Skip to main content
SpringerLink
Log in

Lifts of automorphisms of vertex operator algebras in simple current extensions

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this article, we study isomorphisms between simple current extensions of a simple VOA. For example, we classify the isomorphism classes of simple current extensions of the VOAs \(V_{\sqrt2E_8}^+\) and \(V_{\Lambda_{16}}^{+}\), where Λ16 is the Barnes-Wall lattice of rank 16. Moreover, we consider the same simple current extension and describe the normalizer of the abelian automorphism group associated with this extension. In particular, we regard the moonshine module \(V^\natural\) as simple current extensions of five subVOAs \(V_L^+\) for 2-elementary totally even lattices L, and describe corresponding five normalizers of elementary abelian 2-group in the automorphism group of \(V^\natural\) in terms of \(V_L^+\). By using this description, we show that three of them form a Monster amalgam.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Abe T (2001). Fusion rules for the charge conjugation orbifold. J. Algebra 242: 624–655

    Article  MATH  MathSciNet  Google Scholar 

  2. Abe, T., Dong, C.: Classification of irreducible modules for the vertex operator algebra \(V_+^L\): general case. J. Algebra 273, 657–685 (2004)

    Google Scholar 

  3. Abe, T., Dong, C., Li, H.: Fusion rules for the vertex operator algebras M(1)+ and \(V_L^+\), Comm. Math. Phys. 253, 171–219 (2005)

    Google Scholar 

  4. Borcherds R.E (1986). Vertex algebras, Kac-Moody algebras and the Monster. Proc.Nat’l. Acad. Sci. USA 83: 3068–3071

    Article  MathSciNet  Google Scholar 

  5. Brouwer A.E., Cohen A.M. and Neumaier A (1989). Distance-Regular Graphs. Springer, Berlin Heidelberg Newyork

    MATH  Google Scholar 

  6. Conway J.H. and Sloane N.J.A. (1999). Sphere Packings, Lattices and Groups, 3rd edn. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  7. Conway J.H., Curtis R.T., Norton S.P., Parker R.A. and Wilson R.A. (1985). Atlas of Finite Groups. Oxford University Press, Oxford

    MATH  Google Scholar 

  8. Dong C. and Mason G (1997). On quantum Galois theory. Duke Math. J. 86: 305–321

    Article  MATH  MathSciNet  Google Scholar 

  9. Dong C. and Mason G (2004). Rational vertex operator algebras and the effective central charge.. Int. Math. Res. Not. 56: 2989–3008

    Article  MathSciNet  Google Scholar 

  10. Dong, C., Nagatomo, K.: Representations of vertex operator algebra \(V_L^+\) for rank one lattice L, Comm. Math. Phys. 202, 169–195 (1999)

    Google Scholar 

  11. Dong C., Griess R.L. and Höhn G (1998). Framed vertex operator algebras, codes and Moonshine module. Comm. Math. Phys. 193: 407–448

    Article  MATH  MathSciNet  Google Scholar 

  12. Frenkel, I., Huang, Y., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104 (1993)

  13. Frenkel I., Lepowsky J., Meurman A(1989) Vertex operator algebras and the Monster, Pure and Appl. Math. 134, (1989)

  14. Griess R.L. (1984). Friendly Giant. Invent. Math. 78: 491–499

    Article  MathSciNet  Google Scholar 

  15. Ivanov A.A. (1999). Geometry of Sporadic. Groups I. Encyclopedia of Mathematics and its , Vol. 76. Cambridge University Press, Cambridge

    Google Scholar 

  16. Matsuo A. and Nagatomo K. (1999). A note on free bosonic vertex algebra and its conformal vectors. J. Algebra 212: 395–418

    Article  MATH  MathSciNet  Google Scholar 

  17. Meierfrankenfeld, U.: The maximal 2-local subgroups of the Monster and Baby Monster, II, preprint

  18. Meierfrankenfeld, U., Shpectorov, S.: Maximal 2-local subgroups of the Monster and Baby Monster, preprint

  19. Miyamoto M (1996). Griess algebras and conformal vectors in vertex operator algebra. J. Algebra 179: 528–548

    Article  MathSciNet  Google Scholar 

  20. Shimakura H(2002) Decomposition of the Moonshine module with respect to subVOAs associated to codes over \(\mathbb{Z}_{2k}\). J. Algebra 251:308–322

    Google Scholar 

  21. Shimakura H(2004) The automorphism group of the vertex operator algebra \(V_L^+\) for an even lattice L without roots, J. Algebra 280:29–57

    Google Scholar 

  22. Shimakura H(2006) The automorphism groups of the vertex operator algebras \(V_L^+\): general case, Math. Z. 252:849 - 862

  23. Wilson R.A (1983). The maximal subgroups of Conway’s group Co 1. J. Algebra 85: 144–165

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan

    Hiroki Shimakura

Authors
  1. Hiroki Shimakura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroki Shimakura.

Additional information

The author was supported by the COE grant of Hokkaido University and JSPS Grants-in-Aid for Scientific Research No. 18740001.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shimakura, H. Lifts of automorphisms of vertex operator algebras in simple current extensions. Math. Z. 256, 491–508 (2007). https://doi.org/10.1007/s00209-006-0080-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-006-0080-5

Keywords

Search

Navigation