Transformation Groups

, Volume 15, Issue 3, pp 517–549

A categorical approach to Weyl modules

  • Vyjayanthi Chari
  • Ghislain Fourier
  • Tanusree Khandai


Global and local Weyl modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained. In this paper we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that, unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BN]
    J. Beck, H. Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J. 123 (2004), no. 2, 335–402.MathSciNetMATHGoogle Scholar
  2. [B]
    N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris, 1958.Google Scholar
  3. [C1]
    V. Chari, Integrable representations of affine Lie algebras, Invent. Math. 85 (1986), 317–335.CrossRefMathSciNetMATHGoogle Scholar
  4. [C2]
    V. Chari, On the fermionic formula and the Kirillov-Reshetikhin conjecture, Int. Math. Res. Not. 12 (2001), 629–654.CrossRefMathSciNetGoogle Scholar
  5. [CG1]
    V. Chari, J. Greenstein, Current algebras, highest weight categories and quivers, Adv. Math. 216 (2007), no. 2, 811–840.CrossRefMathSciNetMATHGoogle Scholar
  6. [CG2]
    V. Chari, J. Greenstein, A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras, Adv. Math. 220 (2009), no 4, 1193–1221.CrossRefMathSciNetMATHGoogle Scholar
  7. [CL]
    V. Chari, S. Loktev, Weyl, Demazure and fusion modules for the current algebra of sl r+1 , Adv. Math. 207 (2006), no. 2, 928–960.CrossRefMathSciNetMATHGoogle Scholar
  8. [CM]
    V. Chari, A. Moura, The restricted Kirillov–Reshetikhin modules for the current and twisted current algebras, Comm. Math. Phys. 266 (2006), 431–454.CrossRefMathSciNetMATHGoogle Scholar
  9. [CP1]
    V. Chari, A. Pressley, New unitary representations of loop groups, Math. Ann. 275 (1986), 87–104.CrossRefMathSciNetMATHGoogle Scholar
  10. [CP2]
    V. Chari, A. Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191–223 (electronic).CrossRefMathSciNetMATHGoogle Scholar
  11. [FL]
    B. Feigin, S. Loktev, Multi-dimensional Weyl modules and symmetric functions, Comm. Math. Phys. 251 (2004), 427–445.CrossRefMathSciNetMATHGoogle Scholar
  12. [FoL]
    G. Fourier, P. Littelmann, Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), 566–593.CrossRefMathSciNetMATHGoogle Scholar
  13. [G]
    H. Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480–551.CrossRefMathSciNetMATHGoogle Scholar
  14. [H]
    J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York, 1968. Russian transl.: Дж. Хамфрис, Введение в теорию алгебр Ли и их представлений, МЦНМО, М., 2003.Google Scholar
  15. [K]
    M. Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), no. 1, 117–175.CrossRefMathSciNetMATHGoogle Scholar
  16. [Ku]
    S. Kumar, private correspondence.Google Scholar
  17. [L]
    M. Lau, Representations of multi-loop algebras, to appear in Pacific J. Math., preprint arXiv:0811.2011v2.Google Scholar
  18. [N]
    H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), 145–238,CrossRefMathSciNetMATHGoogle Scholar
  19. [R]
    S. E. Rao, On representations of loop algebras, Comm. Algebra 21 (1993), 2131–2153.CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Vyjayanthi Chari
    • 1
  • Ghislain Fourier
    • 2
  • Tanusree Khandai
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaRiversideUSA
  2. 2.Mathematisches InstitutUniversität zu KölnKölnGermany
  3. 3.Harish-Chandra Research InstituteAllahabadIndia

Personalised recommendations