On the index of certain Lie algebras. (Q1774079)

Translation of the authors' summary (in French): The authors give an upper bound for the index of certain Lie algebras (seaweed subalgebra) introduced by\textit{V. Dergachev} and \textit{A. Kirillov} [J. Lie Theory 10, 331--343 (2000; Zbl 0980.17001)] and \textit{D. Panyushev} [Mosc. Math. J. 1, No. 2, 221--241 (2001; Zbl 0998.17008)]. From this they deduce the proof of a conjecture of D. Panyushev. They also formulate a conjecture concerning the index of these algebras and give a proof in a special case. Finally we state a result on the index of parabolic subalgebras of a semisimple Lie algebra.