Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T01:32:58.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic representations and Drinfeld rational fractions

Part of: Groups and algebras in quantum theory Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  10 July 2012

David Hernandez  and
Michio Jimbo
David Hernandez
Affiliation:
Institut de Mathématiques de Jussieu, Université Paris Diderot (Paris VII), 175 rue du Chevaleret, 75013 Paris, France (email: [email protected])
Michio Jimbo
Affiliation:
Department of Mathematics, Rikkyo University, Toshima-ku, Tokyo 171-8501, Japan (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce and study a category of representations of the Borel algebra associated with a quantum loop algebra of non-twisted type. We construct fundamental representations for this category as a limit of the Kirillov–Reshetikhin modules over the quantum loop algebra and establish explicit formulas for their characters. We prove that general simple modules in this category are classified by n-tuples of rational functions in one variable which are regular and non-zero at the origin but may have a zero or a pole at infinity.

Keywords

quantum affine algebrascategory 𝒪Kirillov–Reshetikhin modulesBorel algebraasymptotic representation theory

MSC classification

Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations 17B10: Representations, algebraic theory (weights) 81R50: Quantum groups and related algebraic methods
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 5 , September 2012 , pp. 1593 - 1623
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[BHK02]Bazhanov, V., Hibberd, A. and Khoroshkin, S., Integrable structure of 𝒲3 conformal field theory, quantum Bouusinesq theory and boundary affine Toda theory, Nucl. Phys. B 622 (2002), 475–547.Google Scholar
[BLZ99]Bazhanov, V., Lukyanov, S. and Zamolodchikov, A., Integrable structure of conformal field theory III. The Yang-Baxter relation, Comm. Math. Phys. 200 (1999), 297–324.CrossRefGoogle Scholar
[BT08]Bazhanov, V. and Tsuboi, Z., Baxter’s Q-operators for supersymmetric chains, Nucl. Phys. B 805 (2008), 451–516.CrossRefGoogle Scholar
[Bec94]Beck, J., Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), 555–568.CrossRefGoogle Scholar
[BCP99]Beck, J., Chari, V. and Pressley, A., An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), 455–487.CrossRefGoogle Scholar
[BK96]Beck, J. and Kac, V. G., Finite-dimensional representations of quantum affine algebras at roots of unity, J. Amer. Math. Soc. 9 (1996), 391–423.CrossRefGoogle Scholar
[BT04]Benkart, G. and Terwilliger, P., Irreducible modules for the quantum affine algebra and its Borel subalgebra , J. Algebra 282 (2004), 172–194.Google Scholar
[Bow07]Bowman, J., Irreducible modules for the quantum affine algebra U q(𝔤) and its Borel subalgebra U q(𝔤)≥0, J. Algebra 316 (2007), 231–253.CrossRefGoogle Scholar
[Cha01]Chari, V., On the fermionic formula and the Kirillov-Reshetikhin conjecture, Int. Math. Res. Not. 2001 (2001), 629–654.Google Scholar
[Cha02]Chari, V., Braid group actions and tensor products, Int. Math. Res. Not. 2002 (2002), 357–382.Google Scholar
[CG05]Chari, V. and Greenstein, J., Filtrations and completions of certain positive level modules of affine algebras, Adv. Math. 194 (2005), 296–331.Google Scholar
[CH10]Chari, V. and Hernandez, D., Beyond Kirillov-Reshetikhin modules, in Quantum affine algebras, extended affine Lie algebras, and their applications, Contemporary Mathematics, vol. 506 (American Mathematical Society, Providence, RI, 2010), 49–81.Google Scholar
[CP94]Chari, V. and Pressley, A., A guide to quantum groups (Cambridge University Press, Cambridge, 1994).Google Scholar
[CP95]Chari, V. and Pressley, A., Quantum affine algebras and their representations, in Representations of groups (Banff, AB, 1994), CMS Conference Proceedings, vol. 16 (American Mathematical Society, Providence, RI, 1995), 59–78.Google Scholar
