Communications in Mathematical Physics

, Volume 316, Issue 1, pp 269–277

\({\mathbb{Z}}\) -Graded Weak Modules and Regularity

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Abstract

It is proved that if any \({\mathbb{Z}}\) -graded weak module for vertex operator algebra V is completely reducible, then V is rational and C2-cofinite. That is, V is regular. This gives a natural characterization of regular vertex operator algebras.

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References

  1. A1.
    Abe T.: A \({\mathbb{Z}_{2}}\) -orbifold model of the symplectic fermionic vertex operator superalgebra. Math. Z. 255, 755–792 (2007)MathSciNetMATHCrossRefGoogle Scholar
  2. A2.
    Abe T.: Rationality of the vertex operator algebra \({V_{L^{+}}}\) for a positive definite even lattice L. Math. Z. 249, 455–484 (2005)MathSciNetMATHCrossRefGoogle Scholar
  3. ABD.
    Abe T., Buhl G., Dong C.: Rationality, Regularity, and C 2-cofiniteness. Trans. Amer. Math. Soc. 356, 3391–3402 (2004)MathSciNetMATHCrossRefGoogle Scholar
  4. B.
    Borcherds R.: Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986)MathSciNetADSCrossRefGoogle Scholar
  5. D.
    Dong C.: Vertex algebras associated with even lattices. J. Alg. 161, 245–265 (1993)MATHCrossRefGoogle Scholar
  6. DGH.
    Dong C., Griess R.L., Höhn G.: Framed vertex operator algebras, codes and the Moonshine module. Commun. Math. Phys. 193, 407–448 (1998)ADSMATHCrossRefGoogle Scholar
  7. DJL.
    Dong, C., Jiang, C., Lin, X.: Rationality of vertex operator algebra \({V_{L^{+}}}\) : higher rank. Proceedings of the London Mathematical Society, vol. 104, pp. 799–826 (2012)Google Scholar
  8. DL.
    Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators. Progress in Math. Vol.112, Boston, MA: Birkhäuser, 1993Google Scholar
  9. DLM1.
    Dong C., Li H., Mason G.: Regularity of rational vertex operator algebra. Adv. Math. 312, 148–166 (1997)MathSciNetCrossRefGoogle Scholar
  10. DLM2.
    Dong C., Li H., Mason G.: Twisted representations of vertex operator algebras. Math. Ann. 310, 571–600 (1998)MathSciNetMATHCrossRefGoogle Scholar
  11. DLM3.
    Dong C., Li H., Mason G.: Modular invariance of trace functions in orbifold theory and generalized moonshine. Commun. Math. Phys. 214, 1–56 (2000)MathSciNetADSMATHCrossRefGoogle Scholar
  12. DMZ.
    Dong C., Mason G., Zhu Y.: Discrete series of the Virasoro algebra and the moonshine module. Proc. Symp. Pure. Math., Amer. Math. Soc. 56(II), 295–316 (1994)MathSciNetGoogle Scholar
  13. FHL.
    Frenkel I., Huang Y., Lepowsky J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104, 494 (1993)MathSciNetGoogle Scholar
  14. FLM.
    Frenkel I., Lepowsky J., Meurman A.: Vertex Operator Algebras and the Monster. Pure and Appl. Math. Vol. 134. Academic Press, Boston, MA (1988)Google Scholar
  15. FZ.
    Frenkel I., Zhu Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992)MathSciNetMATHCrossRefGoogle Scholar
  16. H.
    Huang Y.: Vertex operator algebras and the Verlinde conjecture. Comm. Contemp. Math. 10, 103–154 (2008)MATHCrossRefGoogle Scholar
  17. KL.
    Karel M., Li H.: Certain generating subspaces for vertex operator algebras. J. Alg. 217, 393–421 (1999)MathSciNetMATHCrossRefGoogle Scholar
  18. L.
    Li H.: Some finiteness properties of regular vertex operator algebras. J. Alg. 212, 495–514 (1999)MATHCrossRefGoogle Scholar
  19. LL.
    Lepowsky, J., Li, H.: Introduction to vertex operator algebras and their representations. Progress in Math. 227. Boston, MA: Birkhäuser Boston, Inc., 2004Google Scholar
  20. M.
    Miyamoto M.: Representation theory of code vertex operator algebra. J. Alg. 201, 115–150 (1998)MathSciNetMATHCrossRefGoogle Scholar
  21. W.
    Wang W.: Rationality of Virasoro vertex operator algebras. Int. Math. Res. Not. 1993, 197–211 (1993)MATHCrossRefGoogle Scholar
  22. Z.
    Zhu Y.: Modular Invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSanta CruzUSA

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