Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the sum of the index of a parabolic subalgebra and of its nilpotent
  radical   In this short note, we investigate the following question of Panyushev : ``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
Millet-Fauquant and Joseph, 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 Ra{\"\i}s.

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The categories of this paper are: eess
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: cs
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The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph