Timeline for analogues of Cayley plane as homogenous spaces
Current License: CC BY-SA 2.5
5 events
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| Jan 5, 2011 at 0:33 | comment | added | José Figueroa-O'Farrill | (cont'd) The other cases in the paper you mentioned are surely the ones in the wikipedia page on symmetric spaces to which you linked. I'm afraid that Helgason is the canonical reference for me; although he only discusses the riemannian case. I'm guessing that this is again due to the lack of classification in indefinite signature (except lorentzian). If I come across some "lightweight" treatment, I'll post a link. | |
| Jan 5, 2011 at 0:30 | comment | added | José Figueroa-O'Farrill | There is no classification of pseudo-riemannian symmetric spaces, I'm afraid, which is why the wikipedia page restricts itself to symmetric pairs $(\mathfrak{g},\mathfrak{h})$ where $\mathfrak{g}$ is simple. Indeed, as you point out, there are only two riemannian examples among the cases at hand: the Cayley plane $F_4/\mathrm{Spin}(9)$ and its noncompact dual. Both of these have $\mathrm{Spin}(9) \subset \mathrm{SO}(16)$ isotropy representation, but the transvection group is different: the compact $F_4$ in one case and the maximally split $F_4$ in the other. (TBC) | |
| Jan 4, 2011 at 23:38 | comment | added | Vít Tuček | The proof of these facts as well as a lightweight (as opposed to Helgasson) introduction to symmetric spaces would be most welcome. | |
| Jan 4, 2011 at 23:34 | comment | added | Vít Tuček | The authors of the paper prove that each of the four manifolds is homogeneous and symmetric. I do not know the theory of symmetric spaces and the wikipedia pages are a bit confusing for me. Anyway, if I understand the classification list on wikipedia, I have only two spaces - one homogeneous under compact $F_4$ and the other under $F_4^{(-20)}$ (the hyperbolic plane). I guess the other two could be find among pseudo-Riemannian symmetric in the table here: en.wikipedia.org/wiki/Symmetric_space#Tables By the way, the paper was published in a polished form in DGA 27 (2009). | |
| Jan 4, 2011 at 22:19 | history | answered | José Figueroa-O'Farrill | CC BY-SA 2.5 |