Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T05:48:50.732Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  01 October 2008

XIN TANG  and
YUNGE XU
XIN TANG*
Affiliation:
Department of Mathematics & Computer Science, Fayetteville State University, Fayetteville, NC 28301, USA (email: [email protected])
YUNGE XU
Affiliation:
Faculty of Mathematics & Computer Science, Hubei University, Wuhan 430062, People’s Republic of China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

hyperbolic algebrasspectral theoryWhittaker modelquantum groups

MSC classification

Secondary: 17B10: Representations, algebraic theory (weights) 17B35: Universal enveloping (super)algebras 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 2 , October 2008 , pp. 261 - 284
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

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

References

[1]Bavula, V. V., ‘Generalized Weyl algebras and their representations’, Algebra i Analiz 4(1) (1992), 75–97; (Engl. Transl. St Petersburg Math. J. 4 (1993) 71–93).Google Scholar
[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. 65–92.Google Scholar
[3]Dixmier, J., Enveloping Algebras (North-Holland, Amsterdam, 1977).Google Scholar
[4]Drinfeld, V. G., ‘Hopf algebras and the quantum Yang–Baxter equations’, Soviet math. Dokll 32 (1985), 254–258.Google Scholar
[5]Gabriel, P., ‘Des categories abeliennes’, Bull. Soc. Math. France 90 (1962), 323–449.Google Scholar
[6]Hartwig, J., ‘Hopf structures on ambiskew polynomial rings’, J. Pure Appl. Algebra 212(4) (2008), 863–883.CrossRefGoogle Scholar
[7]Hu, J. and Zhang, Y., ‘Quantum double of U q((sl 2)≤0)’, J. Algebra 317(1) (2007), 87–110.Google Scholar
[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), 201–213.Google Scholar
[9]Jing, N. and Zhang, J., ‘Quantum Weyl algebras and deformations of U(G)’, Pacific J. Math. 171(2) (1995), 437–454.Google Scholar
[10]Kostant, B., ‘On Whittaker vectors and representation theory’, Invent. Math. 48(2) (1978), 101–184.Google Scholar
[11]Lynch, T., ‘Generalized Whittaker vectors and representation theory’, PhD Thesis, MIT, 1979.Google Scholar
[12]Macdowell, E., ‘On modules induced from Whittaker modules’, J. Algebra 96 (1985), 161–177.CrossRefGoogle Scholar
[13]Ondrus, M., ‘Whittaker modules for U q(sl 2)’, J. Algebra 289 (2005), 192–213.Google Scholar
[14]Rosenberg, A., Noncommutative Algebraic Geometry and Representations of Quantized Algebras, Mathematics and its Applications, 330 (Kluwer Academic Publishers, 1995).Google Scholar
[15]Sevostyanov, A., ‘Quantum deformation of Whittaker modules and Toda lattice’, Duke Math. J. 204(1) (2000), 211–238.Google Scholar
[16]Tang, X., ‘Construct irreducible representations of quantum groups U q(f m(K))’, Front. Math. China 3(3) (2008), 371–397.CrossRefGoogle Scholar
[17]Wang, D., Ji, Q. and Yang, S., ‘Finite-dimensional representations of quantum group U q(f(K,H))’, Comm. Algebra 30 (2002), 2191–2211.CrossRefGoogle Scholar