Skip to main content
SpringerLink
Log in

Characters of (relatively) integrable modules over affine Lie superalgebras

  • Original Article
  • Published:
Japanese Journal of Mathematics Aims and scope

Abstract

In the paper we consider the problem of computation of characters of relatively integrable irreducible highest weight modules L over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras \({\mathfrak{g}}\). The problem consists of two parts. First, it is the reduction of the problem to the \({\overline{\mathfrak{g}}}\)-module F(L), where \({\overline{\mathfrak{g}}}\) is the associated to L integral Lie superalgebra and F(L) is an integrable irreducible highest weight \({\overline{\mathfrak{g}}}\)-module. Second, it is the computation of characters of integrable highest weight modules. There is a general conjecture concerning the first part, which we check in many cases. As for the second part, we prove in many cases the KW-character formula, provided that the KW-condition holds, including almost all finite-dimensional \({\mathfrak{g}}\)-modules when \({\mathfrak{g}}\) is basic, and all maximally atypical non-critical integrable \({\mathfrak{g}}\)-modules when \({\mathfrak{g}}\) is affine with non-zero dual Coxeter number.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bernšteĭn I.N., Leĭtes D.A.: A formula for the characters of the irreducible finite-dimensional representations of Lie superalgebras of series Gl and sl. C. R. Acad. Bulgare Sci. 33, 1049–1051 (1980)

    MathSciNet  Google Scholar 

  2. Brundan J.: Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra \({\mathfrak{gl}(m|n)}\). J. Amer. Math. Soc. 16, 185–231 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. S.-J. Cheng and J.-H. Kwon, Kac–Wakimoto character formula for ortho-symplectic Lie superalgebras, preprint, arXiv:1406.6739.

  4. S.-J. Cheng, V. Mazorchuk and W. Wang, Equivalence of blocks for the general linear Lie superalgebra, preprint, arXiv:1301.1204.

  5. M. Chmutov, C. Hoyt and S. Reif, Kac–Wakimoto character formula for the general linear Lie superalgebra, preprint, arXiv:1310.3798.

  6. P. Fiebig, The combinatorics of category \({\mathcal{O}}\) over symmetrizable Kac–Moody algebras, Transform. Groups, 11 (2006), 29–49.

  7. Gorelik M.: Weyl denominator identity for affine Lie superalgebras with non-zero dual Coxeter number. J. Algebra 337, 50–62 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Gorelik, Weyl denominator identity for finite-dimensional Lie superalgebras, In: Highlights in Lie Algebraic Methods, Progr. Math., 295, Birkhaüser/Springer, 2012, pp. 167–188.

  9. Gorelik M., Kac V.G.: On simplicity of vacuum modules. Adv. Math. 211, 621–677 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Gorelik, V.G. Kac, P. Möseneder Frajria and P. Papi, Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs, Jpn. J. Math., 7 (2012), 41–134.

  11. Gorelik M., Reif S.: A denominator identity for affine Lie superalgebras with zero dual Coxeter number. Algebra Number Theory 6, 1043–1059 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Iohara K., Koga Y.: Enright functors for Kac–Moody superalgebras. Abh. Math. Semin. Univ. Hambg. 82, 205–226 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kac V.G.: Lie superalgebras. Advances in Math. 26, 8–96 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  14. V.G. Kac, Representations of classical Lie superalgebras, In: Differential Geometrical Methods in Mathematical Physics. II, Lecture Notes in Math., 676, Springer-Verlag, 1978, pp. 597–626.

  15. V.G. Kac, Infinite-Dimensional Lie Algebras. Third ed., Cambridge Univ. Press, 1990.

  16. Kac V.G., Kazhdan D.A.: Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. in Math. 34, 97–108 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. V.G. Kac, S.-S. Roan and M. Wakimoto, Quantum reduction for affine superalgebras, Comm. Math. Phys., 241(2003), 307–342.

  18. Kac V.G., Wakimoto M.: Modular invariant representations of infinite-dimensional Lie algebras and superalgebras. Proc. Nat. Acad. Sci. U.S.A. 85, 4956–4960 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  19. V.G. Kac and M. Wakimoto, Classification of modular invariant representation of affine algebras, In: Infinite-Dimensional Lie Algebras and Groups, Adv. Ser. Math. Phys., 7, World Sci. Publ., 1989, pp. 138–177.

