Timeline for Looking for interesting actions that are not representations
Current License: CC BY-SA 2.5
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 14, 2011 at 1:25 | answer | added | Jon Bannon | timeline score: 1 | |
| Dec 13, 2011 at 15:55 | comment | added | Robert Bryant | @Theo: Actually, the action of PSO(n) on real projective space does extend quite naturally to a linear action: If you let SO(n) act by conjugation on the vector space of symmetric $n$-by-$n$ matrices, then $\mathbb{RP}^{n-1}$ is identified with the set of symmetric matrices~$p$ that satisfy $p^2=p$ and have trace equal to $1$. One identifies $p$ with its $+1$ eigenspace, which is a line in $\mathbb{R}^n$. | |
| Feb 28, 2011 at 1:10 | vote | accept | Marek | ||
| Feb 23, 2011 at 13:59 | answer | added | S. Carnahan♦ | timeline score: 3 | |
| Feb 23, 2011 at 12:47 | answer | added | Franz Lemmermeyer | timeline score: 5 | |
| Feb 23, 2011 at 12:16 | answer | added | Benoît Kloeckner | timeline score: 5 | |
| Feb 23, 2011 at 10:14 | answer | added | Tom De Medts | timeline score: 3 | |
| Feb 22, 2011 at 21:14 | answer | added | Qiaochu Yuan | timeline score: 8 | |
| Feb 22, 2011 at 19:26 | comment | added | Marek | @Tom: thank you for the references. @Theo: could you be more explicit about the "accident"? For me rotations are about the most natural linear reps there are, so I wonder in what sense this is just accidental. Also $Spin$ groups and Clifford algebras come to mind as another interesting example (and related to other stuff that has been mentioned, I guess). | |
| Feb 22, 2011 at 19:24 | answer | added | Stefan Waldmann | timeline score: 3 | |
| Feb 22, 2011 at 19:19 | comment | added | Marek | @Mariano: I'll look into it. But in physics (whose Hilbert spaces are a natural setting for these reps) they are usually completely ignored (e.g. passing to a covering group or just considering "multi-valued" reps)! Would you have some nice examples in mind where the genuine projective treatment is inevitable (or at least enlightening)? | |
| Feb 22, 2011 at 19:18 | answer | added | Francesco Polizzi | timeline score: 16 | |
| Feb 22, 2011 at 18:59 | comment | added | Theo Johnson-Freyd | Rings, on the other hand, are designed to act on abelian groups, and k-algebras are the things that act on k-modules, i.e. vector spaces. | |
| Feb 22, 2011 at 18:59 | comment | added | Theo Johnson-Freyd | (continuation) On the other hand, its action through PSO(n) on the metrized real projective space (sphere mod antipodal identification) does not extend well to a linear representation, and gives a nice example of Mariano's comment that projective representations are cool. | |
| Feb 22, 2011 at 18:57 | comment | added | Theo Johnson-Freyd | The point of looking for actions on vector spaces is that vector spaces are particularly well-behaved, and so you can study a group by studying its linear representations. But groups most naturally act just on spaces, and Lie groups, being groups of manifolds, naturally act on manifolds. SO(n), for example, by construction acts on the (n-1)-dimensional metrized sphere. That you can extend this to an n-dimensional linear representation is almost an accident (but sure does make studying it easier!). (continued) | |
| Feb 22, 2011 at 18:53 | comment | added | Tom Goodwillie | Making a group act freely (or almost freely in some sense) on some contractible topological space is a standard way of studying the group. I am thinking especially of torsion-free groups, where the space has a chance of being a manifold or at least finite-dimensional. Some keywords: geometric group theory, classifying space, Teichmueller theory. | |
| Feb 22, 2011 at 18:51 | answer | added | J.C. Ottem | timeline score: 5 | |
| Feb 22, 2011 at 18:50 | answer | added | Dick Palais | timeline score: 10 | |
| Feb 22, 2011 at 18:25 | comment | added | Mariano Suárez-Álvarez | Projective representations are pretty useful... | |
| Feb 22, 2011 at 18:21 | history | asked | Marek | CC BY-SA 2.5 |