Cayley plane

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

The Cayley plane was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley 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. 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 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 the complexification of the group E₆ by a parabolic subgroup P₁. 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,^[3] retracting to it.

See also

• Rosenfeld projective plane

Notes

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

References

• Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087.
• Iliev, A.; Manivel, L. (2005). "The Chow ring of the Cayley plane". Compositio Mathematica. 141: 146. arXiv:math/0306329. doi:10.1112/S0010437X04000788.
• 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.
• 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.
• 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. Zbl 1323.55007.
• 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