logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Next article
Dmitri I. Panyushev
An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras
(Extension d'un théorème de Rais et sous-algèbres de Lie simples)
Annales de l'institut Fourier, 55 no. 3 (2005), p. 693-715, doi: 10.5802/aif.2110
Article PDF | Reviews MR 2149399 | Zbl 02171521 | 1 citation in Cedram
Class. Math.: 17B20, 17B70, 14L30
Keywords: field of invariants, generic stabiliser, simple Lie algebra, seaweed subalgebra

Résumé - Abstract

We prove an extension of Rais' theorem on the coadjoint representation of certain graded Lie algebras. As an application, we prove that, for the coadjoint representation of any seaweed subalgebra in a general linear or symplectic Lie algebra, there is a generic stabiliser and the field of invariants is rational. It is also shown that if the highest root of a simple Lie algerba is not fundamental, then there is a parabolic subalgebra whose coadjoint representation do not have a generic stabiliser.

Bibliography

[1] A.A. Arkhangel'skii, “Completely integrable Hamiltonian systems on the group of triangular matrices”, Math. USSR-Sb. 36 (1980), p. 127-134 Article |  Zbl 0433.58015
[2] A.G. Elashvili, “Canonical form and stationary subalgebras of points of general position for simple linear Lie groups”, Funct. Anal. Appl. 6 (1972), p. 44-53 Article |  Zbl 0252.22015
[3] Y. Kosmann & S. Sternberg, “Conjugaison des sous-algèbres d'isotropie”, C. R. Acad. Sci. Paris. Sér. A 279 (1974), p. 777-779  MR 354794 |  Zbl 0297.22013
[4] M.I. Gekhtman & M.Z. Shapiro, “Noncommutative and commutative integrability of generic Toda flows in simple Lie algebras”, Comm. Pure Appl. Math. 52 (1999), p. 53-84 Article |  MR 1648421 |  Zbl 0937.37045
[5] V. Dergachev & A.A. Kirillov, “Index of Lie algebras of seaweed type”, J. Lie Theory 10 (2000), p. 331-343  MR 1774864 |  Zbl 0980.17001
[6] A. Dvorsky, “Index of parabolic and seaweed subalgebras of ${\goth s}{\goth o}_n$”, Lin. Alg. Appl. 374 (2003), p. 127-142 Article |  MR 2008784 |  Zbl 1056.17009
[7] A. Joseph, “A preparation theorem for the prime spectrum of a semisimple Lie algebra”, J. Alg. 48 (1977), p. 241-289 Article |  MR 453829 |  Zbl 0405.17007
[8] D. Panyushev, “Inductive formulas for the index of seaweed Lie algebras”, Moscow Math. J. 1 (2001), p. 221-241  MR 1878277 |  Zbl 0998.17008
[9] D. Panyushev, “The index of a Lie algebra, the centralizer of a nilpotent element, and the normalizer of the centralizer”, Math. Proc. Camb. Phil. Soc. 134 (2003), p. 41-59  MR 1937791 |  Zbl 1041.17022
[10] M. Raïs, “L'indice des produits semi-directs $E\times_{\rho}{\goth g}$”, C.R. Acad. Sc. Paris, Ser. A 287 (1978), p. 195-197  MR 506502 |  Zbl 0387.17002
[11] R.W. Richardson, “Principal orbit types for algebraic transformation spaces in characteristic zero”, Invent. Math. 16 (1972), p. 6-14 Article |  MR 294336 |  Zbl 0242.14010
[12] P. Tauvel & R. Yu, “Indice et formes linéaires stables dans les algèbres de Lie”, J. Alg. 273 (2004), p. 507-516 Article |  MR 2037708 |  Zbl 02055680
[13] P. Tauvel & R. Yu, “Sur l'indice de certaines algèbres de Lie”, Ann. Inst. Fourier 54 (2004) no. 6, p. 1793-1810 Cedram |  MR 2134224 |  Zbl 02162441
[14] E.B. Vinberg & V.L. Popov, Invariant theory, Encyclopaedia Math. Sci., Springer, 1994, p. 123-284  Zbl 0789.14008
top