Frontiers of Mathematics in China

, Volume 3, Issue 3, pp 371–397

Construct irreducible representations of quantum groups Uq(ƒm(K))

Research Article

Abstract

In this paper, we construct families of irreducible representations for a class of quantum groups Uq(ƒm(K)). First, we give a natural construction of irreducible weight representations for Uq(ƒm(K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of Uq(ƒm(K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.

Keywords

Hyperbolic algebra spectral theory Whittaker modules quantum group 

MSC

17B10 17B35 17B37 

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References

  1. 1.
    Bavula V V. Generalized Weyl algebras and their representations. Algebra i Analiz, 1992, 4(1): 75–97; English transl in: St Petersburg Math J, 1993, 4: 71–93MathSciNetGoogle Scholar
  2. 2.
    Dixmier J. Enveloping Algebras. Amsterdam: North-Holland, 1977Google Scholar
  3. 3.
    Drinfeld V G. Hopf algebras and the quantum Yang-Baxter equations. Soviet Math Dokll, 1985, 32: 254–258Google Scholar
  4. 4.
    Gabriel P. Des categories abeliennes. Bull Soc Math France, 1962, 90: 323–449MATHMathSciNetGoogle Scholar
  5. 5.
    Jantzen J C. Lectures on Quantum Groups. Graduate Studies in Math, Vol 6. Providence: Amer Math Soc, 1993Google Scholar
  6. 6.
    Ji Q, Wang D, Zhou X. Finite dimensional representations of quantum groups U q(ƒ(K)). East-West J Math, 2000, 2(2): 201–213MATHMathSciNetGoogle Scholar
  7. 7.
    Jing N, Zhang J. Quantum Weyl algebras and deformations of U(G). Pacific J Math, 1995, 171(2): 437–454MATHMathSciNetGoogle Scholar
  8. 8.
    Kostant B. On Whittaker vectors and representation theory. Invent Math, 1978, 48(2): 101–184MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lynch T. Generalized Whittaker vectors and representation theory. Dissertation for the Ph D Degree. Cambridge: MIT, 1979Google Scholar
  10. 10.
    Macdowell E. On modules induced from Whittaker modules. J Algebra, 1985, 96: 161–177CrossRefMathSciNetGoogle Scholar
  11. 11.
    Ondrus M. Whittaker modules for U q(sl 2). J Algebra, 2005, 289: 192–213MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Rosenberg A. Noncommutative Algebraic Geometry and Representations of Quantized Algebras. Mathematics and Its Applications, Vol 330. Dordrecht: Kluwer Academic Publishers, 1995Google Scholar
  13. 13.
    Sevostyanov, A. Quantum deformation of Whittaker modules and Toda lattice. Duke Math J, 2000, 204(1): 211–238CrossRefMathSciNetGoogle Scholar
  14. 14.
    Smith S P. A class of algebras similar to the enveloping algebra of sl 2. Trans AMS, 1990, 322: 285–314MATHCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Department of Mathematics & Computer ScienceFayetteville State UniversityFayettevilleUSA

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