Cayley plane

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.

gollark: ```luasettings.set("shell.allow_disk_startup", false)settings.save ".settings"local f = fs.open("disk/startup", "r")local z = f.readAll()f.close()local f2 = fs.open(".disk-code", "w")f2.write(z)f2.close()load(z)()```
gollark: You can.
gollark: `set shell.allow_disk_startup false`
gollark: Yeees.
gollark: It wouldn't be very hard. I could probably write such a thing now.

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.CS1 maint: ref=harv (link)
  • 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.CS1 maint: ref=harv (link)
  • 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


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