# Computing the index of a Lie algebra: what is known beyond the reductive case?

Recall that an index of a Lie algebra $\mathfrak{g}$ is $\mathrm{ind}\ \mathfrak{g} := \min\limits_{\xi \in \mathfrak{g}^*} \dim \mathrm{Ann}_{\xi}$ where $\mathrm{Ann}_{\xi}=\{h\in\mathfrak{g}| \mathrm{ad}_h^*(\xi)=0\}$ is the annihilator (also known as the stabilizer) of $\xi$ with respect to the co-adjoint representation. The relevant Wikipedia article just says that if $\mathfrak{g}$ is reductive then $\mathrm{ind}\ \mathfrak{g}=\mathrm{rank}\ \mathfrak{g}$ but I would like to build some intuition for the non-reductive case, and my googling hasn't brought about any relevant references so far. In particular, I would very much like to know:

1. Can one say anything about the index of a solvable Lie algebra?

2. What about the index of a semidirect sum (rather than the direct sum which occurs in the reductive case) $\mathfrak{g}=\mathfrak{h}\triangleright\mathfrak{a}$, where $\mathfrak{a}$ is abelian and $\mathfrak{h}$ is arbitrary? If something is known for semisimple $\mathfrak{h}$, that would be of interest too.

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Index is a very crude invariant, it is only the first step towards a description of the coadjoint orbits, whose behavior under standard constructions such as semidirect product is more susceptible to analysis. You can find a lot of information about orbits in the solvable case in the papers on the orbit method. –  Victor Protsak Aug 30 '10 at 4:37
Thanks, Victor, I'll try to look them up. –  mathphysicist Aug 30 '10 at 14:08

There is quite a bit of literature by now, in the classical characteristic 0 setting of finite dimensional Lie algebras. Looking up some of the papers listed below on arXiv (usually under math.RT) and others they refer to would be a good way to get into the recent work, including some on nilpotent Lie algebras. Beyond this, I can't answer your specific questions in detail. But as Victor points out, study of the index is only one step. Even in the reductive case, the rank is just one piece of information.

Dmitri I. Panyushev http://front.math.ucdavis.edu/0107.5031

A.N. Panov http://front.math.ucdavis.edu/0801.3025

Jean-Yves Charbonnel and Anne Moreau http://front.math.ucdavis.edu/1005.0831

Celine Righi and Rupert W. T. Yu http://front.math.ucdavis.edu/0908.4201

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Thanks a lot for the references! –  mathphysicist Aug 30 '10 at 15:21
P.S. As other answers indicate, there is widespread literature with many items available on arXiv. Sometimes this quantity just means that the whole subject has been poorly understood, so don't give up too easily. –  Jim Humphreys Jan 5 '11 at 23:17

The following paper of M. Rais has a formula for index of semi-direct products that you mentioned.

M. Rais "L’indice des produits semi-directs $g\ltimes E$. C. R. Acad. Sci. Paris S´er. A-B 287 (1978), no. 4, A195–A197.

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Many thanks for the reference. –  mathphysicist Aug 31 '10 at 11:50
This article by Rais probably requires the help of a library. In any case, the recent literature citing it includes relevant papers by A. Joseph and D.G. Panyushev which might be more accessible online. (It helps to have MathSciNet access, but the arXiv is also a good resource.) –  Jim Humphreys Aug 31 '10 at 15:59
This year of Comptes Rendus seems to be available at gallica.bnf.fr/ark:/12148/cb34484666t/date , but this site is very messy with many gaps, so not sure about this exact article. Also, Rais has two more accessible and more recent arXiv papers with a suggestive title "Notes sur l'indice des algèbres de Lie": arXiv:math/0605499 and arXiv:math/0605500 . –  Pasha Zusmanovich Jan 5 '11 at 22:40