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Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

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Abstract

We determine the automorphism groups of the cyclic orbifold vertex operator algebras associated with coinvariant lattices for isometries of the Leech lattice in the conjugacy classes 4C, 6E, 6G, 8E and 10F. As a consequence, we have determined the automorphism groups of all the 10 vertex operator algebras in [Hö], which are useful to analyze holomorphic vertex operator algebras of central charge 24.

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References

  1. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Clarendon Press, Eynsham, 1985.

    Google Scholar 

  2. T. Abe, C. H. Lam and H. Yamada, Extensions of tensor products of ℤp-orbifold models of the lattice vertex operator algebra \({V_{\sqrt 2 {A_{p - 1}}}}\), Journal of Algebra 510 (2018), 24–51.

    Article  MathSciNet  Google Scholar 

  3. B. Bakalov and V. G. Kac, Twisted modules over lattice vertex algebras, in Lie Theory and Its Applications in Physics. V, World Scientific, River Edge, NJ, 2004, pp. 3–26.

    Chapter  Google Scholar 

  4. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, Journal of Symbolic Computation 24 (1997), 235–265.

    Article  MathSciNet  Google Scholar 

  5. R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proceedings of the National Academy of Sciences of the United States of America 83 (1986), 3068–3071.

    Article  MathSciNet  Google Scholar 

  6. S. Carnahan and M. Miyamoto, Regularity of fixed-point vertex operator subalgebras, https://arxiv.org/abs/1603.05645.

  7. H. Y. Chen, C. H. Lam and H. Shimakura, On ℤ3-orbifold construction of the Moonshine vertex operator algebra, Mathematische Zeitschrift 288 (2018), 75–100.

    Article  MathSciNet  Google Scholar 

  8. N. Chigira, C. H. Lam and M. Miyamoto, Orbifold construction and Lorentzian construction of Leech lattice vertex operator algebra, Journal of Algebra 593 (2022), 26–71.

    Article  MathSciNet  Google Scholar 

  9. C. Dong, X. Jiao and F. Xu, Quantum dimensions and quantum Galois theory, Transactions of the American Mathematical Society 365 (2013), 6441–6469.

    Article  MathSciNet  Google Scholar 

  10. C. Dong and J. Lepowsky, The algebraic structure of relative twisted vertex operators, Journal of Pure and Applied Algebra 110 (1996), 259–295.

    Article  MathSciNet  Google Scholar 

  11. C. Dong, H. Li and G. Mason, Modular-invariance of trace functions in orbifold theory and generalized Moonshine, Communications in Mathematical Physics 214 (2000), 1–56.

    Article  MathSciNet  Google Scholar 

  12. C. Dong and K. Nagatomo, Automorphism groups and twisted modules for lattice vertex operator algebras, in Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemporary Mathematics, Vol. 248, American Mathematical Society, Providence, RI, 1999, pp. 117–133.

    Chapter  Google Scholar 

  13. C. Dong, L. Ren and F. Xu, On orbifold theory, Advances in Mathematics 321 (2017), 1–30.

    Article  MathSciNet  Google Scholar 

  14. J. van Ekeren, C. H. Lam, S. Möller and H. Shimakura, Schellekens’ list and the very strange formula, Advances in Mathematics 380 (2021), Article no. 107567.

  15. J. van Ekeren, S. Möller and N. Scheithauer, Construction and classification of holomorphic vertex operator algebras, Journal für die Reine und Angewandte Mathematik 759 (2020), 61–99.

    Article  MathSciNet  Google Scholar 

  16. I. B. Frenkel, Y. Huang and J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Memoirs of the American Mathematical Society 104 (1993).

  17. I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, Boston, MA, 1988.

    Google Scholar 

  18. R. L. Griess, Jr., A vertex operator algebra related to E8with automorphism group O+(10, 2), in The Monster and Lie Algebras (Columbus, OH, 1996), Ohio State University Mathematical Research Institute Publications, Vol. 7, de Gruyter, Berlin, 1998, pp. 43–58.

    Chapter  Google Scholar 

  19. K. Harada and M. L. Lang, On some sublattices of the Leech lattice, Hokkaido Mathematical Journal 19 (1990), 435–446.

  20. G. Höhn, On the genus of the Moonshine module, https://arxiv.org/abs/1708.05990.

  21. G. Höhn and S. Möller, Systematic orbifold constructions of Schellekens’ vertex operator algebras from Niemeier lattices, Journal of the London Mathematical Society 106 (2022), 3162–3207.

    Article  MathSciNet  Google Scholar 

  22. Y. Z. Huang and J. Lepowsky, A theory of tensor product for module category of a vertex operator algebra, III, Journal of Pure and Applied Algebra 100 (1995), 141–171.

    Article  MathSciNet  Google Scholar 

  23. C. H. Lam, Cyclic orbifolds of lattice vertex operator algebras having group-like fusions, Letters in Mathematical Physics 110 (2020), 1081–1112.

    Article  MathSciNet  Google Scholar 

  24. C. H. Lam, Automorphism group of an orbifold vertex operator algebra associated with the Leech lattice, in Vertex Operator Algebras, Number Theory and Related Topics, Contemporary Mathematics, Vol. 753, American Mathematical Society, Providence, RI, 2020, pp. 127–138.

    Chapter  Google Scholar 

  25. C. H. Lam, Some observations about the automorphism groups of certain orbifold vertex operator algebras, RIMS Kôkyûroku Bessatsu, to appear.

  26. C. H. Lam and H. Shimakura, On orbifold constructions associated with the Leech lattice vertex operator algebra, Mathematical Proceedings of the Cambridge Philosophical Society 168 (2020), 261–285.

    Article  MathSciNet  Google Scholar 

  27. C. H. Lam and H. Shimakura, Extra automorphisms of cyclic orbifolds of lattice vertex operator algebras Journal of Pure and Applied Algebra 228 (2024), Article no. 107454.

  28. C. H. Lam and H. Yamada, Sigma involutions associated with parafermion vertex operator algebra \(K({\mathfrak{s}\mathfrak{l}_2},k)\), Liear and Multilinear Algebra 70 (2022), 6780–6819.

    Article  Google Scholar 

  29. C. H. Lam and H. Yamauchi, On 3-transposition groups generated by σ-involutions associated to c = 4/5 Virasoro vectors, Journal of Algebra 416 (2014), 84–121.

    Article  MathSciNet  Google Scholar 

  30. J. Lepowsky, Calculus of twisted vertex operators, Proceedings of the National Academy of Sciences of the United States of America 82 (1985), 8295–8299.

    Article  MathSciNet  Google Scholar 

  31. H. Li, Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, in Moonshine, the Monster, and Related Topics, Contemporary Mathematics, Vol. 193, American Mathematical Society, Providence, RI, 1996, pp. 203–236.

    Chapter  Google Scholar 

  32. M. Miyamoto, C2cofiniteness of cyclic-orbifold models, Communications in Mathematical Physics 335 (2015), 1279–1286.

    Article  MathSciNet  Google Scholar 

  33. S. Möller and N. R. Scheithauer, Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra, Annals of Mathematics 197 (2023), 221–288.

    Article  MathSciNet  Google Scholar 

  34. H. Shimakura, The automorphism group of the vertex operator algebra V +L for an even lattice L without roots, Journal of Algebra 280 (2004), 29–57.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Graduate School of Science and Technology, Hirosaki University, Hirosaki, 036-8561, Japan

    Koichi Betsumiya

  2. Institute of Mathematics, Academia Sinica, Taipei, 10617, Taiwan

    Ching Hung Lam

  3. Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan

    Hiroki Shimakura

Authors
  1. Koichi Betsumiya
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  2. Ching Hung Lam
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  3. Hiroki Shimakura
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Corresponding author

Correspondence to Ching Hung Lam.

Additional information

C. H. Lam was partially supported by a research grant AS-IA-107-M02 of Academia Sinica and MOST grant 107-2115-M-001-003-MY3 of Taiwan.

H. Shimakura was partially supported by JSPS KAKENHI Grant Numbers JP19KK0065 and JP20K03505.

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Betsumiya, K., Lam, C.H. & Shimakura, H. Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries. Isr. J. Math. 259, 621–650 (2024). https://doi.org/10.1007/s11856-023-2552-2

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  • DOI: https://doi.org/10.1007/s11856-023-2552-2

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