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 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.

Notes

References

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