Regular functionals on seaweed Lie algebras

Abstract

The index of a Lie algebra $\mathfrak{g}$ is defined by ind $\mathfrak{g}=$ ${\min }_{f\in {\mathfrak{g}}^{\ast }}\dim (\ker (B_f))$, where $f$ is an element of the linear dual ${\mathfrak{g}}^{\ast }$ 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.