Timeline for Why are parabolic subgroups called "parabolic subgroups"?
Current License: CC BY-SA 2.5
6 events
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| May 18, 2010 at 6:43 | comment | added | BCnrd | @Jim: Wow, the table of contents alone in "Linear Lie Groups" is a barrel of laughs. With crazy-sounding terms like trunks, tools ("Weyl tool"), dressings, wrappings, and virtual reality (really), I can't fathom what those guys were smoking when they wrote the book. | |
| May 17, 2010 at 18:33 | comment | added | Benoît Kloeckner | Another point, in the real hyperbolic space: the stabilizer of a boundary point is isomorphic to the set of similarities of the euclidean space of one less dimension. Inside this set, the translations are the parabolic elements. This makes many of them. More significantly, you form a cusp by quotienting the space by a lattice of this euclidean space: parabolic elements play in this respect a prominent rôle. | |
| May 17, 2010 at 18:32 | comment | added | Benoît Kloeckner | @Timothy Chow: in the stabilizer of a point $p$ of the boundary, there are: -- all hyperbolic elements whose translated geodesic has $p$ as an endpoint; there are many ways to deform the geodesic so that the other endpoint also tends to $p$, and the resulted isometries in the limit are parabolic, -- all elliptic elements whose fixed point set contains $p$ in its closure; for example in the real hyperbolic case, such a fixed point set is a totally geodesic subspace, that can be deformed to $\{p\}$, the deformed isometries in the limit being parabolic. | |
| May 17, 2010 at 17:26 | comment | added | Jim Humphreys | @Timothy: You may be expecting more rationality in the choice of terminology than exists. It's usually hard to come up with just the right word (standard or invented), so people may rely on (1) bland choices like "normal", (2) words transplanted from their original context like "parabolic", (3) names of people (appropriate or not) somehow associated with the concept --- the invented term "K3 surface" is one variant. After a while it's too late to go back and rethink the choices, as Freudentahl-deVries tried to do using highly nonstandard terminology in their 1969 book Linear Lie Groups. | |
| May 17, 2010 at 14:18 | comment | added | Timothy Chow | This is roughly what I have heard before, but the reason it hasn't struck me as being a clincher is that the connection you give between parabolic subgroups and parabolic elements is not as crisp as I would have expected if this were the true motivation for the terminology. Is there a sharper theorem here than "a parabolic subgroup contains many parabolic elements"? | |
| May 17, 2010 at 11:44 | history | answered | Benoît Kloeckner | CC BY-SA 2.5 |