Index of a Lie algebra

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 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.

References

References

1. Panyushev, Dmitri I.. (2003). "The index of a Lie algebra, the centralizer of a nilpotent element, and the normalizer of the centralizer". [[Mathematical Proceedings of the Cambridge Philosophical Society]].