In algebra, let g be a Lie algebra over a field K. Let further formula_1 be a one-form on g. The stabilizer g"ξ" of "ξ" is the Lie subalgebra of elements of g that annihilate "ξ" in the coadjoint representation. The index of the Lie algebra is 
formula_2
Examples.
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 formula_3 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".
References.
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