Skip to main content
SpringerLink
Log in

Spectra, eigenvectors and overlap functions for representation operators ofq-deformed algebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Operators of representations corresponding to symmetric elements of theq-deformed algebrasU q (su1,1),U q (so2,1),U q (so3,1),U q (so n ) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasU q (su1,1) andU q (so2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Barut, A.O., Raczka, R.: Theory of Group Representations and Applications. Warszawa: PWN, 1977

    Google Scholar 

  2. Drinfeld, V.G.: Hopf algebra and quantum Yang-Baxter equation. Sov. Math. Dokl.32, 254–259 (1985)

    Google Scholar 

  3. Jimbo, M.: Aq-difference analogue ofU(g) and the Yang-Baxter equations. Lett. Math. Phys.10, 63–69 (1985)

    Google Scholar 

  4. Vilenkin, N.Ja., Klimyk, A.U.: Representation of Lie Groups and Special Functions. Dordrecht: Kluwer, vol. 1, 1991; vol. 2, 1993

    Google Scholar 

  5. Akhiezer, N.I., Glazman, I.M.: The Theory of Linear Operators in Hilbert Spaces. New York: Ungar, 1961

    Google Scholar 

  6. Berezanskii, Ju.M.: Expansions in Eigenfunctions of Selfadjoint Operators. Providence, R.I.: Am. Math. Soc., 1968

    Google Scholar 

  7. Akhiezer, N.I.: The Classical Moment Problem. Edinburgh: Oliver and Boyd, 1965

    Google Scholar 

  8. Dickinson, D.J., Pollak, H.O., Wannier, G.H.: On a class of polynomials orthogonal over a denumerable set. Pacific J. Math.6, 239–247 (1956)

    Google Scholar 

  9. Goldberg, J.L.: Polynomials orthogonal over a denumerable set. Pacific J. Math.15, 1171–1186 (1965)

    Google Scholar 

  10. Dombrowski, J.: Orthogonal polynomials and functional analysis. In: Orthogonal polynomials: Theory and Practice (ed. P. Nevai). Dordrecht: Kluwer, 1990, pp. 147–161

    Google Scholar 

  11. Askey, R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Memoirs of Am. Math. Soc.54, 1–55 (1985)

    Google Scholar 

  12. Gasper, G., Rahman, M.: Basic Hypergeometric Functions. Cambridge: Cambridge Univ. Press, 1990

    Google Scholar 

  13. Masson, D.R., Repka, J.: Spectral theory of Jacobi matrices inl 2(Z) and thesu(1,1) Lie algebra. SIAM J. Math. Anal.22, 1134–1146 (1991)

    Google Scholar 

  14. Macfarlane, A.J.: Onq-analogues of the quantum harmonic oscillator and the quantum groupSU(2) q . J. Phys. A: Math. Gen.22, 4581–4588 (1989)

    Google Scholar 

  15. Burban, I.M., Klimyk, A.U.: On spectral properties ofq-oscillator operators. Lett. Math. Phys.29, 13–18 (1993)

    Google Scholar 

  16. Askey, R., Ismail, M.: A generalization of ultraspherical polynomials, In: Studies in Pure Mathematics, P. Erdös (ed.) Basel: Birkhäuser, 1983, pp. 55–78

    Google Scholar 

  17. Andrews, G.: The Theory of Partitions. Reading, MA: Addison-Wesley, 1977

    Google Scholar 

  18. Burban, I.M., Klimyk, A.U.: Representations of the quantum algebraU q (su1,1). J. Phys. A: Math. Gen.26, 2139–2151 (1993)

    Google Scholar 

  19. Maksudov, F.G., Allakhverdiev, B.P.: On spectral theory of non-selfadjoint difference operators of the second order with matrix coefficients. Dokl. Russian Akad. Nauk,328, 654–657 (1993) (in Russian)

    Google Scholar 

  20. Gavrilik, A.M., Klimyk, A.U.: Representations ofq-deformed algebrasU q (so2,1) andU q (so3,1). J. Math. Phys.35, 3670–3686 (1994)

    Google Scholar 

  21. Gavrilik, A.M., Klimyk, A.U.:q-Deformed orthogonal and pseudo-orthogonal algebras and their representations. Lett. Math. Phys.21, 215–220 (1991)

    Google Scholar 

  22. Noumi, M.: Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. Adv. in Math. (in press)

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, 252143, Kiev 143, Ukraine

    A. U. Klimyk & I. I. Kachurik

Authors
  1. A. U. Klimyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. I. Kachurik
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. Jimbo

Rights and permissions

Reprints and permissions

About this article

Cite this article

Klimyk, A.U., Kachurik, I.I. Spectra, eigenvectors and overlap functions for representation operators ofq-deformed algebras. Commun.Math. Phys. 175, 89–111 (1996). https://doi.org/10.1007/BF02101625

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02101625

Keywords

Search

Navigation