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.

]]>added this pointer:

- Karl Heinrich Hofmann, Christian Terp,
*Compact Subgroups of Lie Groups and Locally Compact Groups*, Proceedings of the American Mathematical Society Vol. 120, No. 2 (Feb., 1994), pp. 623-634 (jstor:2159906, doi:10.2307/2159906)

Yeah, thanks. I have changed the table to speak about $Pin(p,q)$ for the moment. Should try to sort of the Spin-case…

]]>Re #14: it looks like here too, you’ve written down the maximal compact *connected* subgroup. So the answer might be something like $S(Pin(q) \times Pin(p))/\mathbb{Z}_2$.

Is the line after that correct, the one with

$MaxCompactSubgroup\big(\; Spin(q,p) \;\big) \;=\; \big( Spin(q) \times Spin(p) \big)/\mathbb{Z}_2$?

]]>Thanks! Have added the reference now.

]]>Thanks for catching! Fixed now.

]]>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.

]]>added here a table with the maximal compact subgroups of the real forms of the exceptional $E$-series of Lie groups.

]]>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.

]]>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.

]]>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?

]]>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 :-/ )

]]>Thanks again!

]]>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, yes.

(By “Lie group” I usually mean the finite dimensional case. Otherwise I’d say “Fréchet Lie group” or the like. )

]]>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.

]]>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.

]]>