Duke Mathematical Journal

A quantum analogue of the Capelli identity and an elementary differential calculus on $GL_q(n)$

Masatoshi Noumi, Tôru Umeda, and Masato Wakayama

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J. Volume 76, Number 2 (1994), 567-594.

First available in Project Euclid: 20 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
Secondary: 16W30 33D80: Connections with quantum groups, Chevalley groups, $p$-adic groups, Hecke algebras, and related topics 39A10: Difference equations, additive 58B30 81R50: Quantum groups and related algebraic methods [See also 16T20, 17B37]


Noumi, Masatoshi; Umeda, Tôru; Wakayama, Masato. A quantum analogue of the Capelli identity and an elementary differential calculus on G L q ( n ) . Duke Math. J. 76 (1994), no. 2, 567--594. doi:10.1215/S0012-7094-94-07620-5. http://projecteuclid.org/euclid.dmj/1077286975.

Export citation


  • [A] E. Abe, Hopf algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press, Cambridge, 1980.
  • [AF] A. Yu. Alekseev and L. D. Faddeev, $(T\sp *G)\sb t$: a toy model for conformal field theory, Comm. Math. Phys. 141 (1991), no. 2, 413–422.
  • [B] A. Borel, Hermann Weyl and Lie groups, Hermann Weyl, 1885–1985 ed. K. Chandrasekharan, Eidgenössische Tech. Hochschule, Zürich, 1986, pp. 53–82.
  • [Ca1] A. Capelli, Über die Zurückführung der Cayley'schen Operation $\Omega$ auf gewöhnliche Polar-Operationen, Math. Ann. 29 (1887), 331–338.
  • [Ca2] A. Capelli, Ricerca delle operazioni invariantive fra piu serie di variabili permutabili con ogni altra operazione invariantiva fra le stesse serie, Atti Scienze Fis. Mat. di Napoli (2) I (1888), 1–17.
  • [Ca3] A. Capelli, Sur les opérations dans la théorie des formes algébriques, Math. Ann. 37 (1890), 1–37.
  • [Ca4] A. Capelli, Lezioni sulla Theoria delle Forme Algebriche, Pellerano, Napoli, 1902.
  • [D] V. G. Drinfel'd, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798–820.
  • [Ha] T. Hayashi, $q$-analogues of Clifford and Weyl algebras—spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990), no. 1, 129–144.
  • [HiW] T. Hibi and M. Wakayama, A $q$-analogue of Capelli's identity for $GL(2)$, Adv. Math., to appear.
  • [Ho]1 R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570.
  • [Ho]2 R. Howe, Erratum to: “Remarks on classical invariant theory”, Trans. Amer. Math. Soc. 318 (1990), no. 2, 823.
  • [HoU] R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), no. 3, 565–619.
  • [Ja] F. H. Jackson, On $q$-functions and a certain difference operator, Trans. Roy. Soc. Edinburg. 46 (1908), 253–281.
  • [J1] M. Jimbo, A $q$-analogue of $U(\germ g\germ l(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252.
  • [J2] M. Jimbo, Quantum $R$ matrix for the generalized Toda system, Comm. Math. Phys. 102 (1986), no. 4, 537–547.
  • [KS] P. P. Kulish and E. K. Sklyanin, Solutions of the Yang-Baxter equation, J. Soviet Math. 19 (1982), 1596–1620.
  • [Ma] Yu. I. Manin, Quantum groups and noncommutative geometry, Université de Montréal Centre de Recherches Mathématiques, Montreal, QC, 1988.
  • [My] F. Meyer, Bericht über den gegenwärtingen Stand der Invariantentheorie, Jahresber. d. Deutschen Mathem.-Vereinigung 1 (1892), 79–292.
  • [Na] M. L. Nazarov, Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys. 21 (1991), no. 2, 123–131.
  • [N] M. Noumi, Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math., to appear.
  • [NYM] M. Noumi, H. Yamada, and K. Mimachi, Finite-dimensional representations of the quantum group $\rm GL\sb q(n;\bf C)$ and the zonal spherical functions on $\rm U\sb q(n-1)\backslash\rm U\sb q(n)$, Japan. J. Math. (N.S.) 19 (1993), no. 1, 31–80.
  • [RTF] N. Yu. Reshetikhin, L. A. Takhtadzyan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193–225.
  • [Sw] M. E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.
  • [Tu1] H. W. Turnbull, The theory of determinants, matrices, and invariants, 3rd ed, Dover Publications Inc., New York, 1960.
  • [Tu2] H. W. Turnbull, Symmetric determinants and the Cayley and Capelli operators, Proc. Edinburgh Math. Soc. (2) 8 (1948), 76–86.
  • [Wy] H. Weyl, The Classical Groups, their Invariants and Representations, Princeton University Press, Princeton, 1946.