Mathematische Zeitschrift

, Volume 270, Issue 1–2, pp 559–575 | Cite as

A remark on the C2-cofiniteness condition on vertex algebras

Article

Abstract

We show that a finitely strongly generated, non-negatively graded vertex algebra is C2-cofinite if and only if it is lisse in the sense of Beilinson et al. (preprint). This shows that the C2-cofiniteness is indeed a natural finiteness condition.

Keywords

Vertex algebras C2-cofiniteness 

Mathematics Subject Classification (2000)

17B69 17B65 17B68 

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References

  1. 1.
    Arakawa, T.: Associated varieties of modules over Kac-Moody algebras and C 2-cofiniteness of W-algebras. Preprint arXiv:1004.1554[math.QA]Google Scholar
  2. 2.
    Beilinson A., Drinfeld V.: Chiral Algebras Vol. 51 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (2004)Google Scholar
  3. 3.
    Beilinson, A., Feigin, B., Mazur, B.: Introduction to algebraic field theory on curves (Preprint)Google Scholar
  4. 4.
    Borcherds R.E.: algebras, Kac-Moody algebras, and the Monster. Proc. Nat. Acad. Sci. U.S.A. 83(10), 3068–3071 (1986)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Casselman W., Osborne M.S.: The restriction of admissible representations to n. Math. Ann. 233(3), 193–198 (1978)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Dong C., Li H., Mason G.: Modular-invariance of trace functions in orbifold theory and generalized Moonshine. Commun. Math. Phys. 214(1), 1–56 (2000)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Dong, C., Li, H., Mason, G.: Vertex Lie algebras, vertex Poisson algebras and vertex algebras. In: Recent developments in infinite-dimensional Lie algebras and conformal field theory (Charlottesville, VA, 2000), Vol. 297 of Contemp. Math., pp. 69–96. Amer. Math. Soc., Providence, RI (2002)Google Scholar
  8. 8.
    Dong, C., Mason, G.: Integrability of C 2-cofinite vertex operator algebras. Int. Math. Res. Not., pp. Art. ID 80468, 15 (2006)Google Scholar
  9. 9.
    De Sole A., Kac. V.G.: Freely generated vertex algebras and non-linear Lie conformal algebras. Commun. Math. Phys. 254(3), 659–694 (2005)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    De Sole A., Kac. V.G.: Finite vs. affine W-algebras. Jpn. J. Math. 1(1), 137–261 (2006)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Ein, L., Mustaţă, M.: Jet schemes and singularities. In: Algebraic geometry—Seattle 2005. Part 2, Vol. 80 of Proc. Sympos. Pure Math., pp. 505–546. Amer. Math. Soc., Providence, RI (2009)Google Scholar
  12. 12.
    Frenkel E., Ben-Zvi D.: Vertex Algebras and Algebraic Curves, Vol. 88 of Mathematical Surveys and Monographs. 2nd edn. American Mathematical Society, Providence (2004)Google Scholar
  13. 13.
    Feĭgin, B.L., Fuchs, D.B.: Verma modules over the Virasoro algebra. In: Topology (Leningrad, 1982), Vol. 1060 of Lecture Notes in Math., pp. 230–245. Springer, Berlin (1984)Google Scholar
  14. 14.
    Gorelik M., Kac V.: On simplicity of vacuum modules. Adv. Math. 211(2), 621–677 (2007)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Gaberdiel M.R., Neitzke A.: Rationality, quasirationality and finite W-algebras. Commun. Math. Phys. 238(1-2), 305–331 (2003)MATHMathSciNetGoogle Scholar
  16. 16.
    Huang Y.-Z.: Rigidity and modularity of vertex tensor categories. Commun. Contemp. Math. 10(suppl. 1), 871–911 (2008)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Huang Y.-Z.: Vertex operator algebras and the Verlinde conjecture. Commun. Contemp. Math. 10(1), 103–154 (2008)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Kac V.G.: Infinite-dimensional Lie algebras, and the Dedekind η-function. Funk. Anal. i Priložen. 8(1), 77–78 (1974)CrossRefGoogle Scholar
  19. 19.
    Kac V.G.: Infinite-dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  20. 20.
    Kac V.: Vertex Algebras for Beginners Vol. 10 of University Lecture Series, 2nd edn. American Mathematical Society, Providence (1998)Google Scholar
  21. 21.
    Kac V.G., Wakimoto M.: On rationality of W-algebras. Transform. Groups 13(3–4), 671–713 (2008)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Li H.: Some finiteness properties of regular vertex operator algebras. J. Algebra 212(2), 495–514 (1999)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Li H.: Vertex algebras and vertex Poisson algebras. Commun. Contemp. Math. 6(1), 61–110 (2004)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Li H.: Abelianizing vertex algebras. Commun. Math. Phys. 259(2), 391–411 (2005)CrossRefMATHGoogle Scholar
  25. 25.
    Miyamoto M.: Modular invariance of vertex operator algebras satisfying C 2-cofiniteness. Duke Math. J. 122(1), 51–91 (2004)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Nagatomo K., Tsuchiya A.: Conformal field theories associated to regular chiral vertex operator algebras. I. Theories over the projective line. Duke Math. J. 128(3), 393–471 (2005)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Primc M.: Vertex algebras generated by Lie algebras. J. Pure Appl. Algebra 135(3), 253–293 (1999)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Wang W.: Rationality of Virasoro vertex operator algebras. Int. Math. Res. Notices 7, 197–211 (1993)CrossRefGoogle Scholar
  29. 29.
    Zhu Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9(1), 237–302 (1996)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsNara Women’s UniversityNaraJapan
  2. 2.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

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