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The Cayley projective plane $\mathbb{OP}^2$ can be defined as a homogenous space $\mathrm{F_4/Spin(9)}$, where $\mathrm{F_4}$ is the compact exceptional simple Lie group. The other possible approach is to define it as equivalence classes of (special) triples of octonions, define a Riemannian metric on it and prove that the group of isometries is compact Lie group of type $\mathrm{F_4}$. This is done in this article. There the authors study also other spaces similar to $\mathbb{OP}^2$, namely they consider the octionionic hyperbolic plane,the octonionic projective plane with indefinite signature and analogue of projective plane made up from split octonions.

My question is: What are the "homogeneous presentations" of these spaces?

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1 Answer 1

Disclaimer: I have not read the paper, but I think that this what's going on.

There are three real forms of the complex simple Lie algebra of type $F_4$. Each such real form will have (real) Lie subalgebras whose complexification is $\mathfrak{so}(9;\mathbb{C})$. Each such pair gives rise to homogeneous 16-dimensional pseudoriemannian manifolds.

This construction certainly accounts for the octonionic projective plane and its noncompact dual, as for the octonionic hyperbolic plane. I have not worked out the other cases mentioned in the question.

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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: By the way, the paper was published in a polished form in DGA 27 (2009). – Vít Tuček Jan 4 '11 at 23:34
The proof of these facts as well as a lightweight (as opposed to Helgasson) introduction to symmetric spaces would be most welcome. – Vít Tuček Jan 4 '11 at 23:38
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) – José Figueroa-O'Farrill Jan 5 '11 at 0:30
(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. – José Figueroa-O'Farrill Jan 5 '11 at 0:33

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