# Index of a Lie algebra

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In algebra, let **g** be a Lie algebra over a field **K**. Let further 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

## Examples[edit]

### Reductive Lie algebras[edit]

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[edit]

If ind **g**=0, then **g** is called *Frobenius Lie algebra*. This is equivalent to the fact that the Kirillov form 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[edit]

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 action of *G*.^{[1]}

## References[edit]

**^**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.

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