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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

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.

Rosenfeld projective plane

Notes

Baez (2002).
Iliev and Manivel (2005).

Ahiezer (1983).

References

Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society 39 (2): 145–205. arXiv:math/0105155v4. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087. edit
Iliev, A.; Manivel, L. (2005). "The Chow ring of the Cayley plane". Compositio Mathematica 141: 146. arXiv:math/0306329. doi:10.1112/S0010437X04000788. edit
Ahiezer, D. (1983). "Equivariant completions of homogenous algebraic varieties by homogenous divisors". Annals of Global Analysis and Geometry 1: 49–78. doi:10.1007/BF02329739. edit
Baez, John C. (2005). "Errata for The Octonions" (PDF). Bulletin of the American Mathematical Society 42 (2): 213–213. doi:10.1090/S0273-0979-05-01052-9. edit
McTague, Carl (2014). "The Cayley plane and string bordism". Geometry & Topology 18 (4): 2045–2078. arXiv:1111.4520. doi:10.2140/gt.2014.18.2045. MR 3268773. edit
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