Index of a Lie algebra

In 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

: $\operatorname{ind}\mathfrak{g}:=\min\limits_{\xi\in\mathfrak{g}^*} \dim\mathfrak{g}_\xi.$

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.

=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.{{cite journal|doi=10.1017/S0305004102006230 |last=Panyushev |first= Dmitri I. |year=2003 |title=The index of a Lie algebra, the centralizer of a nilpotent element, and the normalizer of the centralizer |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=134 |number=1 |pages=41–59|s2cid=13138268 }}

References

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Category:Lie algebras