Index of a Lie algebra

In algebra, let g be a Lie algebra over a field K. Let further $$\xi\in\mathfrak{g}^*$$ be a one-form on g. The stabilizer gξ of ξ is the Lie subalgebra of elements of g that annihilate ξ in the coadjoint representation. The index of the Lie algebra is
 * $$\operatorname{ind}\mathfrak{g}:=\min\limits_{\xi\in\mathfrak{g}^*} \dim\mathfrak{g}_\xi.$$

Reductive Lie algebras
If g is reductive then the index of g is also the rank of g, because the adjoint and coadjoint representation are isomorphic and rk g is the minimal dimension of a stabilizer of an element in g. This is actually the dimension of the stabilizer of any regular element in g.

Frobenius Lie algebra
If ind g = 0, then g is called Frobenius Lie algebra. This is equivalent to the fact that the Kirillov form $$K_\xi\colon \mathfrak{g\otimes g}\to \mathbb{K}:(X,Y)\mapsto \xi([X,Y])$$ is non-singular for some ξ in g*. Another equivalent condition when g is the Lie algebra of an algebraic group G, is that g is Frobenius if and only if G has an open orbit in g* under the coadjoint representation.

Lie algebra of an algebraic group
If g is the Lie algebra of an algebraic group G, then the index of g is the transcendence degree of the field of rational functions on g* that are invariant under the (co)adjoint action of G.