Cayley projective plane | Wikipedia audio article
Channel: wikipedia tts
Date: 02/06/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Cayley_plane


00:01:20 1 Properties
00:01:47 2 See also



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SUMMARY
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In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions. It was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley (for his 1845 paper describing the octonions).
More precisely, there are two objects called Cayley planes, namely the real and the complex Cayley plane.
The real Cayley plane is the symmetric space F4/Spin(9), where F4 is a compact form of an exceptional Lie group and Spin(9) is the spin group of nine-dimensional Euclidean space (realized in F4). It admits a cell decomposition into three cells, of dimensions 0, 8 and 16.The complex Cayley plane is a homogeneous space under a noncompact (adjoint type) form of the group E6 by a parabolic subgroup P1. It is the closed orbit in the projectivization of the minimal representation of E6. The complex Cayley plane consists of two F4-orbits: the closed orbit is a quotient of F4 by a parabolic subgroup, the open orbit is the real Cayley plane.