Index of a Lie algebra

Let g be a Lie algebra over a field K. Let further $\xi \in {\mathfrak {g}}^{*}$ 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

$\mathrm {ind} \,{\mathfrak {g}}:=\min \limits _{\xi \in {\mathfrak {g}}^{*}}\mathrm {dim} \,{\mathfrak {g}}_{\xi }.$

Examples

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 $K_{\xi }\colon {\mathfrak {g\otimes g}}\to \mathbb {K} :(X,Y)\mapsto \xi ([X,Y])$ 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.

References

This article incorporates material from index of a Lie algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.