Proceedings of the American Mathematical Society

Author(s): Tôru Umeda
Journal: Proc. Amer. Math. Soc. 126 (1998), 3169-3175.
MSC (1991): Primary 17B35, 15A33
Retrieve article in: PDF
This article is available free of charge

Abstract: This paper presents an explicit relation between the two sets which are well-known generators of the center of the universal enveloping algebra $U(\mathfrak{gl}_{n})$ of the Lie algebra $\mathfrak{gl}_{n}$ : one by Capelli (1890) and the other by Gelfand (1950). Our formula is motivated to give an exact analogy for the classical Newton's formula connecting the elementary symmetric functions and the power sum symmetric functions. The formula itself can be deduced from a more general result on Yangians obtained by Nazarov. Our proof is elementary and has an advantage in its direct accessibility.

[Ca]: A. Capelli, Sur les opérations dans la théorie des formes algébriques, Math. Ann. 37 (1890), 1-37.
[Ge]: I.M. Gelfand, Center of the infinitesimal groups, Mat. Sb. Nov. Ser. 26 (68) (1950), 103-112; English transl. in ``Collected Papers" Vol. II, pp.22-30. MR 11:498a
[GKLLRT]: I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh and J.-Y. Thibon, Noncommutaive symmetric functions, Adv. Math. 112 (1995), 218-348. MR 96e:05175
[H]: R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539-570; Erratum, Trans. Amer. Math. Soc. 318 (1990), 823. MR 90h:22015b
[HU]: R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), 565-619. MR 92j:17004
[I]: M. Itoh, Master's thesis at Kyoto University, 1997 Feb..
[MNO]: A. Molev, M. Nazarov and G. Olshanskii, Yangians and classical Lie algebras, Russian Math. Surveys 51 (1996), 27-104. MR 97f:17019
[Na]: M. Nazarov, Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys. 21 (1991), 123-131. MR 92b:17020
[NaTa]: M. Nazarov and V. Tarasov, Yangians and Gel'fand-Zetlin bases, Publ. RIMS, Kyoto Univ. 30 (1994), 459-478. MR 96m:17033
[Ne]: E. Netto, Vorlesungen über Algebra, Leibzig, 1896.
[N]: I. Newton, Arithmetica universalis (Lugd, ed.), 1707.
[NUW]: M. Noumi, T. Umeda and M. Wakayama, A quantum analogue of the Capelli identity and an elementary differential calculas on $GL_{q}(n)$ , Duke Math. J. 76 (1994), 567-594. MR 95h:17019
[O]: H. Ochiai, Harish-Chandra isomorphism for $\mathfrak{gl}_{n}$ , private notes (1996).
[PP1]: A.M. Perelomov and V.S. Popov, Casimir operators for $\mathrm{U}(n)$ and $\mathrm{SU}(n)$ , Soviet J. Nuclear Phys. 3 (1966), 676-680. MR 34:5446
[PP2]: -, Casimir operators for the orthogonal and symplectic groups, Soviet J. Nuclear Phys. 3 (1966), 819-824. MR 34:5447
[W]: H. Weyl, The Classical Groups, their Invariants and Representations, Princeton Univ. Press, 1946.
[Z]: D.P. \v{Z}elobenko, Compact Lie Groups and their Representations, Transl. Math. Monographs 40 (Amer. Math. Soc.), 1973. MR 57:12776b

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 17B35, 15A33

Tôru Umeda
Affiliation: Department of Mathematics, Kyoto University, Kyoto 606, Japan
Email: umeda@kusm.kyoto-u.ac.jp

DOI: 10.1090/S0002-9939-98-04557-2
PII: S 0002-9939(98)04557-2
Keywords: Center of universal enveloping algebra, Newton's formula, Hamilton-Cayley theorem
Received by editor(s): March 28, 1997
Dedicated: Dedicated to Professor Reiji Takahashi on the occasion of his seventieth birthday
Communicated by: Roe Goodman
Copyright of article: Copyright 1998, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS