Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras.

Karin Baur[1]; Anne Moreau[2]

  • [1] ETH Zürich Departement Mathematik Rämistrasse 101 8092 Zürich (Switzerland)
  • [2] LMA Boulevard Marie et Pierre Curie 86962 Futuroscope Chasseneuil Cedex (France)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 2, page 417-451
  • ISSN: 0373-0956

Abstract

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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 The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso.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.

How to cite

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Baur, Karin, and Moreau, Anne. "Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras.." Annales de l’institut Fourier 61.2 (2011): 417-451. <http://eudml.org/doc/219741>.

@article{Baur2011,
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 The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso.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.},
affiliation = {ETH Zürich Departement Mathematik Rämistrasse 101 8092 Zürich (Switzerland); LMA Boulevard Marie et Pierre Curie 86962 Futuroscope Chasseneuil Cedex (France)},
author = {Baur, Karin, Moreau, Anne},
journal = {Annales de l’institut Fourier},
keywords = {Reductive Lie algebras; quasi-reductive Lie algebras; index; biparabolic Lie algebras; seaweed algebras; regular linear forms; reductive Lie algebras},
language = {eng},
number = {2},
pages = {417-451},
publisher = {Association des Annales de l’institut Fourier},
title = {Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras.},
url = {http://eudml.org/doc/219741},
volume = {61},
year = {2011},
}

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
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 2
SP - 417
EP - 451
AB - 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.
LA - eng
KW - Reductive Lie algebras; quasi-reductive Lie algebras; index; biparabolic Lie algebras; seaweed algebras; regular linear forms; reductive Lie algebras
UR - http://eudml.org/doc/219741
ER -

References

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