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The Intermediate Vertex Subalgebras of the Lattice Vertex Operator Algebras

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Abstract

A notion of intermediate vertex subalgebras of lattice vertex operator algebras is introduced, as a generalization of the notion of principal subspaces. Bases and the graded dimensions of such subalgebras are given. As an application, it is shown that the characters of some modules of an intermediate vertex subalgebra between E 7 and E 8 lattice vertex operator algebras satisfy some modular differential equations. This result is an analogue of the result concerning the “hole” of the Deligne dimension formulas and the intermediate Lie algebra between the simple Lie algebras E 7 and E 8.

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References

  1. Andrews G.E.: The theory of partitions. Encyclopedia of Mathematics and Its Applications, vol. 2. Addison-Wesley, Reading (1976)

    Google Scholar 

  2. Andrews, G.E.: q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Regional Conf. Ser. in Math., no. 66 (1986)

  3. Ardonne E., Kedem R., Stone M.: Fermionic characters of arbitrary highest-weight integrable sl r+1-modules. Commun. Math. Phys. 264, 427–464 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Bannai, E., Koike, M., Munemasa, A., Sekiguti, J.: Klein’s icosahedral equation and modular forms, preprint (1999)

  5. Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)

    MATH  Google Scholar 

  6. Borcherds R.E.: Vertex algebras, Kac–Moody algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83(19), 3068–3071 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  7. Calinescu C., Lepowsky J., Milas A.: Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of type A, D, E. J. Algebra 323(1), 167–192 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cohen A.M., de Man R.: Computational evidence for Deligne’s conjecture regarding exceptional Lie groups. C. R. Acad. Sci. Paris Sér. I Math. 322, 427–432 (1996)

    MATH  Google Scholar 

  9. Capparelli S., Lepowsky J., Milas A.: The Rogers–Selberg recursions, the Gordon–Andrews identities and intertwining operators. Ramanujan J. 12, 379–397 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cook W.J., Li H., Misra K.C.: A recurrence relation for characters of highest weight integrable modules for affine Lie algebras. Commun. Contemp. Math. 9(2), 121–133 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Deligne P.: La série exceptionalle de groupes de Lie. C. R. Acad. Sci. Paris Sér. I Math. 322, 321–326 (1996)

    MATH  MathSciNet  Google Scholar 

  12. Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Mukhin, E.: Principal \({\widehat{\mathfrak{sl}_3}}\) subspaces and quantum Toda Hamiltonian, arXiv:0707.1635 (2007)

  13. Feigin, B.L., Stoyanovsky, A.V.: Quasi-particles models for the representations of Lie algebras and geometry of flag manifold. arXiv:hep-th/9308079 (1993)

  14. Georgiev G.N.: Combinatorial constructions of modules for infinite-dimensional Lie algebras, I. Principal subspace. J. Pure Appl. Algebra 112, 247–286 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gel’fand I.M., Zelevinsky A.V.: Models of representations of classical groups and their hidden symmetries. (Russian) Funkt. Anal. i Prilozhen. 18(3), 14–31 (1984)

    MATH  MathSciNet  Google Scholar 

  16. Ibukiyama T.: Modular forms of rational weights and modular varieties. Abh. Math. Sem. Univ. Hamburg 70, 315–339 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kac V.G.: Infinite dimensional Lie algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  18. Kaneko, M.: On modular forms of weight (6n + 1)/5 satisfying a certain differential equation. In: Number Theory: Tradition and Modernization, pp. 97–102 (2006)

  19. Kaneko M., Koike M.: On modular forms arising from a differential equations of hypergeometric type. Ramanujan J. 7, 145–164 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kaneko M., Nagatomo K., Sakai Y.: Modular forms and second order ordinary differential equations: applications to vertex operator algebras. Lett. Math. Phys. 103, 439–453 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Kaneko M., Zagier D.: Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials. AMS/IP Stud. Adv. Math. 7, 97–126 (1998)

    MathSciNet  Google Scholar 

  22. Kedem R., Klassen T.R., McCoy B.M., Melzer E.: Fermionic sum representations for conformal field theory characters. Phys. Lett. B 307(1-2), 68–76 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. Landsberg J.M., Manivel L.: Triality, exceptional Lie algebras, and Deligne dimension formulas. Adv. Math. 171, 59–85 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Landsberg J.M., Manivel L.: The sextonions and \({E_{7\frac{1}{2}}}\). Adv. Math. 201(1), 143–179 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Matsuo A.: Norton’s trace formulae for the Griess algebra of a vertex operator algebra with larger symmetry. Commun. Math. Phys. 224, 565–591 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  26. Milas A.: Ramanujan’s “Lost Notebook” and the Virasoro algebra. Commun. Math. Phys. 251(3), 567–588 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Milas A.: Modular invariance, Modular identities and supersingular j-invariants. Math. Res. Lett. 13, 729–746 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Milas, A.: On certain automorphic forms associated to rational vertex operator algebras, Moonshine: the first quarter century and beyond, pp. 330-357. London Math. Soc. Lecture Note Ser., vol. 372. Cambridge Univ. Press, Cambridge (2010)

  29. Milas A., Penn M.: Lattice vertex algebras and combinatorial bases: general case and \({\mathcal{W}}\) -algebras. New York J. Math. 18, 621–650 (2012)

    MATH  MathSciNet  Google Scholar 

  30. Mathur S., Mukhi S., Sen A.: On the classification of rational conformal field theories. Phys. lett. B 213(3), 303–308 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  31. Primc M.: Vertex operator construction of standard modules for \({A_n^{(1)}}\). Pacific J. Math. 162, 143–187 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  32. Shtepin V.V.: Intermediate Lie algebras and their finite-dimensional representations. Russian Acad. Sci. Izv. Math. 43(3), 559–579 (1994)

    MathSciNet  Google Scholar 

  33. Stoyanovskii A.V., Feigin B.L.: Functional models for representations of current algebras and semi-infinite Schubert cells. Funct. Anal. Appl. 28(1), 55–72 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  34. Tuite M.P.: Exceptional vertex operator algebras and the Virasoro algebra. Contemp. Math. 497, 213–225 (2009)

    Article  MathSciNet  Google Scholar 

  35. Wakimoto M.: Lectures on infinite dimensional Lie algebras. World Scientific Publishing, Singapore (2001)

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Kazuya Kawasetsu

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  1. Kazuya Kawasetsu
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Correspondence to Kazuya Kawasetsu.

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After finishing this work, we learned of a recent related work by Kaneko et al. [20], where the Kaneko–Zagier equations and the Mathur–Mukhi–Sen classification are studied in detail.

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Kawasetsu, K. The Intermediate Vertex Subalgebras of the Lattice Vertex Operator Algebras. Lett Math Phys 104, 157–178 (2014). https://doi.org/10.1007/s11005-013-0658-x

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  • DOI: https://doi.org/10.1007/s11005-013-0658-x

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