Timeline for Center of the algebraic group $G_{\mathbb{R}}$ for a centerless $G$
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 23, 2012 at 21:42 | comment | added | Jim Humphreys | @Yves: I substituted simpler language in (3), not wanting to add more confusion to the discussion. | |
| Dec 23, 2012 at 21:39 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
deleted 89 characters in body
|
| Dec 23, 2012 at 20:52 | comment | added | Jack | As I have understood from the comments and discussions above, my question about center has nothing to do with connectedness. The center $Z(G)$, commutes with the extension of the base field for arbitrary $G$ (or at least $G$ semisimple) without the assumptions on connectedness (as indicated by Yves, anon and others). | |
| Dec 23, 2012 at 20:16 | comment | added | Jim Humphreys |
Yes, I oversimplified the connectedness issue here. The ideas sketched in (3) seem OK when $G(\mathbb{R})$ is connected in the euclidean topology, but I'm not quite sure what happens otherwise. It seems there is no effect on the center under the conditions placed on $G$, but I'll have to do more thinking about this interplay of algebraic groups and Lie groups (including the Iwasawa decomposition, where the center lies in the maximal compact subgroup).
|
|
| Dec 23, 2012 at 18:17 | comment | added | YCor | I'm confused by (3). In a Lie group context, I would not say that the (non-connected) group $PGL_2(\mathbf{R})$ is a real form of $PGL_2(\mathbf{C})$ (it does not fulfill the categorical definition of complexification using the forgetful functor from complex Lie groups to real Lie group). What did you have in mind? | |
| Dec 23, 2012 at 16:50 | history | answered | Jim Humphreys | CC BY-SA 3.0 |