Timeline for Extensions of Groups
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 13, 2013 at 21:27 | vote | accept | Peter Crooks | ||
| Apr 13, 2013 at 12:16 | answer | added | YCor | timeline score: 1 | |
| Apr 13, 2013 at 8:12 | comment | added | YCor | By Mostow, $G$ has a maximal compact subgroup $K$ and $KG_0=G$. In particular, if you assume in addition that $G_0$ is simply connected, then $G=G_0\rtimes K$ so your extension is split (just assuming $G_0$ nilpotent). In general, since $G$ is nilpotent and $K$ is compact, the action of $K$ on the Lie algebra of $G$ is unipotent and hence trivial, so $[K,G_0]=1$. Note that there are easy non-split extensions, e.g. with $G_0$ the circle and $G/G_0$ noncyclic group of order 4. | |
| Apr 12, 2013 at 22:07 | comment | added | YCor | ... But to understand the general case (with your assumptions: $G$ virtually connected nilpotent Lie group), you can probably reduce to the case when $G_0$ is a torus and in general, I expect that if $W$ is the maximal (compact) torus in $G_0$ then the nilpotent extensions of $G_0$ by a finite nilpotent group $F$ should be classified by the same object as the central extensions $W$-by-$F$. | |
| Apr 12, 2013 at 21:48 | answer | added | Peter Michor | timeline score: 1 | |
| Apr 12, 2013 at 21:36 | history | asked | Peter Crooks | CC BY-SA 3.0 |