Skip to main content
SpringerLink
Log in

Modules for double affine Lie algebras

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

Imaginary Verma modules, parabolic imaginary Verma modules, and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated, and several results are generalized from the affine Lie algebras. In particular, imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.

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. Chang X, Tan S. A class of irreducible integrable modules for the extended baby TKK algebra. Pacific J Math, 2011, 252: 293–312

    Article  MATH  MathSciNet  Google Scholar 

  2. Chari V. Representations of affine and toroidal Lie algebras. In: Geometric Representation Theory and Extended Affine Lie Algebras. Fields Inst Commun, Vol 59. Providence: Amer Math Soc, 2011,169–197

    MathSciNet  Google Scholar 

  3. Chari V, Le T. Representations of double affine Lie algebras. In: A Tribute to C. S. Seshadri, Trends Math. Basel: Birkhäuser, 2003,199–219

    Google Scholar 

  4. Chari V, Pressley A. Weyl modules for classical and quantum affine algebras. Represent Theory, 2001, 5: 191–223

    Article  MATH  MathSciNet  Google Scholar 

  5. Dokuchaev M, Figueiredo L V, Futorny V. Imaginary Verma modules for the extended affine Lie algebra sl2(Cq). Comm Algebra, 2003, 31: 289–308

    Article  MATH  MathSciNet  Google Scholar 

  6. Frenkel I. Representations of Kac-Moody algebras and dual resonance models. In: Applications of Group Theory in Physics and Mathematical Physics (Chicago, 1982). Lectures in Appl Math, 21. Providence: Amer Math Soc, 1985,325–353

    Google Scholar 

  7. Futorny V. Imaginary Verma modules for affine Lie algebras. Canad Math Bull, 1994, 37: 213–218

    Article  MATH  MathSciNet  Google Scholar 

  8. Futorny V, Kashuba I. Verma type modules for toroidal Lie algebras. Comm Algebra, 1999, 27: 3979–3991

    Article  MATH  MathSciNet  Google Scholar 

  9. Garland H. The arithmetic theory of loop algebras. J Algebra 1978, 53: 480–551

    Article  MATH  MathSciNet  Google Scholar 

  10. Kac V G. Infinite Dimensional Lie Algebras. 3rd ed. Cambridge: Cambridge Univ Press, 1990

    Book  MATH  Google Scholar 

  11. Moody R V, Pianzona A. Lie Algebras with Triangular Decompositions. New York: John Wiley & Sons, 1995

    Google Scholar 

  12. Moody R V, Shi Z. Toroidal Weyl groups. Nova J Algebra Geom, 1992, 1: 317–337

    MATH  MathSciNet  Google Scholar 

  13. Rao S E. Classification of irreducible integrable modules for multi-loop algebras with finite dimensional weight spaces. J Algebra, 2001, 246: 215–225

    Article  MATH  MathSciNet  Google Scholar 

  14. Rao S E. Classification of irreducible integrable modules for toroidal Lie algebras with finite dimensional weight spaces. J Algebra, 2004, 277: 318–348

    Article  MATH  MathSciNet  Google Scholar 

  15. Rao S E, Futorny V. Representations of loop Kac-Moody Lie algebras. Comm Algebra, 2013, 41: 3775–3792

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, South China University of Technology, Guangzhou, 510640, China

    Naihuan Jing & Chunhua Wang

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Naihuan Jing

Authors
  1. Naihuan Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunhua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chunhua Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jing, N., Wang, C. Modules for double affine Lie algebras. Front. Math. China 11, 89–108 (2016). https://doi.org/10.1007/s11464-015-0447-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-015-0447-1

Keywords

MSC

Search

Navigation