Advertisement

Algebras and Representation Theory

, Volume 22, Issue 5, pp 1209–1237 | Cite as

Asymptotics of Standard Modules of Quantum Affine Algebras

  • Léa BittmannEmail author
Article
  • 13 Downloads

Abstract

We introduce a sequence of q-characters of standard modules of a quantum affine algebra and we prove it has a limit as a formal power series. For \(\mathfrak {g} = \hat {\mathfrak {s}\mathfrak {l}_{2}}\), we establish an explicit formula for the limit which enables us to construct corresponding asymptotical standard modules associated to each simple module in the category \(\mathcal {O}\) of a Borel subalgebra of the quantum affine algebra. Finally, we prove a decomposition formula for the limit formula into q-characters of simple modules in this category \(\mathcal {O}\).

Keywords

Quantum affine algebra Category \(\mathcal {O}\) Standard modules 

Mathematics Subject Classification (2010)

16T20 17B10 17B37 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baxter, R.: Partition function of the eight-vertex lattice model. Ann. Phys. 70, 193–228 (1972)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beck, J., Chari, V., Pressley, A.: An algebraic characterization of the affine canonical basis. Duke Math. J. 99(3), 455–487 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beck, J.: Braid group action and quantum affine algebras. Comm. Math. Phys. 165(3), 555–568 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y.: Hidden Grassmann structure in the X X Z model. II. Creation operators. Comm. Math. Phys. 286(3), 875–932 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bazhanov, V.V., Lukyanov, S.L., Zamolodchikov, A.B.: Integrable structure of conformal field theory. III,. The Yang-Baxter relation. Comm. Math. Phys. 200(2), 297–324 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chari, V.: Braid group actions and tensor products. Int. Math. Res. Not. 2002 (7), 357–382 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1995). Corrected reprint of the 1994 originalzbMATHGoogle Scholar
  8. 8.
    Damiani, I.: A basis of type Poincaré-Birkhoff-Witt for the quantum algebra of \(\hat {\text {sl}}(2)\). J. Algebra 161(2), 291–310 (1993)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Damiani, I.: La R-matrice pour les algèbres quantiques de type affine non tordu. Ann. Sci. École Norm. Sup. (4) 31(4), 493–523 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Drinfeld, V.G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Berkeley, Calif., 1986), pp. 798–820. Amer. Math. Soc., Providence (1987)Google Scholar
  11. 11.
    Drinfeld, V.G.: A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36, 212–216 (1988)Google Scholar
  12. 12.
    Frenkel, E., Hernandez, D.: Baxter’s relations and spectra of quantum integrable models. Duke Math. J. 164(12), 2407–2460 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Frenkel, E., Mukhin, E.: Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras. Comm. Math. Phys. 216(1), 23–57 (2001)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Frenkel, E., Reshetikhin, N.: The q-characters of representations of quantum affine algebras and deformations of \(\mathscr{W}\)-algebras. In: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), volume 248 of Contemp. Math., pp. 163–205. Amer. Math. Soc., Providence (1999)Google Scholar
  15. 15.
    Hernandez, D.: The Kirillov-Reshetikhin conjecture and solutions of T-systems. J. Reine Angew. Math. 596, 63–87 (2006)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hernandez, D., Jimbo, M.: Asymptotic representations and Drinfeld rational fractions. Compos. Math. 148(5), 1593–1623 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hernandez, D., Leclerc, B.: A cluster algebra approach to q-characters of Kirillov-Reshetikhin modules. J. Eur. Math. Soc. (JEMS) 18(5), 1113–1159 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hernandez, D., Leclerc, B.: Cluster algebras and category \(\mathcal {O}\) for representations of Borel subalgebras of quantum affine algebras. Algebra Number Theory 10(9), 2015–2052 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jantzen, J.C.: Lectures on Quantum Groups. American Mathematical Society, Providence (1996)zbMATHGoogle Scholar
  20. 20.
    Jimbo, M.: A q-difference analogue of \(U({\mathfrak {g}})\) and the Yang-Baxter equation. Lett. Math. Phys. 10(1), 63–69 (1985)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kac, V.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  22. 22.
    Kassel, C.: Quantum Groups, Volume 155 of Graduate Texts in Mathematics. Springer-Verlag, New York (1995)Google Scholar
  23. 23.
    Nakajima, H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Quiver Soc. 14(1), 145–238 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nakajima, H.: t-Analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras. Represent. Theory 7, 259–274 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nakajima, H.: Quiver varieties and t-analogs of q-characters of quantum affine algebras. Ann. Math. (2) 160(3), 1057–1097 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.CNRS Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586Université Paris-DiderotCedex 13France

Personalised recommendations