Dual pairs, spherical harmonics and a Capelli identity in quantum group theory

 Access to full text

 Full (PDF)

1. [Ab] Abe, E.: HopfAlgebras, Cambridge Math. Tracts 74, Cambridge Univ. Press, 1980. Zbl0476.16008MR594432
2. [B] Bergman, G.M.: The diamond lemma for ring theory, Adv. Math.29 (1978) 178-218. Zbl0326.16019MR506890
3. [D] Drinfel'd, V.G.: Quantum groups, in Proc. Int. Cong. Math. Berkeley, 1986, pp. 798-820. Zbl0667.16003MR934283
4. [F] Fischer, E.: Über algebraische Modulsysteme und lineare homogene partielle Differentialgleichungen mit konstaten Koeffizienten, J. Reine Angew. Math.140 (1911) 48-81. Zbl42.0148.01JFM42.0148.01
5. [GK] Gavrilik, A.M. and Klimyk, A.U.: q-Deformed orthogonal and pseudo-orthogonal algebras and their representations, Lett. Math. Phys.21 (1991) 215-220. Zbl0735.17020MR1102131
6. [Ha] Hayashi, T.: Q-analogues of Clifford and Weyl algebras - spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys.127 (1990) 129-144. Zbl0701.17008MR1036118
7. [HiW] Hibi, T. and Wakayama, M.: A q-analogue of Capelli's identity for GL q (2), Adv. in Math. (to appear). Zbl0937.17009
8. [H1] Howe, R.: Remarks on classical invariant theory, Trans. Amer. Math. Soc.313 (1989) 539-570; Erratum, Trans. Amer. Math. Soc.318 (1990) p. 823. Zbl0726.15025MR986027
9. [H2] Howe, R.: Dual pairs in physics: Harmonic oscillators, photons, electrons, and singletons, in Lectures in Applied Math. vol. 21, 1985, pp. 179-207. Zbl0558.22018MR789290
10. [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
11. [Ja] Jackson, F.H.: On q-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh46 (1908) 253-281. 
12. [J1] Jimbo, M.: A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys.10 (1985) 63-69. Zbl0587.17004MR797001
13. [J2] Jimbo, M.: A q-analogue of U(gt(N + 1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys.11 (1986) 247-252. Zbl0602.17005MR841713
14. [Na1] Nazarov, M.L.: Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys.21 (1991) 123-131. Zbl0722.17004MR1093523
15. [Na2] Nazarov, M.L.: private communication. 
16. [N1] Noumi, M.: A remark on semisimple elements in Uq (sl(2, C)), Combinatorial Aspects in Representation Theory and Geometry, RIMS Kôkyûroku765 (1991) 71-78. 
17. [N2] Noumi, M.: A realization of Macdonald's symmetric polynomials on quantum homogeneous spaces, in Proceedings of the 21st International Conference on Differential Geometric Methods in Theoretical Physics, Tianjin China, 1992. 
18. [N3] Noumi, M.: Quantum Grassmannians and q-hypergeometric series, CWI Quarterly5 (1992) 293-307. Zbl0782.33015MR1213744
19. [N4] Noumi, M.: Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, to appear, Adv. Math.. Zbl0874.33011MR1413836
20. [NUW1] Noumi, M., Umeda, T. and Wakayama, M.: A quantum analogue of the Capelli identity and an elementary differential calculas on GLq (n)Duke Math. J.76 (1994) 567-594. Zbl0835.17013MR1302325
21. [NUW2] Noumi, M., Umeda, T. and Wakayama, M.: A quantum dual pair (sl2, On) and the associated Capelli identity, Lett. Math. Phys.34 (1995) 1-8. Zbl0842.17020MR1334029
22. [O] Olshanski, G.I.: Twisted Yangians and infinite-dimensional classical Lie algebras, in Quantum Groups, Lecture Notes in Math. 1510, Springer Verlag (1992), pp. 103-120. Zbl0780.17025MR1183482
23. [RTF] Reshetikhin, N. Yu., Takhtadzyan, L.A. and Faddeev. L.D.: Quantization of the Lie groups and Lie algebras, Leningrad Math. J.1 (1990) 193-225. Zbl0715.17015MR1015339
24. [U] Umeda, T.: Notes on almost homogeneity, preprint (1993). 
25. [UW] Umeda, T. and Wakayama, M.: Another look at the differential operators on quantum matrix spaces and its applications, in preparation. Zbl0957.17023
26. [Wy] Weyl, H.: The Classical Groups, their Invariants and Representations, Princeton Univ. Press, 1946. Zbl1024.20501MR1488158