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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the sum of the index of a parabolic subalgebra and of its nilpotent radical


Author: Rupert W. T. Yu
Journal: Proc. Amer. Math. Soc. 136 (2008), 1515-1522
MSC (2000): Primary 17B20
Published electronically: January 9, 2008
MathSciNet review: 2373578
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Abstract: In this short note, we investigate the following question of Panyushev stated in 2003: ``Is the sum of the index of a parabolic subalgebra of a semisimple Lie algebra $ \mathfrak{g}$ and the index of its nilpotent radical always greater than or equal to the rank of $ \mathfrak{g}$?'' Using the formula for the index of parabolic subalgebras conjectured by Tauvel and the author and proved by Fauquant-Millet and Joseph in 2005 and Joseph in 2006, we give a positive answer to this question. Moreover, we also obtain a necessary and sufficient condition for this sum to be equal to the rank of $ \mathfrak{g}$. This provides new examples of direct sum decomposition of a semisimple Lie algebra verifying the ``index additivity condition'' as stated by Raïs.


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Additional Information

Rupert W. T. Yu
Affiliation: UMR 6086 du C.N.R.S., Département de Mathématiques, Université de Poitiers, Téléport 2 – BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil Cedex, France
Email: yuyu@math.univ-poitiers.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09234-4
PII: S 0002-9939(08)09234-4
Received by editor(s): August 18, 2006
Published electronically: January 9, 2008
Communicated by: Dan M. Barbasch
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.