Skip to main content Accessibility help
×
×
Home

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

  • XIN TANG (a1) and YUNGE XU (a2)
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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      ON REPRESENTATIONS OF QUANTUM GROUPS Uq(fm(K,H))
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      ON REPRESENTATIONS OF QUANTUM GROUPS Uq(fm(K,H))
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      ON REPRESENTATIONS OF QUANTUM GROUPS Uq(fm(K,H))
      Available formats
      ×
Copyright
Corresponding author
For correspondence; e-mail: xtang@uncfsu.edu
Footnotes
Hide All

The second author was partially supported by NSFC, under grant 10501010.

Footnotes
References
Hide All
[1]Bavula, V. V., ‘Generalized Weyl algebras and their representations’, Algebra i Analiz 4(1) (1992), 7597; (Engl. Transl. St Petersburg Math. J. 4 (1993) 71–93).
[2]Benkart, G. and Witherspoon, S., ‘Representations of two-parameter quantum groups and Shur–Weyl duality’, in: Hopf Algebras, Lecture Notes in Pure and Applied Mathematics, 237 (Dekker, New York, 2004), pp. 6592.
[3]Dixmier, J., Enveloping Algebras (North-Holland, Amsterdam, 1977).
[4]Drinfeld, V. G., ‘Hopf algebras and the quantum Yang–Baxter equations’, Soviet math. Dokll 32 (1985), 254258.
[5]Gabriel, P., ‘Des categories abeliennes’, Bull. Soc. Math. France 90 (1962), 323449.
[6]Hartwig, J., ‘Hopf structures on ambiskew polynomial rings’, J. Pure Appl. Algebra 212(4) (2008), 863883.
[7]Hu, J. and Zhang, Y., ‘Quantum double of U q((sl 2)≤0)’, J. Algebra 317(1) (2007), 87110.
[8]Ji, Q., Wang, D. and Zhou, X., ‘Finite dimensional representations of quantum groups U q(f(K))’, East-West J. Math. 2(2) (2000), 201213.
[9]Jing, N. and Zhang, J., ‘Quantum Weyl algebras and deformations of U(G)’, Pacific J. Math. 171(2) (1995), 437454.
[10]Kostant, B., ‘On Whittaker vectors and representation theory’, Invent. Math. 48(2) (1978), 101184.
[11]Lynch, T., ‘Generalized Whittaker vectors and representation theory’, PhD Thesis, MIT, 1979.
[12]Macdowell, E., ‘On modules induced from Whittaker modules’, J. Algebra 96 (1985), 161177.
[13]Ondrus, M., ‘Whittaker modules for U q(sl 2)’, J. Algebra 289 (2005), 192213.
[14]Rosenberg, A., Noncommutative Algebraic Geometry and Representations of Quantized Algebras, Mathematics and its Applications, 330 (Kluwer Academic Publishers, 1995).
[15]Sevostyanov, A., ‘Quantum deformation of Whittaker modules and Toda lattice’, Duke Math. J. 204(1) (2000), 211238.
[16]Tang, X., ‘Construct irreducible representations of quantum groups U q(f m(K))’, Front. Math. China 3(3) (2008), 371397.
[17]Wang, D., Ji, Q. and Yang, S., ‘Finite-dimensional representations of quantum group U q(f(K,H))’, Comm. Algebra 30 (2002), 21912211.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed