Transformation Groups

, Volume 13, Issue 3–4, pp 671–713

On Rationality of W-algebras

Article

Abstract

We study the problem of classification of triples (\( \mathfrak{g} \); f; k), where g is a simple Lie algebra, f its nilpotent element and k\( \mathbb{C} \), for which the simple W-algebra Wk(\( \mathfrak{g} \); f) is rational.

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References

  1. [AM]
    D. Adamović, A. Milas, Vertex operator algebras associated to modular invariant representations for A (1) 1, q-alg/9509025.Google Scholar
  2. [A1]
    T. Arakawa, Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture, Duke Math. J. 130 (2005), 435–478.MATHCrossRefMathSciNetGoogle Scholar
  3. [A2]
    T. Arakawa, Representation theory of W-algebras, Invent. Math. 169 (2007), 219–320.MATHCrossRefMathSciNetGoogle Scholar
  4. [BPZ]
    A. A. Belavin, A. M. Polyakov, A. M. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241 (1984), 333–380.CrossRefMathSciNetGoogle Scholar
  5. [BK]
    J. Brundan, A. Kleshchev, Representations of shifted Yangians and finite W-algebras, math.RT/0508003.Google Scholar
  6. [CM]
    D. Collingwood, W. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostand Reinhold, New York, 1993.MATHGoogle Scholar
  7. [DK]
    A. De Sole, V. G. Kac, Finite vs affine W-algebras, Japan. J. Math. 1 (2006), 137–261.CrossRefGoogle Scholar
  8. [DLM]
    C. Dong, H. Li, G. Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), 571–600.MATHCrossRefMathSciNetGoogle Scholar
  9. [DV]
    K. de Vos, P. van Dreil, The Kazhdan-Lusztig conjecture for finite W-algebras, Lett. Math. Phys. 35 (1995), 333–344.MATHCrossRefMathSciNetGoogle Scholar
  10. [EK]
    A. G. Elashvili, V. G. Kac, Classification of good gradings of simple Lie algebras, in: Amer. Math. Soc. Transl. Ser. 2, Vol. 213, Amer. Math. Soc., Providence, RI, 2005, pp. 85–104.Google Scholar
  11. [FKW]
    E. Frenkel, V. G. Kac, M. Wakimoto, Characters and fusion rules for W-algebras via quantized Drinfeld-Sokolov reduction, Comm. Math. Phys. 147 (1992), 295–328.MATHCrossRefMathSciNetGoogle Scholar
  12. [FZ]
    I. Frenkel, Y. Zhu, Vertex algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123–168.MATHCrossRefMathSciNetGoogle Scholar
  13. [GG]
    W. Gan, V. Ginzburg, Quantization of Slodowy slices, Int. Math. Res. Not. 5 (2002), 243–255.CrossRefMathSciNetGoogle Scholar
  14. [GK1]
    M. Gorelik, V. G. Kac, On simplicity of vacuum modules, Adv. Math. 211 (2007), 621–677.MATHCrossRefMathSciNetGoogle Scholar
  15. [GK2]
    M. Gorelik, V. G. Kac, Characters of highest weight modules over affine Lie algebras are meromorphic functions, Int. Math. Res. Not. (2007), no. 20, Art. ID rnm079, 25 pp., arXiv:0704.2876Google Scholar
  16. [K1]
    V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. Russian transl.: В. Кац, Бесконечномерные алгебры Ли, Мир, М., 1993.Google Scholar
  17. [K2]
    V. G. Kac, Vertex Algebras for Beginners, 2nd ed., University Lecture Notes, Vol. 10, American Mathematical Society, Providence, RI, 1996. Russian transl.: В. Г. Кац, Вертексные алгебры для начинающих, МЦНМО, М., 2005.Google Scholar
  18. [KRW]
    V. G. Kac, S.-S. Roan, M. Wakimoto, Quantum reduction for affine superalgebras, Comm. Math. Phys. 241 (2003), 307–342.MATHMathSciNetGoogle Scholar
  19. [KW1]
    V. G. Kac, M. Wakimoto, Classification of modular invariant representations of affine algebras, in: Infinite-Dimensional Lie Algebras and Groups, Advanced series in Mathematical Physics, Vol. 7, World Scientific, River Edge, NJ, 1989, pp. 138–177.Google Scholar
  20. [KW2]
    V. G. Kac, M. Wakimoto, Branching functions for winding subalgebras and tensor products, Acta Appl. Math. 21 (1990), 3–39.MATHCrossRefMathSciNetGoogle Scholar
  21. [KW3]
    V. G. Kac, M. Wakimoto, Quantum reduction and representation theory of super-conformal algebras, Adv. Math. 185 (2004), 400–458. Corrigendum, Adv. Math. 193 (2005), 453–455.MATHCrossRefMathSciNetGoogle Scholar
  22. [KW4]
    V. G. Kac, M. Wakimoto, Quantum reduction in the twisted case, in: Progress in Mathematics, Vol. 237, Birkhäuser, Boston, 2005, pp. 85–126.Google Scholar
  23. [Kr]
    H. Kraft, Parametrisierung der Konjugationclassen in sn, Math. Ann. 234 (1978), 209–220.CrossRefMathSciNetGoogle Scholar
  24. [M]
    H. Matumoto, Whittaker modules associated with highest weight modules, Duke Math. J. 60 (1990), 59–113.MATHCrossRefMathSciNetGoogle Scholar
  25. [P]
    A. Premet, Special transverse slices and their enveloping algebras, Adv. Math. 170 (2002), 1–55.MATHCrossRefMathSciNetGoogle Scholar
  26. [Z]
    Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237–302.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA

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