In algebra, let g be a Lie algebra over a field K. Let further ξ ∈ g ∗ {\displaystyle \xi \in {\mathfrak {g}}^{*}} be a one-form on g. The stabilizer gξ of ξ is the Lie subalgebra of elements of g that annihilate ξ in the coadjoint representation. The index of the Lie algebra is

ind ⁡ g := min ξ ∈ g ∗ dim ⁡ g ξ . {\displaystyle \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 ξ : g ⊗ g → K : ( X , Y ) ↦ ξ ( [ X , Y ] ) {\displaystyle 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.

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