# Index of a Lie algebra

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

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

## 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 $\displaystyle{ K_\xi\colon \mathfrak{g\otimes g}\to \mathbb{K}:(X,Y)\mapsto \xi([X,Y]) }$ 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.[1]

## References

1. Panyushev, Dmitri I. (2003). "The index of a Lie algebra, the centralizer of a nilpotent element, and the normalizer of the centralizer". Mathematical Proceedings of the Cambridge Philosophical Society 134 (1): 41–59. doi:10.1017/S0305004102006230.