Harmonic analysis on quantum spheres associated with representations of U q ( 𝔰𝔬 N ) and q -jacobi polynomials

Tetsuya Sugitani

Compositio Mathematica (1995)

  • Volume: 99, Issue: 3, page 249-281
  • ISSN: 0010-437X

How to cite


Sugitani, Tetsuya. "Harmonic analysis on quantum spheres associated with representations of $U_q (\mathfrak {so}_N)$ and $q$-jacobi polynomials." Compositio Mathematica 99.3 (1995): 249-281. <http://eudml.org/doc/90416>.

author = {Sugitani, Tetsuya},
journal = {Compositio Mathematica},
keywords = {algebra of regular functions; zonal spherical functions; -orthogonal polynomial},
language = {eng},
number = {3},
pages = {249-281},
publisher = {Kluwer Academic Publishers},
title = {Harmonic analysis on quantum spheres associated with representations of $U_q (\mathfrak \{so\}_N)$ and $q$-jacobi polynomials},
url = {http://eudml.org/doc/90416},
volume = {99},
year = {1995},

AU - Sugitani, Tetsuya
TI - Harmonic analysis on quantum spheres associated with representations of $U_q (\mathfrak {so}_N)$ and $q$-jacobi polynomials
JO - Compositio Mathematica
PY - 1995
PB - Kluwer Academic Publishers
VL - 99
IS - 3
SP - 249
EP - 281
LA - eng
KW - algebra of regular functions; zonal spherical functions; -orthogonal polynomial
UR - http://eudml.org/doc/90416
ER -


