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 The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso.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.
Une algèbre de Lie de dimension finie est dite quasi-réductive si elle possède une forme linéaire dont le stablisateur pour la représentation coadjointe, modulo le centre, est une algèbre de Lie réductive avec un centre formé d’éléments semi-simples. Les sous-algèbres paraboliques d’une algèbre de Lie semi-simple ne sont pas toujours quasi-réductives (sauf en types A ou C d’après un résultat de Panyushev). Récemment, Duflo, Khalgui and Torasso ont terminé la classification des sous-algèbres paraboliques quasi-réductives dans le cas classique. Dans cet article nous étudions la quasi-réductivité des sous-algèbres biparaboliques des algèbres de Lie réductives. Les sous-algèbres biparaboliques sont les intersections de deux sous-algèbres paraboliques dont la somme est l’algèbre de Lie ambiante. Notre principal résultat est la complétion de la classification des sous-algèbres paraboliques quasi-réductives des algèbres de Lie réductives.
Keywords: Reductive Lie algebras, quasi-reductive Lie algebras, index, biparabolic Lie algebras, seaweed algebras, regular linear forms
Mot clés : algèbres de Lie réductives, algèbres de Lie quasi-réductives, algèbres de Lie biparaboliques, formes linéaires régulières
@article{AIF_2011__61_2_417_0, author = {Baur, Karin and Moreau, Anne}, title = {Quasi-reductive (bi)parabolic subalgebras in reductive {Lie} algebras.}, journal = {Annales de l'Institut Fourier}, pages = {417--451}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {2}, year = {2011}, doi = {10.5802/aif.2619}, zbl = {1246.17010}, mrnumber = {2895063}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2619/} }
TY - JOUR AU - Baur, Karin AU - Moreau, Anne TI - Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras. JO - Annales de l'Institut Fourier PY - 2011 SP - 417 EP - 451 VL - 61 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2619/ DO - 10.5802/aif.2619 LA - en ID - AIF_2011__61_2_417_0 ER -
%0 Journal Article %A Baur, Karin %A Moreau, Anne %T Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras. %J Annales de l'Institut Fourier %D 2011 %P 417-451 %V 61 %N 2 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2619/ %R 10.5802/aif.2619 %G en %F AIF_2011__61_2_417_0
Baur, Karin; Moreau, Anne. Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras.. Annales de l'Institut Fourier, Volume 61 (2011) no. 2, pp. 417-451. doi : 10.5802/aif.2619. http://www.numdam.org/articles/10.5802/aif.2619/
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