Bulletin of the Australian Mathematical Society

Research Article

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

XIN TANGa1 c1 and YUNGE XUa2

a1 Department of Mathematics & Computer Science, Fayetteville State University, Fayetteville, NC 28301, USA (email: xtang@uncfsu.edu)

a2 Faculty of Mathematics & Computer Science, Hubei University, Wuhan 430062, People’s Republic of China (email: xuy@hubu.edu.cn)

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.

(Received December 28 2007)

2000 Mathematics subject classification

  • 17B10;
  • 17B35;
  • 17B37

Keywords and phrases

  • hyperbolic algebras;
  • spectral theory;
  • Whittaker model;
  • quantum groups

Correspondence:

c1 For correspondence; e-mail: xtang@uncfsu.edu

Footnotes

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