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Mathematische Zeitschrift

, Volume 249, Issue 2, pp 455–484 | Cite as

Rationality of the vertex operator algebra V L + for a positive definite even lattice L

  • Toshiyuki AbeEmail author
Article

Abstract.

The lattice vertex operator algebra V L associated to a positive definite even lattice L has an automorphism of order 2 lifted from −1-isometry of L. The fixed point set V L + of V L for the automorphism is naturally a vertex operator algebra. We prove that any ℤ≥0-graded weak V L +-module is completely reducible.

Keywords

Vertex Operator Operator Algebra Vertex Operator Algebra Lattice Vertex Lattice Vertex Operator Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abe, T.: Fusion rules for the free bosonic orbifold vertex operator algebra. J. Algebra 229, 333–374 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Abe, T.: The charge conjugation orbifold Vℤα+ is rational when 〈α,α〉/2 is prime. Internat. Math. Res. Notices 12, 647–665 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Abe, T., Buhl, G., Dong, C.: Rationality, Regularity, and C2 Co-finiteness. Trans. Amer. Math. Soc. 356, 3391–3402 (2004)CrossRefGoogle Scholar
  4. 4.
    Abe, T., Dong, C.: Classification of irreducible modules for the vertex operator algebra VL+: General case. J. Algebra 273, 657–685 (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    Borcherds, R.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986)Google Scholar
  6. 6.
    Buhl, G.: A spanning set for VOA modules. J. Algebra 254 (1), 125–151 (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Dong, C.: Vertex algebras associated with even lattices. J. Algebra 160, 245–265 (1993)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Dong, C.: Twisted modules for vertex algebras associated with even lattices. J. Algebra 165, 90–112 (1994)CrossRefGoogle Scholar
  9. 9.
    Dong, C., Griess, R.L.: Rank one lattice type vertex operator algebras and their automorphism groups. J. Algebra 208, 262–275 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Dong, C., Griess, R.L., Höhn, G.: Framed vertex operator algebras, codes and the Moonshine module. Comm. Math. Phys. 193 (2), 407–448 (1998)CrossRefzbMATHGoogle Scholar
  11. 11.
    Dong, C., Griess, R.L., Ryba, A.: Rank one lattice type vertex operator algebras and their automorphism groups II. E-series. J. Algebra 17, 701–710 (1999)Google Scholar
  12. 12.
    Dong, C., Lepowsky, J.: Generalized vertex algebras and relative vertex operators. Progress in Math., Vol.112, Birkhäuser, Boston, 1993Google Scholar
  13. 13.
    Dong, C., Lin, Z.: Induced modules for vertex operator algebras. Comm. Math. Phys. 179, 157–184 (1996)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Dong, C., Li, H.-S., Mason, G.: Twisted representations of vertex operator algebras. Math. Ann. 310, 571–600 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Dong, C., Li, H.-S., Mason, G.: Vertex operator algebras and associative algebras. J. Algebra 206, 67–96 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Dong, C., Mason, G.: On quantum Galois theory. Duke Math. J. 86 (2), 305–321 (1997)zbMATHGoogle Scholar
  17. 17.
    Dong, C., Nagatomo, K.: Classification of irreducible modules for the vertex operator algebra M(1)+. J. Algebra 216, 384–404 (1999)CrossRefzbMATHGoogle Scholar
  18. 18.
    Dong, C., Nagatomo, K.: Representations of Vertex operator algebra VL+ for rank one lattice L. Comm. Math. Phys. 202, 169–195 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Dong, C., Nagatomo, K.: Classification of irreducible modules for the vertex operator algebra M(1)+ II. Higher Rank. J. Algebra 240, 389–325 (2001)Google Scholar
  20. 20.
    Dong, C., Yamskulna, G.: Vertex operator algebras, generalized doubles and dual pairs. Math. Z. 241 (2), 397–423 (2002)CrossRefzbMATHGoogle Scholar
  21. 21.
    Frenkel, I., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Am. Math. Soc. 104, (1993)Google Scholar
  22. 22.
    Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Appl. Math., Vol. 134, Academic Press, Boston, 1988Google Scholar
  23. 23.
    Frenkel, I., Zhu, Y.-C.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ginsparg, P.: Curiosities at c=1. Nucl. Phys. B 295, 153–170 (1988)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Gaberdiel, M., Neitzke, A.: Rationality, quasirationality and finite W-algebras. Comm. Math. Phys. 238, 305–331 (2003)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Harris, G.: SU(2) current algebra orbifolds of the Gaussian model. Nucl. Phys. 300, 588–610 (1988)CrossRefGoogle Scholar
  27. 27.
    Huang, Y.-Z.: Differential equations and intertwining operators. math.QA/0206206Google Scholar
  28. 28.
    Huang, Y.-Z.: Riemann surfaces with boundaries and the theory of vertex operator algebras. Vertex operator algebras in mathematics and physics (Toronto, ON, 2000), Fields Inst. Commun. 39, Amer. Math. Soc., Providence, RI, 2003, pp. 109–125Google Scholar
  29. 29.
    Huang, Y.-Z.: Differential equations, duality and modular invariance. math.QA/0303049Google Scholar
  30. 30.
    Huang, Y.-Z.: Differential equations and conformal field theories. Nonlinear evolution equations and dynamical systems, 61–71 World Sci. (2003)Google Scholar
  31. 31.
    Hanaki, A., Miyamoto, M., Tambara, D.: Quantum Galois theory for finite groups. Duke Math. J. 97 (3), 541–544 (1999)zbMATHGoogle Scholar
  32. 32.
    Kiritsis, E.: Proof of the completeness of the classification of rational conformal field theories with c=1. Phys. Lett. B 217, 427–430 (1989)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Karel, M., Li, H.-S.: Certain generating subspaces for vertex operator algebras. J. Algebra 217, 393–421 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Li, H.-S.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure. Appl. Algebra 96, 279–297 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Li, H.-S.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure. Appl. Algebra 109, 143–195 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Miyamoto, M.: Modular invariance of vertex operator algebras satisfying C2-cofiniteness. Duke Math. J. 122 (1), 51–91 (2004)Google Scholar
  37. 37.
    Miyamoto, M.: A theory of tensor products for vertex operator algebra satisfying C2-cofiniteness. math.QA/0309350Google Scholar
  38. 38.
    Matsuo, A., Nagatomo, K.: Axioms for a Vertex Algebra and the Locality of Quantum Fields. MSJ Memoirs No.4, Mathematical Society of Japan, 1999Google Scholar
  39. 39.
    Moody, R., Pianzola, A.: Lie algebras with triangular decompositions. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley and Sons, Inc., New York, 1995Google Scholar
  40. 40.
    Miyamoto, M., Tanabe, K.: Uniform product of Ag,n(V) for an orbifold model V and G-twisted Zhu algebra. J. Algebra 274, 80–96 (2004)CrossRefzbMATHGoogle Scholar
  41. 41.
    Nagatomo, K., Tsuchiya, A.: Conformal field theories associated to regular chiral vertex operator algebras I: theories over the projective line. math.QA/0206223Google Scholar
  42. 42.
    Wang, W.: Rationality of Virasoro vertex operator algebras. Internat. Math. Res. Notices 7, 197–211 (1993)Google Scholar
  43. 43.
    Yamskulna, G.: C2-cofiniteness of the vertex operator algebra VL+ when L is a rank one lattice. math.QA/0202056. To appear Commun. Alg.Google Scholar
  44. 44.
    Zhu, Y.-C.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9, 237–302 (1996)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, Faculty of ScienceEhime UniversityMatsuyama, EhimeJapan

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