Cayley plane

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.

Properties

In the Cayley plane, lines and points may be defined in a natural way so that it becomes a 2-dimensional projective space, that is, a projective plane. It is a non-Desarguesian plane, where Desargues' theorem does not hold.

See also

• Rosenfeld projective plane

References

• Helmut Salzmann et al. "Compact projective planes. With an introduction to octonion geometry"; de Gruyter Expositions in Mathematics, 21. Walter de Gruyter & Co., Berlin, 1995. xiv+688 pp. ISBN 3-11-011480-1