Skip to main content
SpringerLink
Log in

Integrable Modules Over Affine Lie Superalgebras \({\mathfrak{sl}(1|n)^{(1)}}\)

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We describe the category of integrable \({\mathfrak{sl}(1|n)^{(1)}}\)-modules with the positive level and show that the irreducible modules provide the full set of irreducible representations for the corresponding simple vertex algebra.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adamović D., Milas A.: Vertex operator algebras associated to modular invariant representations for \({\it A}_{1}^{(1)}\). Math. Res. Lett. 2(5), 563–575 (1995)

    Article  MathSciNet  Google Scholar 

  2. Chari V.: Integrable representation of affine Lie algebras. Invent. Math. 85(2), 317–335 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  3. De Sole A., Kac V.G.: Finite vs affine W-algebras. Jpn. J. Math. 1(1), 137–261 (2006)

    Article  MathSciNet  Google Scholar 

  4. Dong C., Li H., Mason G.: Regularity of rational vertex algebras. Adv. Math. 132(1), 148–166 (1997)

    Article  MathSciNet  Google Scholar 

  5. Duflo, M., Serganova, V.: On associated varieties for Lie superalgebras. arXiv:math/0507198

  6. Frenkel I., Zhu Y.-C.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992)

    Article  MathSciNet  Google Scholar 

  7. Futorny V., Eswara Rao S.: Integrable representation of affine Lie superalgebras. Trans. AMS 361(10), 5435–5455 (2009)

    Article  Google Scholar 

  8. Germoni J.: Indecomposable representations of general linear Lie superalgebras. J. Algebra 209, 367–401 (1998)

    Article  MathSciNet  Google Scholar 

  9. Gorelik M., Kac V.G.: On complete reducibility for infinite-dimensional Lie alegbars. Adv. Math. 226(2), 1911–1972 (2011)

    Article  MathSciNet  Google Scholar 

  10. Gorelik M., Kac V.G.: Characters of (relatively) integrable modules over affine Lie superalgebras. Jpn. J. Math. 10(2), 135–235 (2015)

    Article  MathSciNet  Google Scholar 

  11. Kac V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)

    Article  Google Scholar 

  12. Kac V.G.: Laplace opeartors of infinite-dimensional Lie algebras and theta functions. Proc. Natl. Acad. Sci. USA 81, 645–647 (1984)

    Article  ADS  Google Scholar 

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

    Book  Google Scholar 

  14. Kac V.G., Wakimoto M.: Integrable highest weight modules over affine Lie superalgebras and number theory. Lie Theory Geom. Birkhauser Prog. Math. 123, 415–456 (1994)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  16. Serganova, V.: On the superdimension of an irreducible representation of a basic classical Lie superalgebra. In: Ferrara, S., Fioresi, R., Varadarajan V.S. (eds.) Supersymmetry in Mathematics and Physics. Lecture Notes in Mathematics, pp. 253–273. Springer, Berlin (2011)

    Google Scholar 

  17. Serganova, V.: Kac–Moody superalgebras and integrability. In: Neeb, K.H., Pianzola, A. (eds.) Developments and Trends Infinite-Dimensional Lie Theory. Progress in Mathematics, vol. 288, pp. 169–218. Birkhäuser Boston, Inc., Boston (2011)

    Google Scholar 

  18. Zhu Y.: Modular invariance of characters of vertex operator algebras. JAMS 9(1), 237–302 (1996)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to V. Kac for helpful discussions. We also thank the referees for very useful comments and suggestions.

Author information

Authors and Affiliations

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

    Maria Gorelik

  2. Department of Mathematics, University of California at Berkeley, Berkeley, CA, 94720, USA

    Vera Serganova

Authors
  1. Maria Gorelik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vera Serganova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Gorelik.

Additional information

Communicated by Y. Kawahigashi

Supported in part by BSF Grant 2012227. The second author was partially supported by NSF Grant DMS-1701532.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gorelik, M., Serganova, V. Integrable Modules Over Affine Lie Superalgebras \({\mathfrak{sl}(1|n)^{(1)}}\). Commun. Math. Phys. 364, 635–654 (2018). https://doi.org/10.1007/s00220-018-3246-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-018-3246-1

Search

Navigation