Ukrainian Mathematical Journal

, Volume 47, Issue 5, pp 793–797 | Cite as

On representations of the Heisenberg relations for the quantumE(2) group

  • V. L. Ostrovs'kyi
  • Yu. S. Samoilenko
Article
Received:

Abstract

All irreducible* -representations of the involutive algebraA generated by the so-called Heisenberg relations for the quantum deformation of the group of motions of the Euclidean plane are described.

Keywords

Euclidean Plane Quantum Deformation Heisenberg Relation Involutive algebraA 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Yu. M. Berezanskii, V. L. Ostrovskii, and Yu. S. Samoilenko, “Expansion of families of commuting operators in eigenfunctions and representations of commutation relations,”Ukr. Mat. Zh.,40, No. 1, 106–109 (1988).Google Scholar
  2. 2.
    Yu. M. Berezanskii and Yu. G. Kondrat'ev,Spectral Methods in Infinite-Dimensional Analysis [in Russian], Naukova Dumka, Kiev (1988).English translation: Kluwer AP, Dordrechtf. (1995)Google Scholar
  3. 3.
    V. L. Ostrovskii and Yu. S. Samoilenko, “Unbounded operators satisfying non-Lie commutation relations,”Repts. Math. Phys.,28, No. 1, 91–103 (1989).Google Scholar
  4. 4.
    V. L. Ostrovskii and Yu. S. Samoilenko, “On pairs of self-adjoint operators,”Seminar Sophus Lie,3, No. 2, 185–218 (1993).Google Scholar
  5. 5.
    E. E. Vaisleb and Yu. S. Samoilenko, “Representations of operator relations by unbounded operators and multi-dimensional dynamical systems,”Ukr. Math. Zh.,42, No. 9, 1011–1019 (1990).Google Scholar
  6. 6.
    V. L. Ostrovskii and L. B. Turovskaya, “Representations of* -algebras and multi-dimensional dynamical systems,”Ukr. Math. Zh.,47, No. 4, 488–497 (1995).Google Scholar
  7. 7.
    G. W. MacKey, “Induced representations of locally compact groups,”Ann. Math.,55, No. 1, 101–139 (1952).Google Scholar
  8. 8.
    S. L. Woronowicz, “Unbounded elements affiliated withC *-algebras and non-compact quantum groups,”Commun. Math. Phys.,136, 399–432 (1991).Google Scholar
  9. 9.
    K. Schmüdgen, “Integrable operator representations of ℝq2 andSL q(2R),”Seminar Sophus Lie,2, 107–114 (1992).Google Scholar
  10. 10.
    L. L. Vaksman and L. I. Korogodskii, “Algebra of bounded functions on the quantum group of motions of the plane andq-analog of Bessel functions,”Dokl. Akad. Nauk SSSR,304, No. 5, 1036–1040 (1989).Google Scholar
  11. 11.
    A. M. Gavrilik and A. U. Klimyk, “q-Deformed polynomials and pseudo-orthogonal algebras and their representations,”Lett. Math. Phys.,21, 215–220 (1990).Google Scholar
  12. 12.
    P. Podleś, “Quantum spheres,”Lett. Math. Phys.,14, 193–202 (1987).Google Scholar
  13. 13.
    W. Pusz and S. L. Woronowicz, “Twisted second quantization,”Repts. Math. Phys.,27, 231–257 (1989).Google Scholar
  14. 14.
    S. Klimek and A. Lesniewski, “Quantum Riemann surfaces. I. The unit disc,”Commun. Math. Phys.,146, 103–122 (1992).Google Scholar
  15. 15.
    S. L. Woronowicz, “QuantumE(2) group and its Pontryagin dual,”Lett. Math. Phys.,23, 251–263 (1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • V. L. Ostrovs'kyi
    • 1
  • Yu. S. Samoilenko
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesUSSR

Personalised recommendations