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On the Superdimension of an Irreducible Representation of a Basic Classical Lie Superalgebra

  • Vera SerganovaEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2027)

Abstract

In this paper we prove the Kac-Wakimoto conjecture that a simple module over a basic classical Lie superalgebra has non-zero superdimension if and only if it has maximal degree of atypicality. The proof is based on the results of [Duflo and Serganova, On associated variety for Lie superalgebras, math/0507198] and [Gruson and Serganova, Proceedings of the London Mathematical Society, doi:10.1112/plms/pdq014].We also prove the conjecture in [Duflo and Serganova, On associated variety for Lie superalgebras, math/0507198] about the associated variety of a simple module and the generalized Kac-Wakimoto conjecture in [Geer, Kujawa and Patureau-Mirand, Generalized trace and modified dimension functions on ribbon categories, arXiv:1001.0985v1] for the general linear Lie superalgebra.

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Notes

Acknowledgements

The author thanks Michel Duflo and Jonathan Kujawa for stimulating discussions and the referee for helpful suggestions. This work was partially supported by NSF grant 0901554.

References

  1. 1.
    J. Brundan, C. Stroppel, Highest weight categories arising from Khovanov’s diagram algebra IV: The general linear supergroup. Newton Institute (Preprint, 2009)Google Scholar
  2. 2.
    M. Duflo, V. Serganova, On associated variety for Lie superalgebras, math/0507198Google Scholar
  3. 3.
    N. Geer, B. Patureau-Mirand, An invariant supertrace for the category of representations of Lie supealgebras. Pacific J. Math. 238(2), 331–348 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    N. Geer, J. Kujawa, B. Patureau-Mirand, Generalized trace and modified dimension functions on ribbon categories. arXiv:1001.0985v1Google Scholar
  5. 5.
    M. Gorelik, The Kac construction of the centre of U(g) for Lie superalgebras. J. Nonlinear Math. Phys. 11(3), 325–349 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    C. Gruson, V. Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras. in Proceedings of the London Mathematical Society, doi:10.1112/plms/pdq014Google Scholar
  7. 7.
    V. Kac, Representations of classical Lie superalgebras. Lect. Notes Math. 676, 597–626 (1978)CrossRefGoogle Scholar
  8. 8.
    V. Kac, Laplace operators of infinite-dimensional Lie algebras and theta functions. Proc. Nat. Acad. Sci. U.S.A. 81(2), 645–647 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    V. Kac, M. Wakimoto, in Integrable Highest Weight Modules Over Affine Superalgebras and Number Theory. Lie Theory and Geometry. Progr. Math., vol. 123 (Birkhauser, MA, 1994), pp. 415–456Google Scholar
  10. 10.
    I. Musson, V. Serganova, Combinatorics of Character Formulas for the Lie Superalgebra gl(m,n), /pantherfile.uwm.edu/musson/www/ (Preprint)Google Scholar
  11. 11.
    A. Sergeev, The invariant polynomials on simple Lie superalgebras. Represent. Theor. 3, 250–280 (1999)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

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