index of a Lie algebra

Let $\\mathfrak{q}$ be a Lie algebra over $\\mathbb{K}$ and $\\mathfrak{q}^{*}$ its vector space dual. For $\\xi\\in\\mathfrak{q}^{*}$ let $\\mathfrak{q}_{\\xi}$ denote the stabilizer of $\\xi$ with respect to the co-adjoint representation.

The index of $\\mathfrak{q}$ is defined to be

\\operatorname{ind\\,}\\mathfrak{q}:=\\min\\limits_{\\xi\\in\\mathfrak{g}^{*}}\\dim%\\mathfrak{q}_{\\xi}

Examples

1.
If $\\mathfrak{q}$ is reductive then $\\operatorname{ind\\,}\\mathfrak{q}=\\operatorname{rank\\,}\\mathfrak{q}$ . Indeed, $\\mathfrak{q}$ and $\\mathfrak{q}^{*}$ are isomorphic asrepresentations for $\\mathfrak{q}$ and so the index is the minimal dimension among stabilizers of elements in $\\mathfrak{q}$ . In particular the minimum is realized in the stabilizer of any regular element of $\\mathfrak{q}$ . These elemtents have stabilizer dimension equal to the rank of $\\mathfrak{q}$ .
2.
If $\\operatorname{ind\\,}\\mathfrak{q}=0$ then $\\mathfrak{q}$ is called aFrobenius Lie algebra. This is equivalent to condition thatthe Kirillov form $K_{\\xi}\\colon\\mathfrak{q}\\times\\mathfrak{q}\\to\\mathbb{K}$ given by $(X,Y)\\mapsto\\xi([X,Y])$ is non-singular for some $\\xi\\in\\mathfrak{q}^{*}$ . Another equivalent condition when $\\mathfrak{q}$ is the Lie algebra of an algebraic group $Q$ is that $\\mathfrak{q}$ is Frobenius if and only if $Q$ has an open orbit on $\\mathfrak{q}^{*}$ .