Index of a Lie algebra explained

In algebra, let g be a Lie algebra over a field K. Let further

\xi\in ak{g} ^{*} 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 \operatorname{ind} ak{g}:=min\limits ^{*}} _{\xi\in ak{g} \dim ak{g} _{\xi.}

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

K _{\xi\colon} ak{g ⊗ g}\to 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.^[1]

Notes and References

1. 10.1017/S0305004102006230 . 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 . 134 . 1 . 41–59. 13138268 .