Timeline for Simplicity of (complex) orthogonal groups
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 3, 2010 at 10:10 | vote | accept | JGis | ||
| Jun 22, 2010 at 0:52 | comment | added | Skip | To reinforce what Jim wrote in the preceding comment, if you assume the group is isotropic, you can typically prove results that depend only on the type of the group and not on the arithmetic of the ground field $K$. But if you allow the group to be isotropic, results typically involve heavily the arithmetic of $K$. To underline this distinction, one can compare the Kneser-Tits Problem (for isotropic groups) with the Margulis-Platonov Conjecture (same issue, but for anisotropic groups over number fields). These are the modern context for Jakub's question. | |
| Jun 21, 2010 at 22:00 | comment | added | Jim Humphreys |
Yes, I was following your formulation at first over $\mathbb{C}$. Once you get into the forms of Witt index 0 it gets more complicated. This is why the theory got developed further around isotropic and anisotropic forms over a given field, which gets reformulated in appropriate generality in the general theory of algebraic groups. After a while that setting seems more natural for simplicity questions.
|
|
| Jun 21, 2010 at 21:33 | comment | added | JGis | In many places they assume "isotropicity" of the group (i.e. Witt index > 0) to get simplicity of the projectivisation of the derived subgroup (L. Grove, thm 6.31, page 58). In the case of usual SO_n, the Witt index is 0, so the simplicity is not clear. | |
| Jun 21, 2010 at 20:53 | history | edited | Jim Humphreys | CC BY-SA 2.5 |
deleted 1 characters in body
|
| Jun 21, 2010 at 20:40 | history | answered | Jim Humphreys | CC BY-SA 2.5 |