Inductive formulas for the index of seaweed Lie algebras

A seaweed subalgebra of a semisimple Lie algebra $\mathfrak{g}$ is a generalization of the notion of parabolic subalgebra. In the case $\mathfrak{g}=\mathfrak{sl}(V)$, seaweed subalgebras were recently introduced by Dergachev and Kirillov. We give an inductive procedure for computing the index of seaweed subalgebras of classical Lie algebras. This allows us to prove that the index of any seaweed in $\mathfrak{sl}(V)$ or $\mathfrak{sp}(V)$ is at most the rank of $\mathfrak{g}$. For $\mathfrak{so}(V)$, the problem is reduced to the case of parabolic subalgebras.