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  • Bulletin of the Australian Mathematical Society, Volume 78, Issue 2
  • October 2008, pp. 261-284

ON REPRESENTATIONS OF QUANTUM GROUPS Uq(fm(K,H))

  • XIN TANG (a1) and YUNGE XU (a2)
  • DOI: http://dx.doi.org/10.1017/S0004972708000701
  • Published online: 01 October 2008
Abstract
Abstract

We construct families of irreducible representations for a class of quantum groups Uq(fm(K,H). First, we realize these quantum groups as hyperbolic algebras. Such a realization yields natural families of irreducible weight representations for Uq(fm(K,H)). Second, we study the relationship between Uq(fm(K,H)) and Uq(fm(K)). As a result, any finite-dimensional weight representation of Uq(fm(K,H)) is proved to be completely reducible. Finally, we study the Whittaker model for the center of Uq(fm(K,H)), and a classification of all irreducible Whittaker representations of Uq(fm(K,H)) is obtained.

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Copyright
Corresponding author
For correspondence; e-mail: xtang@uncfsu.edu
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The second author was partially supported by NSFC, under grant 10501010.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[3]J. Dixmier , Enveloping Algebras (North-Holland, Amsterdam, 1977).

[6]J. Hartwig , ‘Hopf structures on ambiskew polynomial rings’, J. Pure Appl. Algebra 212(4) (2008), 863883.

[7]J. Hu and Y. Zhang , ‘Quantum double of Uq((sl2)≤0)’, J. Algebra 317(1) (2007), 87110.

[9]N. Jing and J. Zhang , ‘Quantum Weyl algebras and deformations of U(G)’, Pacific J. Math. 171(2) (1995), 437454.

[10]B. Kostant , ‘On Whittaker vectors and representation theory’, Invent. Math. 48(2) (1978), 101184.

[12]E. Macdowell , ‘On modules induced from Whittaker modules’, J. Algebra 96 (1985), 161177.

[13]M. Ondrus , ‘Whittaker modules for Uq(sl2)’, J. Algebra 289 (2005), 192213.

[14]A. Rosenberg , Noncommutative Algebraic Geometry and Representations of Quantized Algebras, Mathematics and its Applications, 330 (Kluwer Academic Publishers, 1995).

[16]X. Tang , ‘Construct irreducible representations of quantum groups Uq(fm(K))’, Front. Math. China 3(3) (2008), 371397.

[17]D. Wang , Q. Ji and S. Yang , ‘Finite-dimensional representations of quantum group Uq(f(K,H))’, Comm. Algebra 30 (2002), 21912211.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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