Timeline for A geometric property of certain Lie groups
Current License: CC BY-SA 4.0
16 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Aug 18, 2019 at 9:46 | comment | added | YCor | One case where things are maybe already known might be the case of the complex hyperbolic space? | |
| Aug 18, 2019 at 9:46 | comment | added | Ali Taghavi | @ThiKu Thank you very much for this very helpful information. | |
| Aug 18, 2019 at 9:18 | comment | added | Ali Taghavi | @YCor I sincerely thank you very much for your revision.You realy make it clear. Yes it is exactly what I meant. I apologize you and other participants for my inappropriate writting of the question. | |
| Aug 18, 2019 at 9:07 | comment | added | ThiKu | The natural generalization of your Poincaré half-space groups would be the well.studied notion of Heintze groups, which are certain sem-direct groups of nilpotent groups (horospheres) with R and which are known to be exactly the homogeneous spaces of negative curvature. Examples include the noncompact noneuclidean symmetric spaces of rank 1. You may try to generalize computations from hyperbolic plane to this setting, replacing the horosphere by a more general nilpotent group. | |
| Aug 18, 2019 at 8:17 | comment | added | YCor | Ali, I've completely rewritten the question to make it clearer. Please check it means what you intended. | |
| Aug 18, 2019 at 8:16 | history | edited | YCor | CC BY-SA 4.0 |
rewrote the question for clarification
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| Aug 18, 2019 at 8:03 | comment | added | Venkataramana | @paul garrett, I agree. After all, the isometry group of the upper half space is not the parabolic subgroup $P$ but is $G$. The isometries from $K$ fix a point, but are non-trivial! | |
| Aug 18, 2019 at 5:04 | comment | added | paul garrett | @YCor, I can understand that there could be reasons to take that viewpoint, but for my purposes, anyway, it is far more useful to see that a physical space is $G/K$, rather than (via an Iwasawa decomposition) $P/(P\cap K)$. Yes, the fact that parabolics have some semi-direct product structure is very useful, and I do use "Iwasawa coordinates" on $G/K$, but I don't ignore $G$ at all. Probably other people have other goals, etc. | |
| Aug 18, 2019 at 4:17 | comment | added | YCor | @paulgarrett I don't really agree... the upper half space is precisely a model of hyperbolic geometry for which there is a canonical choice of point at infinity. It sounds like saying that the real $\mathbf{R}$ line has no zero, because the real line is the affine line, which has a transitive isometry group etc. The upper half plane is perfectly suited to describe the semidirect product described by the OP. | |
| Aug 18, 2019 at 3:20 | comment | added | paul garrett | Upper half-spaces' geometries are not the geometry coming from the semi-direct product of $\mathbb R^n$ and $\mathbb R^+$. That semi-direct product is just a parabolic subgroup of the isometry group $O(n,1)$, which determines the geometry. Then the upper-half space is an $O(n,1)$-space on which $O(n,1)$ acts transitively, isomorphic to the quotient $O(n,1)/(O(n)\times O(1))$. | |
| Aug 18, 2019 at 1:59 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
added 4 characters in body
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| Aug 18, 2019 at 1:58 | comment | added | Ali Taghavi | The later desription is written in de Carmo's book:Riemannian Geometry.(An exercise in this book). | |
| Aug 18, 2019 at 1:55 | comment | added | Ali Taghavi | @Venkataramana It is a Lie group beacuse it is semi direct product of $\mathbb{R}^{n-1}$ and $\mathbb{R}^+$. An alternative description is that every element is in the form of an affine function on $\mathbb{R}^{n-1}$ in the form of $x \mapsto ax+b,\quad a>0, b\in \mathbb{R}^{n-1}$. (With composition) | |
| Aug 18, 2019 at 1:47 | comment | added | Venkataramana | the Poincare upper half space is not a LIe group in a natural sense. | |
| Aug 18, 2019 at 1:40 | history | asked | Ali Taghavi | CC BY-SA 4.0 |