Index of a Lie algebra
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|Group theory → Lie groups
Let g be a Lie algebra over a field K. Let further 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
Reductive Lie algebras
If g is reductive then ind g = rk 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 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.