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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.

Many thanks in advance!

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    $\begingroup$ 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. $\endgroup$
    – Victor Protsak
    Commented Aug 30, 2010 at 4:37

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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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    $\begingroup$ 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.) $\endgroup$
    – Jim Humphreys
    Commented Aug 31, 2010 at 15:59
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    $\begingroup$ 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 . $\endgroup$
    – Pasha Zusmanovich
    Commented Jan 5, 2011 at 22:40
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Additional references:

V. Dergachev, Some properties of index of Lie algebras, arXiv:math/0001042

O. Yakimova, The index of centralisers of elements in classical Lie algebras, Funct. Anal. Appl. 40 (2006), N1, 42-51, arXiv:math/0407065

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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.

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  • $\begingroup$ Thanks a lot for the references! $\endgroup$
    – mathphysicist
    Commented Aug 30, 2010 at 15:21
  • $\begingroup$ 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. $\endgroup$
    – Jim Humphreys
    Commented Jan 5, 2011 at 23:17

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