[CP01]Chari, V. and Pressley, A., Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191–223 (electronic).Google Scholar
[Dam98]Damiani, I., La ℛ-matrice pour les algèbres quantiques de type affine non tordu, Ann. Sci. Éc. Norm. Supér. 31 (1998), 493–523.Google Scholar
[Dri87]Drinfel’d, V., Quantum groups, in Proceedings of the International Congress of Mathematicians (Berkeley, August 3–11, 1986) (American Mathematical Society, Providence, RI, 1987), 798–820.Google Scholar
[Dri88]Drinfel’d, V., A new realization of Yangians and of quantum affine algebras, Soviet Math. Dokl. 36 (1988), 212–216.Google Scholar
[FL07]Fourier, G. and Littelmann, P., Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions, Adv. Math. 211 (2007), 566–593.CrossRefGoogle Scholar
[Fre85]Frenkel, I., Representations of affine Kac-Moody algebras and dual resonance models, in Applications of group theory in physics and mathematical physics (Chicago, 1982), Lectures in Applied Mathematics, vol. 21 (American Mathematical Society, Providence, RI, 1985), 325–353.Google Scholar
[FM01]Frenkel, E. and Mukhin, E., Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), 23–57.Google Scholar
[FR99]Frenkel, E. and Reshetikhin, N., The q-characters of representations of quantum affine algebras and deformations of W-algebras, in Recent Developments in Quantum Affine Algebras and related topics, Contemporary Mathematics, vol. 248 (American Mathematical Society, Providence, RI, 1999), 163–205.CrossRefGoogle Scholar
[Her05]Hernandez, D., Representations of quantum affinizations and fusion product, Transform. Groups 10 (2005), 163–200.CrossRefGoogle Scholar
[Her06]Hernandez, D., The Kirillov-Reshetikhin conjecture and solutions of T-systems, J. Reine Angew. Math. 596 (2006), 63–87.Google Scholar
[Her07]Hernandez, D., Drinfeld coproduct, quantum fusion tensor category and applications, Proc. Lond. Math. Soc. 95 (2007), 567–608.CrossRefGoogle Scholar
[Her10]Hernandez, D., Simple tensor products, Invent. Math. 181 (2010), 649–675.CrossRefGoogle Scholar
[HL10]Hernandez, D. and Leclerc, B., Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), 265–341.Google Scholar
[Jan96]Jantzen, J., Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6 (American Mathematical Society, Providence, RI, 1996).Google Scholar
[Jim85]Jimbo, M., A q-difference analogue of 𝒰(𝔤) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63–69.CrossRefGoogle Scholar
[Kac90]Kac, V., Infinite dimensional Lie algebras, third edition (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[Kas02]Kashiwara, M., On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), 117–175.CrossRefGoogle Scholar
[KS95]Kazhdan, D. and Soibelman, Y., Representations of quantum affine algebras, Selecta Math. (N.S.) 1 (1995), 537–595.CrossRefGoogle Scholar
[Koj08]Kojima, T., The Baxter’s Q-operator for the W-algebra W N, J. Phys. A 41 (2008), 355206.Google Scholar
[Lec11]Leclerc, B., Quantum loop algebras, quiver varieties, and cluster algebras, in Representations of algebras and related topics, EMS Series of Congress Reports, vol. 5, eds Skowroński, A. and Yamagata, K. (European Mathematical Society, Zürich, 2011), 117–152.CrossRefGoogle Scholar
[Nak01]Nakajima, H., Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), 145–238.Google Scholar
[Nak03]Nakajima, H., t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Theory 7 (2003), 259–274.CrossRefGoogle Scholar
[VV02]Varagnolo, M. and Vasserot, E., Standard modules of quantum affine algebras, Duke Math. J. 111 (2002), 509–533.Google Scholar
[Ver03]Vershik, A., Two lectures on the asymptotic representation theory and statistics of Young diagrams, in Asymptotic combinatorics with applications to mathematical physics (St. Petersburg, 2001), Lecture Notes in Mathematics, vol. 1815 (Springer, Berlin, 2003), 161–182.CrossRefGoogle Scholar