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Patrice Tauvel; Rupert W.T. Yu
Sur l'indice de certaines algèbres de Lie
(On the index of certain Lie algebras)
Annales de l'institut Fourier, 54 no. 6 (2004), p. 1793-1810, doi: 10.5802/aif.2066
Article PDF | Reviews MR 2134224 | Zbl 02162441 | 1 citation in Cedram
Class. Math.: 17B05, 17B20
Keywords: semisimple lie algebras, seaweed lie algebras, index

Résumé - Abstract

We give an upper bound for the index of certain Lie algebras, called of seaweed type, introduced by V. Dergachev, A. Kirillov and D. Panyushev. We deduce from this a conjecture of D. Panyushev. We conjecture that our upper bound gives in fact a formula for the index of such Lie algebras, and we prove it in certain cases. Finally, we obtain a result concerning the index of parabolic subalgebras of a semisimple Lie algebra.

Bibliography

[1] D.V. Aleskseevsky & M. Perolomov 1, “Poisson and symplectic structures on Lie algebras. I”, J. Geom. Phys 22 (1987), p. 191-211  MR 1453001 |  Zbl 0874.22018
[2] N. Bourbaki, Groupes et algèbres de Lie Chap. 4-5-6, Masson, 1981  MR 647314 |  Zbl 0483.22001
[3] V. Dergachev & A. Kirillov, “Index of Lie algebras of seaweed type”, J. of Lie Theory 10 (2000), p. 331-343  MR 1774864 |  Zbl 0980.17001
[4] J. Dixmier, Enveloping algebras, Graduate Studies in Math. 11, AMS, 1996  MR 1393197 |  Zbl 0867.17001
[5] A.G. Elashvili, “On the index of parabolic subalgebras of semisimple Lie algebras”, Preprint, 1990
[6] J.C. Jantzen, Einhüllenden Algebren halbeinfacher Lie-Algebren, Ergebnisse der Mathematik and iher Grenzgebiete 3, Springer-Verlag, 1983  MR 721170 |  Zbl 0541.17001
[7] A. Joseph, “A preparation theorem for the prime spectrum of a semisimple Lie algebra”, J. of Algebra 48 (1977), p. 241-289  MR 453829 |  Zbl 0405.17007
[8] D. Panysushev, “Inductive formulas for the index of seaweed Lie algebras”, Moscow Math. Journal 1 (2001), p. 221-241  MR 1878277 |  Zbl 0998.17008
[9] P. Tauvel, “Introduction à la théorie des algèbres de Lie”, Diderot, 1998
[10] P. Tauvel & R.W.T. Yu, “Indice et formes linéaires stables dans les algèbres de Lie”, J. of Algebra 273 (2004), p. 507-516  MR 2037708 |  Zbl 02055680
[11] V.V. Trofimov, “Semi-invariants of the coadjoint representation of Borel subalgebras of semisimple Lie algebras”, Selecta Math. Sovietica 8 (1989), p. 31-56  Zbl 0659.17009
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