Communications in Mathematical Physics

, Volume 177, Issue 2, pp 399–415 | Cite as

Spectral analysis and the Haar functional on the quantumSU (2) group

  • H. T. Koelink
  • J. Verding


The Haar functional on the quantumSU(2) group is the analogue of invariant integration on the groupSU(2). If restricted to a subalgebra generated by a self-adjoint element the Haar functional can be expressed as an integral with a continuous measure or with a discrete measure or by a combination of both. These results by Woronowicz and Koornwinder have been proved by using the corepresentation theory of the quantumSU(2) group and Schur's orthogonality relations for matrix elements of irreducible unitary corepresentations. These results are proved here by using a spectral analysis of the generator of the subalgebra. The spectral measures can be described in terms of the orthogonality measures of orthogonal polynomials by using the theory of Jacobi matrices.


Neural Network Statistical Physic Matrix Element Complex System Spectral Analysis 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Askey, R., Ismail, M.E.H., A generalization of ultraspherical polynomials. Studies in Pure Mathematics, Erdős, P. (ed.), Basel: Birkhäuser, 1983, 55–78Google Scholar
  2. 2.
    Askey R., Ismail M.E.H.: Recurrence relations, continued fractions and orthogonal polynomials. Mem. Am. Math. Soc.49, no. 300, 1984Google Scholar
  3. 3.
    Askey, R.A., Rahman, M., Suslov S.K.: On a generalq-Fourier transformation with non-symmetric kernels. Preprint 21, Series 2, Carleton University (1994)Google Scholar
  4. 4.
    Askey R., Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc.54, no. 319, (1985)Google Scholar
  5. 5.
    Berezanskii, J.M.: Expansions in Eigenfunctions of Selfadjoint Operators. Transl. Math. Monographs17, Providenc RI: Am. Math. Soc., 1968Google Scholar
  6. 6.
    Bressoud, D.M.: A simple proof of Mehler's formula forq-Hermite polynomials. Indiana Univ. Math. J.29, 577–580 (1980)Google Scholar
  7. 7.
    Chihara, T.S.: An Introduction to Orthogonal Polynomials. math. and Appl.13, New York; Gordon and Breach, 1978Google Scholar
  8. 8.
    Dombrowski, J.: Orthogonal polynomials and functional analysis. Orthogonal Polynomials: Theory and Practice, Nevai, P. (ed.), NATO ASI series C, vol.294, Dordrecht; Kluwer, 1990, pp. 147–161Google Scholar
  9. 9.
    Dunford, N., Schwartz, J.T.: Linear Operators. Part II: Spectral Theory. New York: Interscience, 1963Google Scholar
  10. 10.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications35, Cambridge: Cambridge University Press, 1990Google Scholar
  11. 11.
    Koelink, H.T.: Addition formula for bigq-Legendre polynomials from the quantumSU(2) group. Canad. J. Math.47, 436–448 (1995)Google Scholar
  12. 12.
    Koelink, H.T.: Askey-Wilson polynomials and the quantumSU(2) group: Survey and applications. Acta Appl. Math. (to appear)Google Scholar
  13. 13.
    Koelink, H.T.: Addition formula for 2-parameter family of Askey-Wilson polynomials. Preprint (1994)Google Scholar
  14. 14.
    Koornwinder, T.H.: Representations of the twistedSU(2) quantum group and someq-hypergeometric orthogonal polynomials. Proc. Kon. Ned. Akad. van Wetensch., Ser. A92 (Indag. Math.51), 97–117 (1989)Google Scholar
  15. 15.
    Koornwinder, T.H.: Orthogonal polynomials in connection with quantum groups. Orthogonal Polynomials: Theory and Practice, Nevai, P. (ed.), NATO ASI series C, vol.294, Dordrecht; Kluwer, 1990, pp. 257–292Google Scholar
  16. 16.
    Koornwinder, T.H.: Askey-Wilson polynomials as zonal spherical functions on theSU(2) quantum group. SIAM J. Math. Anal.24, 795–813 (1993)Google Scholar
  17. 17.
    Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Ueno, K.: Representations of the quantum groupSU q(2) and the littleq-Jacobi polynomials. J. Funct. Anal.99, 357–386 (1991)Google Scholar
  18. 18.
    Noumi, M.: Quantum groups andq-orthogonal polynomials. Towards a realization of Askey-Wilson polynomials onSU q(2). Special Functions, Kashiwara, M., Miwa, T. (eds.), ICM-90 Satellite Conference Proceedings, Berlin, Heidelberg, New York: Springer, 1991Google Scholar
  19. 19.
    Noumi, M., Mimachi, K.: Spherical functions on a family of quantum 3-spheres. Compos. Math.83, 19–42 (1992)Google Scholar
  20. 20.
    Vaksman, L.L., Soibelman, Y.S.: Algebra of functions on the quantum groupSU(2). Funct. Anal. Appl.22, 170–181 (1988)Google Scholar
  21. 21.
    Woronowicz, S.L.: TwistedSU(2) group. An example of non-commutative differential calculus. Publ. Res. Inst. Math. Sci., Kyoto Univ.23, 117–181 (1987)Google Scholar
  22. 22.
    Woronowicz, S.L.: Compact matrix pseudo groups. Commun. Math. Phys.111, 613–665 (1987)Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • H. T. Koelink
    • 1
  • J. Verding
    • 2
  1. 1.Vakgroep WiskundeUniversiteit van AmsterdamAmsterdamThe Netherlands
  2. 2.Departement WiskundeKatholieke Universiteit LeuvenLeuven (Heverlee)Belgium

Personalised recommendations