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I have tried to start a table at maximal compact subgroup listing Lie groups and their max compact subgroups. But once again the table does not want to typeset properly.
Have to run now, will try to fiddle with this later.
I thought the existence and uniqueness up to conjugation of maximal compact subgroups was a theorem about finite-dimensional Lie groups, so I put that in.
Thanks, yes.
(By “Lie group” I usually mean the finite dimensional case. Otherwise I’d say “Fréchet Lie group” or the like. )
Added a comment about Fréchet Lie group to Lie group, and also added the crucial connectedness hypothesis to the statement of existence and uniqueness of maximal compact subgroups.
Apropos of that, I also started Prüfer group, and added an example to counterexamples in algebra.
Thanks again!
Ah, stupid me. I finally realized why the tables wouldn’t do what I want. More often than not I would type
a | b
------
c | d
e | f
Instead of
a | b
--|---
c | d
e | f
It’s only the second syntax that is recognized as a table. (This was clear to you all. I am just saying it for my own sake :-/ )
I now had some time to work on the entry.
I have added a statement of the Malcev-Iwasawa theorem, and of a recent refinement by Antonyan.
So “almost connectedness” is sufficient for the existence of a maximal compact subgroup.
Todd, I have moved your counterexample of the Prüfer group to section “Examples”, subsection “Counterexamples”. Okay?
More than okay, Urs – I really like what you’ve done! Nice work.
I have half a mind to write up something on Hilbert’s fifth problem, which we don’t have an article on.
Thanks, Todd.
I have further edited (corrected and expanded) the table of Lie groups and their max compact subgroups. I would like it to be more extensive, still. But I’l have to call it quits for tonight.
added here a table with the maximal compact subgroups of the real forms of the exceptional $E$-series of Lie groups.
It looks like there’s a slight mistake in the row with $SO(p, q)$: it needs to be $\{T, U) \in O(p) \times O(q)|\;\det(T) = \det(U)\}$. The group $SO(p) \times SO(q)$ is of index $2$ in this.
Thanks for catching! Fixed now.
Is the line after that correct, the one with
$MaxCompactSubgroup\big(\; Spin(q,p) \;\big) \;=\; \big( Spin(q) \times Spin(p) \big)/\mathbb{Z}_2$?
Re #14: it looks like here too, you’ve written down the maximal compact connected subgroup. So the answer might be something like $S(Pin(q) \times Pin(p))/\mathbb{Z}_2$.
Yeah, thanks. I have changed the table to speak about $Pin(p,q)$ for the moment. Should try to sort of the Spin-case…
added this pointer:
This page really ought to be split into “compact subgroup” and “maximal compact subgroup”. Or maybe at least it ought to just be renamed to “compact subgroup”. But I don’t have the energy now. Maybe later.
Why was the statement of Antonyan’s theorem changed to say “Lie group” instead of “topological group”? In his paper, Theorem 1.2 is stated for topological groups.
Also, is the new definition really correct? For example, Antonyan’s paper defines maximal compact subgroups like in the previous revision, not the current one.
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