Cayley plane

In mathematics, the Cayley plane (or octonionic projective plane) OP² is a projective plane over the octonions. It was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley (for his 1845 paper describing the octonions).

As a symmetric space, the Cayley plane is 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). As a homogeneous space it is also the quotient of a noncompact form of the group E₆ by a parabolic subgroup P₁.

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.

See also

• Rosenfeld projective plane

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. 
• 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. 
• 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