# zbMATH — the first resource for mathematics

Monomials of $$q$$ and $$q,t$$-characters for non simply-laced quantum affinizations. (English) Zbl 1098.17010
Author’s abstract: H. Nakajima [Proceedings of the Nagoya 2000 2nd international workshop, Nagoya, Japan, 2000, 196–219 (2001; Zbl 1011.17013), (*) Ann. Math. (2) 160, No. 3, 1057–1097 (2005; Zbl 1140.17015), see also arXiv:math.QA/0105173] 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 [D. Hernandez, Adv. Math. 187, No. 1, 1–52 (2004; Zbl 1098.17006)] we generalized the construction of $$q$$, $$t$$-characters for non simply-laced quantum affine algebras. First in this paper we prove a conjecture of [Adv. Math., loc. cit.]: 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 [(*) loc. cit.]) and the coefficients are nonnegative. 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 [E. Frenkel and N. Reshetikhin, Contemp. Math. 248, 163–205 (1999; Zbl 0973.17015), E. Frenkel and E. Mukhin, Commun. Math. Phys. 216, 23–57 (2001; Zbl 1051.17013), H. Nakajima, J. Am. Math. Soc. 14, No. 1, 145–238 (2001; Zbl 0981.17016)]: 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.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
deformations
Full Text:
##### References:
 [1] Akasaka et, T., Kashiwara, M.: Finite-dimensional representations of quantum affine algebras. Publ. Res. Inst. Math. Sci. 33(5), 839–867 (1997) · Zbl 0915.17011 [2] Beck, J.: Braid group action and quantum affine algebras. Comm. Math. Phys. 165(3), 555–568 (1994) · Zbl 0807.17013 · doi:10.1007/BF02099423 [3] Bourbaki, N.: Groupes et algèbres de Lie, Chapitres IV-VI, Hermann (1968) · Zbl 0186.33001 [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) · Zbl 0855.17009 [5] Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge, 1994 · Zbl 0839.17009 [6] Chari, V., Pressley, A.: Weyl modules for classical and quantum affine algebras, Represent. Theory 5, 191–223 (electronic) (2001) · Zbl 0989.17019 [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 2001 · Zbl 1034.17008 [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) [9] Drinfel’d, V.G.: A new realization of Yangians and of quantum affine algebras. Soviet Math. Dokl. 36(2), 212–216 (1988) [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) · Zbl 1051.17013 · doi:10.1007/s002200000323 [11] Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and W-algebras. Comm. Math. Phys. 178(1), 237–264 (1996) · Zbl 0869.17014 · doi:10.1007/BF02104917 [12] Frenkel, E., Reshetikhin, N.: Deformations of W-algebras associated to simple Lie algebras. Comm. Math. Phys. 197(1), 1–32 (1998) · Zbl 0939.17011 [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) · Zbl 0973.17015 [14] Hernandez, D.: t-analogues des opérateurs d’écrantage associés aux q-caractères. Int. Math. Res. Not. 2003(8), 451–475 (2003) · Zbl 1098.17008 · doi:10.1155/S107379280320605X [15] Hernandez, D.: Algebraic Approach to q,t-Characters. Adv. Math. 187(1), 1–52 (2004) · Zbl 1098.17009 · doi:10.1016/j.aim.2003.07.016 [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) · Zbl 1127.17015 · doi:10.1016/j.jalgebra.2004.02.022 [17] Hernandez, D.: Representations of Quantum Affinizations and Fusion Product, to appear in Transform. Groups (preprint arXiv:math.QA/0312336) [18] Jimbo, M.: A q-difference analogue of and the Yang-Baxter equation. Lett. Math. Phys. 10(1), 63–69 (1985) · Zbl 0587.17004 · doi:10.1007/BF00704588 [19] Jing, N.: Quantum Kac-Moody algebras and vertex representations. Lett. Math. Phys. 44(4), 261–271 (1998) · Zbl 0911.17006 · doi:10.1023/A:1007493921464 [20] Knight, H.: Spectra of tensor products of finite-dimensional representations of Yangians. J. Algebra 174(1), 187–196 (1995) · Zbl 0868.17009 · doi:10.1006/jabr.1995.1123 [21] Miki, K.: Representations of quantum toroidal algebra Uq( sln+1, tor) (n2). J. Math. Phys. 41(10), 7079–7098 (2000) · Zbl 1028.17011 · doi:10.1063/1.1287436 [22] Nakajima, H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc. 14(1), 145–238 (2001) · Zbl 0981.17016 · doi:10.1090/S0894-0347-00-00353-2 [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) · Zbl 1011.17013 [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) · Zbl 1140.17015 [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) [26] Nakajima, H.: Geometric construction of representations of affine algebras. In: Proceedings of the International Congress of Mathematicians, Volume I, 2003, pp 423–438 · Zbl 1049.17014 [27] Varagnolo, M., Vasserot, E.: Standard modules of quantum affine algebras. Duke Math. J. 111(3), 509–533 (2002) · Zbl 1011.17012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.