mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Cayley plane (or octonionic projective plane) P²(O) is a

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...

over the

octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...

s.Baez (2002). The Cayley plane was discovered in 1933 by Ruth Moufang, and is named after

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...

for his 1845 paper describing the octonions.

Properties

In the Cayley plane, lines and points may be defined in a natural way so that it becomes a 2-dimensional

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

, that is, a

. It is a

non-Desarguesian plane In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective ...

, where

Desargues' theorem In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and tho ...

does not hold. More precisely, as of 2005, there are two objects called Cayley planes, namely the real and the complex Cayley plane. The real Cayley plane is the

symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geome ...

F₄/Spin(9), where F₄ is a compact form of an exceptional Lie group and Spin(9) is the

spin group In mathematics the spin group, denoted Spin(''n''), page 15 is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathbb_2 \to \o ...

of nine-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

(realized in F₄). It admits a cell decomposition into three cells, of dimensions 0, 8 and 16.Iliev and Manivel (2005). The complex Cayley plane is a

homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...

under the complexification of the group E₆ by a

parabolic subgroup Parabolic subgroup may refer to: * a parabolic subgroup of a reflection group * a subgroup of an algebraic group that contains a Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zarisk ...

''P''₁. It is the closed orbit in the projectivization of the minimal complex representation of E₆. The complex Cayley plane consists of two complex F₄-orbits: the closed orbit is a quotient of the complexified F₄ by a parabolic subgroup, the open orbit is the complexification of the real Cayley plane,Ahiezer (1983). retracting to it.

Notes

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. Projective geometry {{geometry-stub

Properties

See also

Notes

References