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Karin Baur; Anne Moreau
Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras.
(Sous-algèbres (bi)paraboliques quasi-réductives des algèbres de Lie réductives)
Annales de l'institut Fourier, 61 no. 2 (2011), p. 417-451, doi: 10.5802/aif.2619
Article PDF | Reviews MR 2895063 | Zbl 1246.17010
Class. Math.: 17B20, 17B45, 22E60
Keywords: Reductive Lie algebras, quasi-reductive Lie algebras, index, biparabolic Lie algebras, seaweed algebras, regular linear forms

Résumé - Abstract

We say that a finite dimensional Lie algebra is quasi-reductive if it has a linear form whose stabilizer for the coadjoint representation, modulo the center, is a reductive Lie algebra with a center consisting of semisimple elements. Parabolic subalgebras of a semisimple Lie algebra are not always quasi-reductive (except in types A or C by work of Panyushev). The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso. In this paper, we investigate the quasi-reductivity of biparabolic subalgebras of reductive Lie algebras. Biparabolic (or seaweed) subalgebras are the intersection of two parabolic subalgebras whose sum is the total Lie algebra. As a main result, we complete the classification of quasi-reductive parabolic subalgebras of reductive Lie algebras by considering the exceptional cases.

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