Abstract.
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akasaka et, T., Kashiwara, M.: Finite-dimensional representations of quantum affine algebras. Publ. Res. Inst. Math. Sci. 33(5), 839–867 (1997)
Beck, J.: Braid group action and quantum affine algebras. Comm. Math. Phys. 165(3), 555–568 (1994)
Bourbaki, N.: Groupes et algèbres de Lie, Chapitres IV-VI, Hermann (1968)
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)
Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge, 1994
Chari, V., Pressley, A.: Weyl modules for classical and quantum affine algebras, Represent. Theory 5, 191–223 (electronic) (2001)
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 2001
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)
Drinfel’d, V.G.: A new realization of Yangians and of quantum affine algebras. Soviet Math. Dokl. 36(2), 212–216 (1988)
Frenkel, E., Mukhin, E.: Combinatorics of q-Characters of Finite-Dimensional Representations of Quantum Affine Algebras. Comm. Math. Phy. 216(1), 23–57 (2001)
Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and W-algebras. Comm. Math. Phys. 178(1), 237–264 (1996)
Frenkel, E., Reshetikhin, N.: Deformations of W-algebras associated to simple Lie algebras. Comm. Math. Phys. 197(1), 1–32 (1998)
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)
Hernandez, D.: t-analogues des opérateurs d’écrantage associés aux q-caractères. Int. Math. Res. Not. 2003(8), 451–475 (2003)
Hernandez, D.: Algebraic Approach to q,t-Characters. Adv. Math. 187(1), 1–52 (2004)
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)
Hernandez, D.: Representations of Quantum Affinizations and Fusion Product, to appear in Transform. Groups (preprint arXiv:math.QA/0312336)
Jimbo, M.: A q-difference analogue of
and the Yang-Baxter equation. Lett. Math. Phys. 10(1), 63–69 (1985)Jing, N.: Quantum Kac-Moody algebras and vertex representations. Lett. Math. Phys. 44(4), 261–271 (1998)
Knight, H.: Spectra of tensor products of finite-dimensional representations of Yangians. J. Algebra 174(1), 187–196 (1995)
Miki, K.: Representations of quantum toroidal algebra U q ( sln+1, tor) (n≥ 2). J. Math. Phys. 41(10), 7079–7098 (2000)
Nakajima, H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14(1), 145–238 (2001)
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)
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)
Nakajima, H.: t-analogs of q-characters of quantum affine algebras of type A n , D n , in Combinatorial and geometric representation theory (Seoul, 2001), 141–160, Contemp. Math. 325, Amer. Math. Soc. Providence, RI (2003)
Nakajima, H.: Geometric construction of representations of affine algebras. In: Proceedings of the International Congress of Mathematicians, Volume I, 2003, pp 423–438
Varagnolo, M., Vasserot, E.: Standard modules of quantum affine algebras. Duke Math. J. 111(3), 509–533 (2002)
and the Yang-Baxter equation. Lett. Math. Phys. 10(1), 63–69 (1985)Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hernandez, D. Monomials of q and q, t-characters for non simply-laced quantum affinizations. Math. Z. 250, 443–473 (2005). https://doi.org/10.1007/s00209-005-0762-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-005-0762-4