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

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

Masatoshi Noumi, Tôru Umeda, and Masato Wakayama

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

First Page: Show

Primary Subjects: 17B37

Secondary Subjects: 16W30, 33D80, 39A10, 58B30, 81R50

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1077286975
Mathematical Reviews number (MathSciNet): MR1302325
Zentralblatt MATH identifier: 0835.17013
Digital Object Identifier: doi:10.1215/S0012-7094-94-07620-5

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References

[A] E. Abe, Hopf algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press, Cambridge, 1980.

Mathematical Reviews (MathSciNet): MR83a:16010

Zentralblatt MATH: 0476.16008

[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.

Mathematical Reviews (MathSciNet): MR93g:81138

Zentralblatt MATH: 0767.17024

Digital Object Identifier: doi:10.1007/BF02101512

Project Euclid: euclid.cmp/1104248306

[B] A. Borel, Hermann Weyl and Lie groups, Hermann Weyl, 1885–1985 ed. K. Chandrasekharan, Eidgenössische Tech. Hochschule, Zürich, 1986, pp. 53–82.

Mathematical Reviews (MathSciNet): MR875268

Zentralblatt MATH: 0599.01014

[Ca1] A. Capelli, Über die Zurückführung der Cayley'schen Operation $\Omega$ auf gewöhnliche Polar-Operationen, Math. Ann. 29 (1887), 331–338.

Zentralblatt MATH: 19.0151.01

[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.

Zentralblatt MATH: 20.0093.01

[Ca3] A. Capelli, Sur les opérations dans la théorie des formes algébriques, Math. Ann. 37 (1890), 1–37.

Zentralblatt MATH: 22.0137.01

[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.

Mathematical Reviews (MathSciNet): MR89f:17017

Zentralblatt MATH: 0667.16003

[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.

Mathematical Reviews (MathSciNet): MR91a:17015

Zentralblatt MATH: 0701.17008

Digital Object Identifier: doi:10.1007/BF02096497

Project Euclid: euclid.cmp/1104180043

[HiW] T. Hibi and M. Wakayama, A $q$-analogue of Capelli's identity for $GL(2)$, Adv. Math., to appear.

Mathematical Reviews (MathSciNet): MR1351325

Zentralblatt MATH: 0937.17009

Digital Object Identifier: doi:10.1006/aima.1995.1049

[Ho]¹ R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570.

Mathematical Reviews (MathSciNet): MR90h:22015a

Zentralblatt MATH: 0674.15021

Digital Object Identifier: doi:10.2307/2001418

[Ho]² R. Howe, Erratum to: “Remarks on classical invariant theory”, Trans. Amer. Math. Soc. 318 (1990), no. 2, 823.

Mathematical Reviews (MathSciNet): MR90h:22015b

Zentralblatt MATH: 0726.15025

Digital Object Identifier: doi:10.2307/2001333

[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.

Mathematical Reviews (MathSciNet): MR92j:17004

Zentralblatt MATH: 0733.20019

Digital Object Identifier: doi:10.1007/BF01459261

[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.

Mathematical Reviews (MathSciNet): MR87k:17011

Zentralblatt MATH: 0602.17005

Digital Object Identifier: doi:10.1007/BF00400222

[J2] M. Jimbo, Quantum $R$ matrix for the generalized Toda system, Comm. Math. Phys. 102 (1986), no. 4, 537–547.

Mathematical Reviews (MathSciNet): MR87h:58086

Zentralblatt MATH: 0604.58013

Digital Object Identifier: doi:10.1007/BF01221646

Project Euclid: euclid.cmp/1104114539

[KS] P. P. Kulish and E. K. Sklyanin, Solutions of the Yang-Baxter equation, J. Soviet Math. 19 (1982), 1596–1620.

Zentralblatt MATH: 0553.58039

[Ma] Yu. I. Manin, Quantum groups and noncommutative geometry, Université de Montréal Centre de Recherches Mathématiques, Montreal, QC, 1988.

Mathematical Reviews (MathSciNet): MR91e:17001

Zentralblatt MATH: 0724.17006

[My] F. Meyer, Bericht über den gegenwärtingen Stand der Invariantentheorie, Jahresber. d. Deutschen Mathem.-Vereinigung 1 (1892), 79–292.

Zentralblatt MATH: 24.0045.01

[Na] M. L. Nazarov, Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys. 21 (1991), no. 2, 123–131.

Mathematical Reviews (MathSciNet): MR92b:17020

Zentralblatt MATH: 0722.17004

Digital Object Identifier: doi:10.1007/BF00401646

[N] M. Noumi, Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math., to appear.

Mathematical Reviews (MathSciNet): MR1413836

Digital Object Identifier: doi:10.1006/aima.1996.0066

[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.

Mathematical Reviews (MathSciNet): MR94i:17023

Zentralblatt MATH: 0806.17016

[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.

Zentralblatt MATH: 0715.17015

Mathematical Reviews (MathSciNet): MR1015339

[Sw] M. E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.

Mathematical Reviews (MathSciNet): MR40:5705

Zentralblatt MATH: 0194.32901

[Tu1] H. W. Turnbull, The theory of determinants, matrices, and invariants, 3rd ed, Dover Publications Inc., New York, 1960.

Mathematical Reviews (MathSciNet): MR24:A123

Zentralblatt MATH: 0103.00702

[Tu2] H. W. Turnbull, Symmetric determinants and the Cayley and Capelli operators, Proc. Edinburgh Math. Soc. (2) 8 (1948), 76–86.

Mathematical Reviews (MathSciNet): MR10,347i

Zentralblatt MATH: 0036.15002

Digital Object Identifier: doi:10.1017/S0013091500024822

[Wy] H. Weyl, The Classical Groups, their Invariants and Representations, Princeton University Press, Princeton, 1946.

Zentralblatt MATH: 0020.20601

Mathematical Reviews (MathSciNet): MR1488158

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