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Cayley plane

Contents

Contents

Idea

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

Definition

The Cayley plane can’t be constructed using homogeneous coordinates the way it works 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

General

The octonionic projective plane is a non-Desarguesian plane, that is, Desargues' theorem does not hold. See projective plane.

Isometries

Cell structure

Proposition

There is a homeomorphism

𝕆P 2S 15h 𝕆𝕆P 1 \mathbb{O}P^2 \,\simeq\, S^{15} \underset{h_{\mathbb{O}}}{\cup} \mathbb{O}P^1

between the octonionic projective plane and the attaching space obtained from the octonionic projective line along the octonionic Hopf fibration.

(Lackman 19, Lemma 3.4)

See also at cell structure of projective spaces.

AKM-Theorem

Proposition

(octonionic AKM-theorem)

The 13-sphere is the quotient space of the (right-)octonionic projective plane by the left multiplication action by Sp(1):

𝕆P 2/Sp(1)S 13 \mathbb{O}P^2 / \mathrm{Sp}(1) \simeq S^{13}

(Atiyah-Berndt 02)

References

General

See also:

  • Wikipedia, Cayley plane

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

The AKM-theorem for 𝕆P 2\mathbb{O}P^2:

In string theory

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

Indications that M-theory in 10+1 dimensions may be understood as the KK-compactification on Cayley-plane fibers of some kind of bosonic M-theory in 26+1 dimensions:

Last revised on February 7, 2021 at 00:02:49. See the history of this page for a list of all contributions to it.