analogues of Cayley plane as homogenous spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T07:45:30Z http://mathoverflow.net/feeds/question/51157 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51157/analogues-of-cayley-plane-as-homogenous-spaces analogues of Cayley plane as homogenous spaces robot 2011-01-04T20:40:39Z 2011-01-04T23:46:43Z <p>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 <a href="http://arxiv.org/abs/math/0702631" rel="nofollow">article</a>. 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.</p> <p>My question is: What are the "homogeneous presentations" of these spaces?</p> http://mathoverflow.net/questions/51157/analogues-of-cayley-plane-as-homogenous-spaces/51161#51161 Answer by José Figueroa-O'Farrill for analogues of Cayley plane as homogenous spaces José Figueroa-O'Farrill 2011-01-04T22:19:34Z 2011-01-04T22:19:34Z <p>Disclaimer: I have not read the paper, but I think that this what's going on.</p> <p>There are <a href="http://en.wikipedia.org/wiki/List_of_simple_Lie_groups" rel="nofollow">three real forms of the complex simple Lie algebra of type $F_4$</a>. 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.</p> <p>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.</p>