Letters in Mathematical Physics

, Volume 78, Issue 1, pp 1–16 | Cite as

Verma Modules for Yangians

  • Y. Billig
  • V. Futorny
  • A. MolevEmail author


We study the Verma modules M((μu)) over the Yangian Y \((\mathfrak{a})\) associated with a simple Lie algebra \(\mathfrak{a}\). We give necessary and sufficient conditions for irreducibility of M(μ(u)). Moreover, regarding the simple quotient L((μu)) of M((μu)) as an \(\mathfrak{a}\)-module, we give necessary and sufficient conditions for finite-dimensionality of the weight subspaces of L((μu)).


Yangian Verma module simple quotient 

Mathematics Subject Classification



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  1. 1.
    Arakawa T. (1999) Drinfeld functor and finite-dimensional representations of Yangian. Comm. Math. Phys. 205, 1–18zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Beck J. (1994) Braid group action and quantum affine algebras. Comm. Math. Phys. 165, 555–568zbMATHMathSciNetCrossRefADSGoogle Scholar
  3. 3.
    Billig Y., Zhao K. (2004) Weight modules over exp-polynomial Lie algebras. J. Pure Appl. Algebra 191, 23–42zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chari V., Pressley A. (1994) A Guide to Quantum Groups. Cambridge University Press, CambridgezbMATHGoogle Scholar
  5. 5.
    Chari, V., Pressley, A.: Quantum affine algebras and their representations. In: Representations of Groups (Banff, AB, 1994), pp. 59–78. CMS Conference Proceedings, vol. 16. American Mathematical Society, Providence (1995)Google Scholar
  6. 6.
    Dixmier J. (1974) Algèbres Enveloppantes. Gauthier-Villars, PariszbMATHGoogle Scholar
  7. 7.
    Drinfeld V.G. (1985) Hopf algebras and the quantum Yang–Baxter equation. Sov. Math. Dokl. 32, 254–258Google Scholar
  8. 8.
    Drinfeld V.G. (1988) A new realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36, 212–216zbMATHMathSciNetGoogle Scholar
  9. 9.
    Levendorskii S.Z. (1993) On PBW bases for Yangians. Lett. Math. Phys. 27, 37–42zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Molev A.I. (1998) Finite-dimensional irreducible representations of twisted Yangians. J. Math. Phys. 39, 5559–5600zbMATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Molev A., Nazarov M., Olshanski G. (1996) Yangians and classical Lie algebras. Russ. Math. Surv. 51(2): 205–282zbMATHCrossRefGoogle Scholar
  12. 12.
    Tarasov V.O. (1984) Structure of quantum L-operators for the R-matrix of the XXZ-model. Theor. Math. Phys. 61, 1065–1071CrossRefGoogle Scholar
  13. 13.
    Tarasov V.O. (1985) Irreducible monodromy matrices for the R-matrix of the XXZ-model and lattice local quantum Hamiltonians. Theor. Math. Phys. 63, 440–454MathSciNetCrossRefGoogle Scholar
  14. 14.
    Vasserot E. (1998) Affine quantum groups and equivariant K-theory. Transform. Groups 3, 269–299zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  2. 2.Instituto de Matematica e EstatisticaUniversidade de São PauloSao PauloBrazil
  3. 3.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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