In the case of a Lie algebra over a non-archimedean local field of positive characteristic (I have been led to believe that) it is not necessarily true that all Cartan subalgebras have the same dimension. So what would be a reasonable definition of the rank of a Lie group over such a field?
$\begingroup$
$\endgroup$
3
-
1$\begingroup$ Please me more clear on whether your question concerns Lie algebras or Lie groups. If you talk about Lie algebras, there is one definition of Cartan subalgebra (as in Bourbaki): nilpotent subalgebra equal to its normalizer. Is this the one you use? Or do you implicitly assume any kind of semisimplicity? If so, which definition of semisimple do you use? It certainly matters in positive characteristic. $\endgroup$– YCorCommented Apr 24, 2015 at 8:47
-
$\begingroup$ I don't want to assume semisimplicity, and yes I was wanting to use that definition of Cartan subalgebra. My question was about Lie groups, I just recently checked out Serre's book "Lie algebras and Lie groups" which describes how to associate a Lie algebra with any Lie group over a non-archimedean local field. With an arbitrary Lie group over a local field of positive characteristic, not necessarily semisimple in any sense, I am wondering how you define the rank of that Lie group. $\endgroup$– RupertCommented Apr 24, 2015 at 8:49
-
$\begingroup$ Then I'd suggest to consider germs of subgroups instead of Lie subalgebras, and in the non-archimedean case germs can be represented by compact subgroups. So we could define a Cartan subgroup to be a compact nilpotent Lie subgroup over the given field that is open in its normalizer. But I don't know if all such guys have the same dimension. $\endgroup$– YCorCommented Apr 24, 2015 at 9:43
Add a comment
|