  1. [A] Abe, E.: Hopf algebras, Cambridge University Press, 1980. Zbl0476.16008MR594432
  2. [AW] Askey, E. and Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalizeJacobi polynomials54 (1985), Memoirs Amer. Math. Soc., No. 319. Zbl0572.33012MR783216
  3. [B] Bergman, G.M.: The Diamond lemma for ring theory, Adv. in Math.29 (1978), 178-218. Zbl0326.16019MR506890
  4. [D1] Drinfel'd, V.G.: Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk. SSSR283 (1985), 1060-1064. Zbl0588.17015MR802128
  5. [D2] Drinfel'd, V.G.: Quantum groups, Proceedings of the International Congress of Mathematicians, Berkeley, California, USA, 1986, pp. 798-820 Zbl0667.16003MR934283
  6. [GR] Gasper, G. and Rahman, M.: Basic hypergeometric series, Encyclopedia of Mathematics and its applications, Cambridge University Press, 1990. Zbl1129.33005MR1052153
  7. [H] Howe, R.: Remarks on classical invariant theory, Trans. Amer. Math.318 (1989), 539-570; Erratum, ibid. 318 (1990) p. 823. Zbl0726.15025MR986027
  8. [HU] Howe, R. and Umeda, T.: The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann.290 (1991), 565-619. Zbl0733.20019MR1116239
  9. [J1] Jimbo, M.: A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys.10 (1985), 63-69. Zbl0587.17004MR797001
  10. [J2] Jimbo, M.: Quantum R matrix for the generalized Toda system, Comm. Math. Phys.102 (1986), 537-548. Zbl0604.58013MR824090
  11. [K1] Koornwinder, T.H.: Continuous q-Legendre polynomials as spherical matrix elements of irreducible representations of the quantum SU(2) group, CWI Quarterly2 (1989), 171-173. Zbl0677.33008
  12. [K2] Koornwinder, T.H.: Orthogonal polynomials in connection with quantum groups, in: Orthogonal Polynomials: Theory and Practice (P. Nevai, ed.), NATO ASI Series, Kluwer Academic Publishers, 1990, pp. 257-292. Zbl0697.42019MR1100297
  13. [L] Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math.70 (1988), 237-249. Zbl0651.17007MR954661
  14. [Na] Nakashima, T.: Crystal base and a generalization of the Littlewood-Richardson Rule for the classical Lie algebras, Comm. Math. Phys.154 (1993), 215-243. Zbl0795.17016MR1224078
  15. [N1] Noumi, M.: Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. in Math. (to appear). Zbl0874.33011
  16. [N2] Noumi, M.: A realization of Macdonald's symmetric polynomials on quantum homogeneous spaces, Int. J. Mod. Phys.A (Proc. Suppl.) 3A, World Scientific, 1993, pp. 218-223. 
  17. [N3] Noumi, M.: Quantum groups and q-orthogonal polynomials - Towards a realization of Askey-Wilson polynomials on SUq(2), in "Special Functions" (M. Kashiwara and T. Miwa, eds.), ICM-90 Satellite Conference Proceedings, Springer-Verlag, 1991, pp. 260-288. Zbl0784.33010MR1166821
  18. [NM1] Noumi, M. and Mimachi, K.: Quantum 2-spheres and big q-Jacobi polynomials, Comm. Math. Phys.128 (1990), 521-531. Zbl0699.33005MR1045882
  19. [NM2] Noumi, M. and Mimachi, K.: Spherical functions on a family of quantum 3-spheres, Compositio Mathematica83 (1992), 19-42. Zbl0760.33009MR1168121
  20. [NM3] Noumi, M. and Mimachi, K.: Rogers's q-ultraspherical polynomials on a quantum 2-sphere, Duke. Math. J.63 (1991), 65-80. Zbl0780.33011MR1106938
  21. [NM4] Noumi, M. and Mimachi, K.: Askey-Wilson polynomials and the quantum group SUq(2), Proc. Japan Acad. 66 (1990), 146-149. Zbl0707.33010MR1065793
  22. [NUW1] Noumi, M., Umeda, T. and Wakayama, M.: A quantum analogue of the Capelli identity and an elementary differential calculus on GLq(n), preprint J-Tokyo-Math91-16 (1991). Zbl0835.17013
  23. [NUW2] Noumi, M., Umeda, T. and Wakayama, M.: A quantum dual pair (sl2, On) and the associated Capelli identity, preprint (1993). 
  24. [NUW3] Noumi, M., Umeda, T. and Wakayama, M.: Dual pairs, spherical harmonics and a Capelli identity in quantum group theory, preprint (1993). Zbl0930.17012
  25. [NYM] Noumi, M., Yamada, H. and Mimachi, K.: Finite dimensional representations of the quantum group GLq(n;C) and the zonal spherical functions on U q(n - 1)(n), Japanese J. Math.19 (1993), 31-80. Zbl0806.17016MR1231510
  26. [P1] Podles, P.: Differential calculus on quantum spheres, Lett. Math. Phys.18 (1989), 107-119. Zbl0702.53073MR1010990
  27. [P2] Podles, P.: Quantum spheres, Lett. Math. Phys.14 (1987), 193-202. Zbl0634.46054MR919322
  28. [R] Rosso, M.: Finite representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys.117 (1988), 581-593. Zbl0651.17008MR953821
  29. [RTF] Reshetikhin, N. Yu., Takhtajan, L.A. and Faddeev, L.D.: Quantization of Lie groups and Lie algebras, Algebra and Analysis1 (1989), 178-206, English transl., in Leningrad Math. J.1 (1990), 193-225. Zbl0715.17015MR1015339
  30. [T] Tanisaki, T.: Killing forms, Harish-Chandra isomorphisms, and universal R-matrices for quantum algebras, Int. J. Mod. Phys. A vol. 7 suppl. 1B (1992), 142-147. Zbl0870.17007MR1187582
  31. [V] Vilenkin, N.J.: Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs vol.22, 1968. Zbl0172.18404MR229863
  32. [W1] Woronowicz, S.L.: Compact matrix pseudogroups, Comm. Math. Phys.111 (1987), 613-665. Zbl0627.58034MR901157
  33. [W2] Woronowicz, S.L.: Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys.122 (1980), 125-170. Zbl0751.58042MR994499
  34. [WSW] Watamura, U.C., Schlieker, M. and Watamura, S.: SOq(N) covariant differential calculus on quantum spaces and quantum deformation of Schrödinger equation, Zeitschrift für Physik C - Particles and Fields49 (1991), 439-446. MR1097174

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.