Index of a Lie algebra

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 ξ : g ⊗ g → K : ( X , Y ) ↦ ξ ( [ 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.