MR2427540 Tôru UMEDA Tôru UMEDA On the Proof of the Capelli Identities Funkcialaj Ekvacioj. Serio Internacia 51 2008 1--15 http://fe.math.kobe-u.ac.jp/FE/FullPapers/51-1/51_1.pdf http://www.ams.org/mathscinet-getitem?mr=MR2427540 Using exterior calculus, we present detailed proofs of the classical Capelli identities in a purely computational manner. The proof of the fact that the Capelli elements are central is also given in a similar way. In the course of proofs of these two facts, one can easily see the mechanism of the multiplication formula of determinant with non-commutative entries. Simple treatments of the related facts from the Appendix of [HU] are also given. Center of universal enveloping algebra, Capelli identity. 17B35, 15A33. 51-1 2008 On the Proof of the Capelli Identities Tôru UMEDA Tôru UMEDA
1 Capelli, A. Über die Zurückführung der Cayley'schen Operation $\Omega$ auf gewöhnliche Polar-Operationen Math. Ann. 29 1887 331--338 MR1510419
2 Capelli, A. Ricerca delle operazioni invariantive fra piu serie di variabili permutabili con ogni altra operazione invariantiva fra le stesse serie Atti delle Scinze Fis. e Mat. di Napoli (2) I 1888 1--17
3 Capelli, A. Sur les opérations dans la théorie des formes algébriques Math. Ann. 37 1890 1--37 MR1510639
4 Howe, R. Remarks on classical invariant theory Trans. Amer. Math. Soc. 313 1989 539--570. Erratum Trans. Amer. Math. Soc., 318 (1990), 823. MR0986027
5 Howe, R.; Umeda, T. The Capelli identity, the double commutant theorem, and multiplicity-free actions Math. Ann. 290 1991 565--619 MR1116239
6 Itoh, M. Capelli elements for the orthogonal Lie algebras J. Lie Theory 10 2000 463--489 MR1774874
7 Itoh, M. A Cayley-Hamilton theorem for the skew Capelli elements J. Algebra 242 2001 740--761 MR1848969
8 Itoh, M. Capelli identities for the dual pair $({\roman{O}_M, \roman{Sp}_N)$ Math. Z. 246 2004 125--154 MR2031449
9 Itoh, M. Capelli identities for the reductive dual pairs Adv. Math. 194 2005 345--397 MR2139918
10 Itoh, M., Two determinants in the universal enveloping algebras of the orthogonal Lie algebras, preprint (2006) MR2331772 11 Itoh, M., Two permanents in the universal enveloping algebras of the symplectic Lie algebras, preprint (2006)
12 Itoh, M.; Umeda, T. On central elements in the universal enveloping algebras of the orthogonal Lie algebras Compositio Math. 127 2001 333--359 MR1845042
13 Koszul, J-L. Les algèbre de Lie graduée de type ${\frak s}{\frak l}(n,1)$ et l'opérateur de A. Capelli C. R. Acad. Sc. Paris 292 1981 139--141 MR0610304
14 Meyer, F. Bericht über den gegenwärtingen Stand der Invariantentheorie Jber. d. Dt. Math.-Verein 1 1892 79--292
15 Molev, A.; Nazarov, M.; Olshanskii, G. Yangians and classical Lie algebras Russian Math. Surveys 51 1996 205--282 MR1401535
16 Nazarov, M. Quantum Berezinian and the classical Capelli identity Lett. Math. Phys. 21 1991 123--131 MR1093523
17 Noumi, M.; Umeda, T.; Wakayama, M. A quantum analogue of the Capelli identity and an elementary differential calculus on $GL_q(n)$ Duke Math. J. 76 1994 567--594 MR1302325
18 Umeda, T. The Capelli identities, a century after Sugaku 46 1994 206--227; (in Japanese); English transl. in "Selected Papers on Harmonic Analysis, Groups, and Invariants", AMS Translations, Series 2, vol. 183 (1998), pp. 51--78, ed. by K. Nomizu MR1300359
19 Umeda, T. Newton's formula for $\frak{gl}_n$ Proc. Amer. Math. Soc. 126 1998 3169--3175 MR1468206
20 Umeda, T. On Turnbull identity for skew-symmetric matrices Proc. Edinburgh. Math. Soc. (2) 43 2000 379--393 MR1762897
21 Umeda, T. Application of Koszul complex to Wronski relations for $U(\frak{gl}_n)$ Comment. Math. Helv. 73 2003 663--680 MR2016689
22 Wachi, A. Capelli type identities on certain scalar generalized Verma modules J. Math. Kyoto Univ. 40 2000 705--727 MR1802842
23 Wachi, A. Capelli type identities on certain scalar generalized Verma modules II J. Math. Soc. Japan 56 2004 447--473 MR2048468
24 Wachi, A. Central elements in the universal enveloping algebaras for the split realization of the orthogonal Lie algebras Lett. Math. Phys. 77 2006 155--168 MR2251303
25 Weyl, H. The Classical Groups, their Invariants and Representations Princeton Univ. Press 1946 MR0000255 26 Želobenko, D. P. Compact Lie Groups and their Representations Transl. Math. Monographs 40, Amer. Math. Soc. 1973 MR0473098