index of a Lie algebra
Let $\U0001d52e$ be a Lie algebra^{} over $\mathbb{K}$ and ${\U0001d52e}^{*}$ its vector space dual. For $\xi \in {\U0001d52e}^{*}$ let ${\U0001d52e}_{\xi}$ denote the stabilizer^{} of $\xi $ with respect to the coadjoint representation^{}.
The index of $\U0001d52e$ is defined to be
$$\mathrm{ind}\U0001d52e:=\underset{\xi \in {\U0001d524}^{*}}{\mathrm{min}}dim{\U0001d52e}_{\xi}$$ 
Examples

1.
If $\U0001d52e$ is reductive then $\mathrm{ind}\U0001d52e=\mathrm{rank}\U0001d52e$. Indeed, $\U0001d52e$ and ${\U0001d52e}^{*}$ are isomorphic^{} as representations for $\U0001d52e$ and so the index is the minimal^{} dimension among stabilizers of elements in $\U0001d52e$. In particular the minimum is realized in the stabilizer of any regular^{} element of $\U0001d52e$. These elemtents have stabilizer dimension equal to the rank of $\U0001d52e$.

2.
If $\mathrm{ind}\U0001d52e=0$ then $\U0001d52e$ is called a Frobenius Lie algebra. This is equivalent^{} to condition that the Kirillov form ${K}_{\xi}:\U0001d52e\times \U0001d52e\to \mathbb{K}$ given by $(X,Y)\mapsto \xi ([X,Y])$ is nonsingular for some $\xi \in {\U0001d52e}^{*}$. Another equivalent condition when $\U0001d52e$ is the Lie algebra of an algebraic group $Q$ is that $\U0001d52e$ is Frobenius if and only if $Q$ has an open orbit on ${\U0001d52e}^{*}$.
Title  index of a Lie algebra 

Canonical name  IndexOfALieAlgebra 
Date of creation  20130322 15:30:47 
Last modified on  20130322 15:30:47 
Owner  benjaminfjones (879) 
Last modified by  benjaminfjones (879) 
Numerical id  6 
Author  benjaminfjones (879) 
Entry type  Definition 
Classification  msc 17B05 
Defines  index of a Lie algebra 
Defines  Frobenius Lie algebra 
Defines  Kirillov form 