Mathematische Zeitschrift

, Volume 250, Issue 2, pp 443–473

Monomials of q and q, t-characters for non simply-laced quantum affinizations



Nakajima [23][24] introduced the morphism of q, t-characters for finite dimensional representation of simply-laced quantum affine algebras: it is a t-deformation of the Frenkel-Reshetikhin’s morphism of q-characters (sum of monomials in infinite variables). In [15] we generalized the construction of q, t-characters for non simply-laced quantum affine algebras. First in this paper we prove a conjecture of [15]: the monomials of q and q, t-characters of standard representations are the same in non simply-laced cases (the simply-laced cases were treated in [24]) and the coefficients are non negative. In particular those q, t-characters can be considered as t-deformations of q-characters. In the proof we show that for quantum affine algebras of type A, B, C and quantum toroidal algebras of type A(1) the l-weight spaces of fundamental representations are of dimension 1. Eventually we show and use a generalization of a result of [13][10][22]: for general quantum affinizations we prove that the l-weights of a l-highest weight simple module are lower than the highest l-weight in the sense of monomials.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akasaka et, T., Kashiwara, M.: Finite-dimensional representations of quantum affine algebras. Publ. Res. Inst. Math. Sci. 33(5), 839–867 (1997)Google Scholar
  2. 2.
    Beck, J.: Braid group action and quantum affine algebras. Comm. Math. Phys. 165(3), 555–568 (1994)Google Scholar
  3. 3.
    Bourbaki, N.: Groupes et algèbres de Lie, Chapitres IV-VI, Hermann (1968)Google Scholar
  4. 4.
    Chari, V., Pressley, A.: Quantum affine algebras and their representations, in Representations of groups (Banff, AB, 1994), 59–78, CMS Conf. Proc, 16, Amer. Math. Soc., Providence, RI (1995)Google Scholar
  5. 5.
    Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge, 1994Google Scholar
  6. 6.
    Chari, V., Pressley, A.: Weyl modules for classical and quantum affine algebras, Represent. Theory 5, 191–223 (electronic) (2001)Google Scholar
  7. 7.
    Chari, V., Pressley, A.: Integrable and Weyl modules for quantum affine sl2, Quantum groups and Lie theory (Durham, 1999), 48–62, London Math. Soc. Lecture Note Ser., 290, Cambridge Univ. Press, Cambridge 2001Google Scholar
  8. 8.
    Drinfel’d, V.G.: Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 798–820, Amer. Math. Soc., Providence, RI, (1987)Google Scholar
  9. 9.
    Drinfel’d, V.G.: A new realization of Yangians and of quantum affine algebras. Soviet Math. Dokl. 36(2), 212–216 (1988)Google Scholar
  10. 10.
    Frenkel, E., Mukhin, E.: Combinatorics of q-Characters of Finite-Dimensional Representations of Quantum Affine Algebras. Comm. Math. Phy. 216(1), 23–57 (2001)Google Scholar
  11. 11.
    Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and W-algebras. Comm. Math. Phys. 178(1), 237–264 (1996)Google Scholar
  12. 12.
    Frenkel, E., Reshetikhin, N.: Deformations of W-algebras associated to simple Lie algebras. Comm. Math. Phys. 197(1), 1–32 (1998)Google Scholar
  13. 13.
    Frenkel, E., Reshetikhin, N.: The q-Characters of Representations of Quantum Affine Algebras and Deformations of W-Algebras, Recent Developments in Quantum Affine Algebras and related topics. Cont. Math. 248, 163–205 (1999)Google Scholar
  14. 14.
    Hernandez, D.: t-analogues des opérateurs d’écrantage associés aux q-caractères. Int. Math. Res. Not. 2003(8), 451–475 (2003)Google Scholar
  15. 15.
    Hernandez, D.: Algebraic Approach to q,t-Characters. Adv. Math. 187(1), 1–52 (2004)Google Scholar
  16. 16.
    Hernandez, D.: The t-analogs of q-characters at roots of unity for quantum affine algebras and beyond. J. Algebra 279(2), 514–557 (2004)Google Scholar
  17. 17.
    Hernandez, D.: Representations of Quantum Affinizations and Fusion Product, to appear in Transform. Groups (preprint arXiv:math.QA/0312336)Google Scholar
  18. 18.
    Jimbo, M.: A q-difference analogue of Open image in new window and the Yang-Baxter equation. Lett. Math. Phys. 10(1), 63–69 (1985)Google Scholar
  19. 19.
    Jing, N.: Quantum Kac-Moody algebras and vertex representations. Lett. Math. Phys. 44(4), 261–271 (1998)Google Scholar
  20. 20.
    Knight, H.: Spectra of tensor products of finite-dimensional representations of Yangians. J. Algebra 174(1), 187–196 (1995)Google Scholar
  21. 21.
    Miki, K.: Representations of quantum toroidal algebra Uq( sln+1, tor) (n≥ 2). J. Math. Phys. 41(10), 7079–7098 (2000)Google Scholar
  22. 22.
    Nakajima, H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14(1), 145–238 (2001)Google Scholar
  23. 23.
    Nakajima, H.: t-Analogue of the q-Characters of Finite Dimensional Representations of Quantum Affine Algebras, “Physics and Combinatorics”, Proc. Nagoya 2000 International Workshop, World Scientific, pp 181–212 (2001)Google Scholar
  24. 24.
    Nakajima, H.: Quiver Varieties and t-Analogs of q-Characters of Quantum Affine Algebras. To appear in Ann. of Math. (preprint arXiv:math.QA/0105173)Google Scholar
  25. 25.
    Nakajima, H.: t-analogs of q-characters of quantum affine algebras of type An, Dn, in Combinatorial and geometric representation theory (Seoul, 2001), 141–160, Contemp. Math. 325, Amer. Math. Soc. Providence, RI (2003)Google Scholar
  26. 26.
    Nakajima, H.: Geometric construction of representations of affine algebras. In: Proceedings of the International Congress of Mathematicians, Volume I, 2003, pp 423–438Google Scholar
  27. 27.
    Varagnolo, M., Vasserot, E.: Standard modules of quantum affine algebras. Duke Math. J. 111(3), 509–533 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.École Normale Supérieure – DMAParis, Cedex 05France

Personalised recommendations