Timeline for Is there any useful literature about $ \mathbb R $-compactness of almost simple factors of $ \operatorname{Spin}(p, q) $?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 13, 2023 at 13:20 | comment | added | Tempestas Ludi | @paulgarrett Thank you! The dimension $ \geq 5 $ case unfortunately does not apply here, but it is good to know. Also, isotropicity at all but finitely many points does not give me what I need, but may be good to look into. Lastly, I actually know Theorem 5.10.6 from Platonov-Rapinchuk, but I did not read the rest of the book. I will look into that. | |
| Jan 13, 2023 at 13:18 | comment | added | Tempestas Ludi | @LSpice Thank you! I did not know about anisotropicity, nor about the article and book you mentioned. | |
| Jan 13, 2023 at 6:21 | comment | added | paul garrett | And the Platonov-Rapinchuk book "Algebraic groups and number theory" is fairly encyclopedic... though sometimes making things more complicated in order to be sure to treat general cases. :) There is also the "Book of Involutions", by Knus, Merkurjev, Rost, and Tignol. Maybe overkill. | |
| Jan 13, 2023 at 5:15 | comment | added | paul garrett | ... also, if this is something you're wanting, all p-adic orthogonal groups for spaces of dimension $\ge 5$ are non-compact... This and my previous remark boil down to looking at quaternion algebras over global and local fields (for example), which is well documented... | |
| Jan 13, 2023 at 2:45 | comment | added | paul garrett | For any quaternary (non-degenerate) quadratic form over a number field, at all but finitely-many places the quadratic form is isotropic, so the local points of the orthogonal group are non-compact. Is this the sort of thing you want, or am I failing to understand some subtleties...? :) | |
| Jan 13, 2023 at 2:35 | comment | added | LSpice | (Re, Tits's article: Classification of algebraic semisimple groups; Springer's book: Linear algebraic groups.) | |
| Jan 13, 2023 at 2:31 | comment | added | LSpice | You probably already know this, but, just in case you don't, the purely algebraic version of "compact group of rational points" is anisotropic(-ity): a reductive $\mathbb R$-group has compact (in the analytic topology) group of $\mathbb R$-points if and only if it possesses no non-trivial $\mathbb R$-split torus. Unfortunately, I'm not familiar enough with real groups to comment on this particular problem, but I'll bet the tables in Tits's article or Springer's book can answer. | |
| Jan 13, 2023 at 2:30 | history | edited | LSpice | CC BY-SA 4.0 |
`\DeclareMathOperator`
|
| S Jan 13, 2023 at 0:00 | review | First questions | |||
| Jan 13, 2023 at 1:18 | |||||
| S Jan 13, 2023 at 0:00 | history | asked | Tempestas Ludi | CC BY-SA 4.0 |