Cayley plane

In mathematics, the Cayley plane (or octonionic projective plane) OP² 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).

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 projective space. It is one of the rare projective spaces, where the Desargues' theorem does not hold.

Notes

1. ^ Baez (2002).

References

• John C. Baez, "The Octonions", Bull. Amer. Math. Soc. 39 (2002), 145–205. Available electronically.[1]
• 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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