Abstract
We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of Frenkel and Hernandez (Math Ann, to appear) and Frenkel and Reshetikhin (Commun Math Phys 197(1):1–32, 1998). We prove this duality for the Kirillov–Reshetikhin modules and their irreducible tensor products. In the course of the proof we introduce and construct “interpolating (q, t)-characters” depending on two parameters which interpolate between the q-characters of a quantum affine algebra and its Langlands dual.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bouwknegt P., Pilch K.: On deformed W-algebras and quantum affine algebras. Adv. Theor. Math. Phys. 2(2), 357–397 (1998)
Chari V., Hernandez D.: Beyond Kirillov–Reshetikhin modules. Contemp. Math. 506, 49–81 (2010)
Chari V., Pressley A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)
Chari V., Pressley A.: Factorization of representations of quantum affine algebras, Modular interfaces (Riverside CA 1995). AMS/IP Stud. Adv. Math. 4, 33–40 (1997)
Frenkel, E., Hernandez, D.: Langlands duality for representations of quantum groups. Math. Ann. (to appear)
Frenkel E., Mukhin E.: Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras. Commun. Math. Phys. 216(1), 23–57 (2001)
Frenkel E., Reshetikhin N.: Deformations of W-algebras associated to simple Lie algebras. Commun. 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 (Raleigh, NC, 1998), pp. 163–205. Contemp. Math., vol. 248. American Mathematical Society, Providence (1999)
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.: The Kirillov–Reshetikhin conjecture and solutions of T-systems. J. Reine Angew. Math. 596, 63–87 (2006)
Hernandez D.: On minimal affinizations of representations of quantum groups. Commun. Math. Phys. 277, 221–259 (2007)
Hernandez D.: Kirillov–Reshetikhin conjecture: the general case. Int. Math. Res. Not. 1, 149–193 (2010)
Hernandez D.: Simple tensor products. Invent. Math. 181(3), 649–675 (2010)
Hernandez D., Leclerc B.: Cluster algebras and quantum affine algebras. Duke Math. J. 154(2), 265–341 (2010)
Kuniba A., Nakamura S., Hirota R.: Pfaffian and determinant solutions to a discretized Toda equation for B r , C r and D r . J. Phys. A 29(8), 1759–1766 (1996)
Kuniba A., Ohta Y., Suzuki J.: Quantum Jacobi–Trudi and Giambelli formulae for \({\mathcal{U}_q(B_r^{(1)})}\) from analytic Bethe Ansatz. J. Phys. A 28(21), 6211–6226 (1995)
Kuniba A., Suzuki S.: Analytic Bethe Ansatz for fundamental representations of Yangians. Commun. Math. Phys. 173, 225–264 (1995)
McGerty K.: Langlands duality for representations and quantum groups at a root of unity. Commun. Math. Phys. 296(1), 89–109 (2010)
Nakajima H.: Quiver varieties and t-analogs of q-characters of quantum affine algebras. Ann. Math. 160, 1057–1097 (2004)
Nakajima H.: t-Analogs of q-characters of Kirillov–Reshetikhin modules of quantum affine algebras. Represent. Theory 7, 259–274 (2003) (electronic)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
To V. B. Matveev on his 65th birthday
E. Frenkel’s work was supported in part by DARPA through the grant HR0011-09-1-0015 and by Fondation Sciences mathématiques de Paris.
D. Hernandez’s work was supported partially by ANR through Project “Géométrie et Structures Algébriques Quantiques”.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Frenkel, E., Hernandez, D. Langlands Duality for Finite-Dimensional Representations of Quantum Affine Algebras. Lett Math Phys 96, 217–261 (2011). https://doi.org/10.1007/s11005-010-0426-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-010-0426-0