mathbb{Z}}_3\)-Orbifold Theory of Lattice Vertex Operator Algebra and \({\mathbb{Z}}_3\)-Orbifold Constructions, Symmetries, Integrable Systems and representations, Springer Proceedings in Mathematics & Statistics, vol. 40. Springer, Heidelberg (2013)Google Scholar
  23. 23.
    Sakuma, S., Yamauchi, H.: Vertex operator algebra with two Miyamoto involutions generating \(S_3\). J. Algebra 267(1), 272–297 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Salarian, M.R., Stroth, G.: An identification of the Monster group. J. Algebra 323(4), 1186–1195 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Shimakura, H.: The automorphism group of the vertex operator algebra \(V_L^+\) for an even lattice \(L\) without roots. J. Algebra 280, 29–57 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Shimakura, H.: Lifts of automorphisms of vertex operator algebras in simple current extensions. Math. Z. 256(3), 491–508 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Shimakura, H.: An \(E_8\)-approach to the moonshine vertex operator algebra. J. Lond. Math. Soc. 83, 493–516 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shimakura, H.: The automorphism group of the \({{\mathbb{Z}}}_2\)-orbifold of the Barnes–Wall lattice vertex operator algebra of central charge 32. Math. Proc. Camb. Philos. Soc. 156, 343–361 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tanabe, K., Yamada, H.: Fixed point subalgebras of lattice vertex operator algebras by an automorphism of order three. J. Math. Soc. Jpn. 6(5), 1169–1242 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tits, J., Griess’, O.R.: Friendly giant. Invent. Math. 7(8), 491–499 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Hsian-Yang Chen
    • 1
  • Ching Hung Lam
    • 2
    • 3
  • Hiroki Shimakura
    • 4
  1. 1.National University of TainanTainanTaiwan
  2. 2.Institute of Mathematics, Academia SinicaTaipeiTaiwan
  3. 3.National Center for Theoretical SciencesHsinchuTaiwan
  4. 4.Graduate School of Information SciencesTohoku UniversitySendaiJapan

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