On the Structure of \(\mathbb{N}\)-Graded Vertex Operator Algebras

  • Geoffrey MasonEmail author
  • Gaywalee Yamskulna
Part of the Developments in Mathematics book series (DEVM, volume 38)


We consider the algebraic structure of \(\mathbb{N}\)-graded vertex operator algebras with conformal grading \(V = \oplus _{n\geq 0}V _{n}\) and \(\mathrm{dim}V _{0} \geq 1\). We prove several results along the lines that the vertex operators Y (a, z) for a in a Levi factor of the Leibniz algebra V 1 generate an affine Kac–Moody subVOA. If V arises as a shift of a self-dual VOA of CFT-type, we show that V 0 has a “de Rham structure” with many of the properties of the de Rham cohomology of a complex connected manifold equipped with Poincaré duality.

Key words

Vertex operator algebra Lie algebra Leibniz algebra. 

Mathematics Subject Classification (2010):

17B65 17B69. 


  1. B.
    D. Barnes, On Levi’s Theorem for Leibniz algebras, Bull. Aust. Math. Soc. 86(2) (2012), 184–185.Google Scholar
  2. Br.
    P. Bressler, Vertex algebroids I, arXiv: math. AG/0202185.Google Scholar
  3. DL.
    C. Dong and J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Progress in Mathematics Vol. 112, Birkhäuser, Boston, 1993.Google Scholar
  4. DLM1.
    C. Dong, H. Li and G. Mason, Regularity of vertex operator algebras, Adv. Math. 132 (1997) No. 1, 148–166.MathSciNetCrossRefzbMATHGoogle Scholar
  5. DLM2.
    C. Dong, H. Li and G. Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998) No. 3, 571–600.Google Scholar
  6. DM1.
    C. Dong and G. Mason, Rational vertex operator algebras and the effective central charge, Int, Math. Res. Not. 56 (2004), 2989–3008.Google Scholar
  7. DM2.
    C. Dong and G. Mason, Local and semilocal vertex operator algebras, J. Algebra 280 (2004), 350–366.Google Scholar
  8. DM3.
    C. Dong and G. Mason, Shifted vertex operator algebras, Math. Proc. Cambridge Phil. Soc. 141 (2006), 67–80.Google Scholar
  9. DM4.
    C. Dong and G. Mason, Integrability of C 2-cofinite Vertex Operator Algebra, IMRN, Article ID 80468 (2006), 1–15.Google Scholar
  10. FHL.
    I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches to Vertex Operator Algebras and Modules, Mem. A.M.S. Vol. 104 No. 4, 1993.Google Scholar
  11. GN.
    M. Gaberdiel and A. Neitzke, Rationality, quasirationality and finite W-algebras, Comm. Math. Phys. 238 (2008), 305–331.Google Scholar
  12. GMS.
    V. Gorbounov, F. Malikov and V. Schechtman, Gerbes of chiral differential operators II, Vertex algebroids, Invent. Math. 155 (2004), 605–680.Google Scholar
  13. L.
    H. Li, Symmetric invariant bilinear forms on vertex operator algebras, J. Pure and Appl. Math. 96 (1994), no.1, 279–297.Google Scholar
  14. LL.
    J. Lepowsky and H.-S. Li, Introduction to Vertex Operator Algebras and Their Representations, Progress in Math. Vol. 227, Birkhäuser, Boston, 2003.Google Scholar
  15. LY.
    H. Li and G. Yamskulna, On certain vertex algebras and their modules associated with vertex algebroids, J. Alg 283 (2005), 367–398.Google Scholar
  16. M.
    G. Mason, Lattice subalgebras of vertex operator algebras, to appear in Proceedings of the Heidelberg Conference, Springer, arXiv: 1110.0544.Google Scholar
  17. MS1.
    F. Malikov and V. Schechtman, Chiral Poincaré duality, Math. Res. Lett. 6 (1999), 533–546, in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc.Transl. Ser. 2 194 (1999), 149–188.Google Scholar
  18. MS2.
    F. Malikov and V. Schechtman, Chiral de Rham complex II, in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc.Transl. Ser. 2 194 (1999), 149–188.Google Scholar
  19. MSV.
    F. Malikov, V. Schechtman and A. Vaintrob, Chiral de Rham complex, Comm. Math. Phys. 204 (1999), 439–473.Google Scholar
  20. MY.
    G. Mason and G. Yamskulna, Leibniz algebras and Lie algebras, SIGMA 9 (2013), no 63, 10 pages.Google Scholar
  21. Z.
    Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996) No. 1, 237–302.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSanta CruzUSA
  2. 2.Department of Mathematical SciencesIllinois State UniversityNormalUSA

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