Timeline for faithful adjoint representation
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Nov 8, 2010 at 15:31 | comment | added | BCnrd | Ah, but ${\rm{PGL}}_n$ is connected for Zariski topology over any field. For a connected adjoint semisimple gp $G$ over a field $k$ then its adjoint representation has trivial kernel (so it's even a closed immersion). For $k = \mathbf{R}$, the functor $G \rightsquigarrow G(\mathbf{R})$ from (perhaps disconnected) group varieties to Lie groups is clearly compatible with the formation of adjoint representations, so when $G$ is connected adjoint semisimple (connected for Zariski topology!) then $G(\mathbf{R})$ has faithful adjoint repn even if it is disconnected (for the classical topology). QED | |
| Nov 8, 2010 at 13:59 | answer | added | S. Carnahan♦ | timeline score: 4 | |
| Nov 8, 2010 at 13:23 | comment | added | user9552 | NO. I know that for connected lie groups, if it has a trivial center then it has a faithful adjoint representation. Are you saying that the same statement holds for nonconnected lie groups? | |
| Nov 8, 2010 at 10:02 | comment | added | S. Carnahan♦ | Are you asking why $PGL_n(\mathbb{R})$ has trivial center? | |
| Nov 8, 2010 at 9:41 | history | asked | user9552 | CC BY-SA 2.5 |