Timeline for Jacobson-Morozov on the algebraic group level
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 30, 2020 at 10:39 | answer | added | Strichcoder | timeline score: 2 | |
| Apr 22, 2010 at 16:30 | answer | added | Jim Humphreys | timeline score: 6 | |
| Apr 22, 2010 at 15:15 | comment | added | BCnrd | You mean $\varphi$ is hom. ${\rm{SL}}_ 2 \rightarrow G$ as $k$-gps, not merely homomorphism of gps of $k$-points. Zariski closure $U$ of $\langle g rangle$ is comm. and unip., hence vector gp since in char 0. Then get $\mathbf{G}_ a$ containing $g$ and must be $U$ if $g \ne 1$ (as we may and do assume). Apply J-M to $\mathfrak{u} = {\rm{Lie}}(U)$ to get $\mathfrak{sl}_ 2$ in $\mathfrak{g}$ containing $\mathfrak{u}$. By Cor 7.9 in Borel's alg. gp. book, in char. 0 perfect Lie subalg "integrates" to closed $k$-subgp. That's ${\rm{SL}}_ 2$ or ${\rm{PGL}}_ 2$ containing $U$, and does the job. | |
| Apr 22, 2010 at 14:08 | history | asked | Guntram | CC BY-SA 2.5 |