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Cayley plane

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Idea

The Cayley plane is the projective plane over the octonions, 𝕆P 2\mathbb{O} P^2.

Definition

It can’t be constructed using homogeneous coordinates the way we do for projective planes over division rings, since multiplication of octonions is not associative. However, it can be constructed in several other ways that generalize the approach for division rings:

  • One can start with 𝕆 2\mathbb{O}^2 and give it the structure of an affine plane? in the obvious way, and then add “points at infinity” in the usual way to obtain a projective plane. This is the most straightforward approach, but as always it has the defect that it makes the line at infinity appear special.

  • One can consider the space of 3×33\times 3 matrices over 𝕆\mathbb{O} that are “Hermitian” and idempotent, hence can be imagined as “projections onto dimension-1 subspaces of 𝕆 3\mathbb{O}^3”.

  • Writing out the components of such a matrix explicitly, one obtains a Veronese vector (x 1,x 2,x 3;ξ 1,ξ 2,ξ 3)(x_1,x_2,x_3;\xi_1,\xi_2,\xi_3) where x i𝕆x_i\in \mathbb{O} and ξ i\xi_i\in \mathbb{R}, such that ξ ix i¯=x jx k\xi_i \overline{x_i} = x_j x_k and x i 2=ξ jξ k\Vert x_i\Vert^2 = \xi_j \xi_k for all cyclic permutations (i,j,k)(i,j,k) of (1,2,3)(1,2,3). The Cayley plane can be identified with the space of nonzero such vectors modulo the scalar action of \mathbb{R}.

Properties

  • F4 is the isometry group of 𝕆P 2\mathbb{O} P^2, with the stabilizer of a point being Spin(9). Hence 𝕆P 2F 4/Spin(9)\mathbb{O} P^2 \cong F_4/Spin(9).

  • It is a non-Desarguesian plane, that is, Desargues’ theorem does not hold. See projective plane.

References

  • Wikipedia, Cayley plane

  • Salzmann et. al., Compact Projective Planes, with an introduction to Octonion Geometry

Discussion of the Witten genus of Cayley plane-fiber bundles is in

Last revised on May 17, 2019 at 08:33:44. See the history of this page for a list of all contributions to it.