In
mathematics, the Cayley plane (or octonionic projective plane) P
2(O) is a
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...
over the
octonions.
[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 prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, Cayley enjoyed solving complex maths problems ...
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, that is, a
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...
. It is a
non-Desarguesian plane, where
Desargues' theorem 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 symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
F4/Spin(9), where F
4 is a compact form of an
exceptional Lie group and Spin(9) is the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As ...
of nine-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
(realized in F
4). 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, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ...
under the complexification of the group
E6 by a
parabolic subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
''P''
1. It is the closed orbit in the projectivization of the minimal complex representation of E
6. The complex Cayley plane consists of two complex F
4-orbits: the closed orbit is a quotient of the complexified F
4 by a parabolic subgroup, the open orbit is the complexification of the real Cayley plane,
[Ahiezer (1983).] retracting to it.
See also
*
Rosenfeld projective plane
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
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