The index of a Lie algebra $\mathfrak{ g}$ is defined by ind $\mathfrak{ g}=$ $\min_{ f\in \mathfrak{ g}^*}\dim(\ker (B_f))$, where $f$ is an element of the linear dual $\mathfrak{ g}^*$ and $B_f(x,y)=f([x,y])$ is the associated skew-symmetric Kirillov form. We develop a broad general framework for the explicit construction of regular (index realizing) functionals for seaweed subalgebras of $\mathfrak{ gl}(n)$ and the classical Lie algebras: $A_n=\mathfrak{ sl}(n+1),$ $B_n=\mathfrak{ so}(2n+1)$, and $C_n=\mathfrak{ sp}(2n)$. Until now, this problem has remained open in $\mathfrak{ gl}(n)$ -- and in all the classical types.
Categories: math.RA