  20. V.G. Kac and M. Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, In: Lie Theory and Geometry, Progr. Math., 123, Birkhäuser Boston, Boston, MA, 1994, pp. 415–456.

  21. Kac V.G., Wakimoto M.: Integrable highest weight modules over affine superalgebras and Appell’s function. Comm. Math. Phys. 215, 631–682 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kac V.G., Wakimoto M.: Representations of affine superalgebras and mock theta functions. Transform. Groups. 19, 383–455 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. V.G. Kac and M. Wakimoto, Representations of affine superalgebras and mock theta functions. II, preprint, arXiv:1402:0727, to appear in Adv. Math.

  24. V.G. Kac and M. Wakimoto, Representations of affine superalgebras and mock theta functions. III, preprint, arXiv:1505.01047.

  25. M. Kashiwara and T. Tanisaki, Kazhdan–Lusztig conjecture for symmetrizable Kac–Moody algebras. III. Positive rational case, In: Mikio Sato: A Great Japanese Mathematician of the Twentieth Century, Asian J. Math., 2, International Press, 1998, pp. 779–832.

  26. M. Kashiwara and T. Tanisaki, Characters of the irreducible modules with non-critical highest weights over affine Lie algebras, In: Representations and Quantizations, Shanghai, 1998, China High. Educ. Press, Beijing, 2000, pp. 275–296.

  27. S. Reif, Denominator Identity for twisted affine Lie superalgebras, Int. Math. Res. Not. IMRN, 2014, 4146–4178.

  28. V. Serganova, Kazhdan–Lusztig polynomials and character formula for the Lie superalgebra \({\mathfrak{gl}(m|n)}\), Selecta Math. (N.S.), 2 (1996), 607–651.

  29. Serganova V.: On generalizations of root systems. Comm. Algebra 24, 4281–4299 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. V. Serganova, Characters of irreducible representations of simple Lie superalgebras, In: Proceedings of the International Congress of Mathematicians. Vol. II, Berlin, 1998, Doc. Math., 1998, pp. 583–593.

  31. V. Serganova, Kac–Moody superalgebras and integrability, In: Developments and Trends in Infinite-Dimensional Lie Theory, Progr. Math., 288, Birkhäuser Boston, Boston, MA, 2011, pp. 169–218.

  32. A. Shaviv, On the correspondence of affine generalized root systems and symmetrizable affine Kac–Moody superalgebras, M. Sc. thesis, 2014.

  33. Su Y., Zhang R.B.: Character and dimension formulae for general linear superalgebra. Adv. Math. 211, 1–33 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Van der Jeugt J.: Irreducible representations of the exceptional Lie superalgebras \({D(2,1;\alpha)}\). J. Math. Phys. 26, 913–924 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  35. Van der Jeugt J.: Character formulae for the Lie superalgebra C(n). Comm. Algebra 19, 199–222 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  36. Van der Jeugt J., Hughes J.W.B., King R.C., Thierry-Mieg J.: Character formulas for irreducible modules of the Lie superalgebras \({\mathfrak{sl}(m/n)}\). J. Math. Phys., 31, 2278–2304 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  37. S. Zwegers, Mock theta functions, preprint, arXiv:0807.4834.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Maria Gorelik

  2. Department of Mathematics, 2-178, Massachusetts Institute of Technology, Cambridge, MA, 02139-4307, USA

    Victor G. Kac

Authors
  1. Maria Gorelik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Victor G. Kac
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Gorelik.

Additional information

Communicated by: Yasuyuki Kawahigashi

Maria Gorelik: Supported in part by BSF Grant No. 711623.

Victor G. Kac: Supported in part by Simons fellowship.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gorelik, M., Kac, V.G. Characters of (relatively) integrable modules over affine Lie superalgebras. Jpn. J. Math. 10, 135–235 (2015). https://doi.org/10.1007/s11537-015-1464-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11537-015-1464-2

Keywords and phrases

Mathematics Subject Classification (2010)

Search

Navigation