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

Yu, Rupert

doi:10.1090/S0002-9939-08-09234-4

On the sum of the index of a parabolic subalgebra and of its nilpotent radical
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by Rupert W. T. Yu PDF

Proc. Amer. Math. Soc. 136 (2008), 1515-1522 Request permission

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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