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I would like to find a list (or at least a description) of the maximal closed connected subgroups of $\mathrm{SL}(n, \mathbb{R})$ , and also of $\mathrm{SU}(p,q)$.

In the following MO discussion is indicated a link to a nice paper of Dynkin where he classifies the closed Lie subgroups of $\mathrm{SL}(n, \mathbb{C})$, but I'm not sure if one can deduce the answer to my question from this classification.

Thanks

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For $su(p,q)$, I found

Selim, Taufik Mohamed On maximal subalgebras in classical real Lie algebras. Selected translations. Selecta Math. Soviet. 6 (1987), no. 2, 163–176.

See also this MO problem.

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  • $\begingroup$ I can't find the paper anywhere, any clue? $\endgroup$ – Selim G May 24 '16 at 7:12
  • $\begingroup$ Are you still in Paris? There, three libraries are carrying the journal (ENS, Jussieu, Orsay). See cfp.mathdoc.fr/periodique.php?id=4604. $\endgroup$ – Friedrich Knop May 24 '16 at 10:09
  • $\begingroup$ Thank you very much! I didn't know about this catalogue, very useful indeed! $\endgroup$ – Selim G May 24 '16 at 12:56
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Maximal closed connected subgroups of positive dimension in $\mathrm{SL}(n,\mathbb{R})$ are parabolic, the normalizer of a connected semisimple subgroup, or the normalizer of a maximal torus. There are finitely many conjugation classes of each type, and so you could try to work out the maximal ones.

General References

According to the MathSciNet review (see also MO answers here and here) this paper addresses the problem (without proof):

Komrakov, B. P. Maximal subalgebras of real Lie algebras and a problem of Sophus Lie. Dokl. Akad. Nauk SSSR 311 (1990), no. 3, 528--532; translation in Soviet Math. Dokl. 41 (1990), no. 2, 269–273 (1991)

Another paper that might be helpful is:

Mostow, G. D. On maximal subgroups of real Lie groups. Ann. of Math. (2) 74 1961 503–517.

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    $\begingroup$ I am not sure why these papers would contain a solution since a maximal subalgebra doesn't have to be simple and a simple subalgebra doesn't have to be maximal. These problems are related but not the same. $\endgroup$ – Friedrich Knop May 17 '16 at 17:14
  • $\begingroup$ @FriedrichKnop Yes, I had in mind using the Karpelevič papers with Dynkin's classification. I edited the answer accordingly and added some other references that claim to directly deal with maximal subgroups/subalgebras. Thanks for the comment. $\endgroup$ – Sean Lawton May 17 '16 at 19:04

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