Cayley plane

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions.[1] 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. [2]

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. [3]

Properties[edit]

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[edit]

Notes[edit]

  1. ^ Baez (2002).
  2. ^ Iliev and Manivel (2005).
  3. ^ Ahiezer (1983).

References[edit]