4

You can use octonion algebra's $\mathbb{O}$ over a field to coordinatize projective planes. They are called Cayley planes as far as I know.

You can't use the usual approach with homogeneous coordinates, but as for explicit constructions, I only found some very vague descriptions (for example here using phrases like "turning $\mathbb{O}\oplus\mathbb{O}$ into an affine space in the usual way" and "adding points at infinity").

Can someone explain or give me the reference to somewhere where it is constructed in an understandable and rigorous way?

I also know that you can use planar ternary rings to coordinatise planes, but are octonion algebra's always planar ternary rings? Edit: I don't think they are.

Any help would be appreciated.

Travis Willse
  • 108,056
Vincent Batens
  • 1,959

1 Answers1

2

For an explicit construction of $\Bbb O \Bbb P^2$ using triality and the exceptional Jordan algebra, $\mathfrak{h}_3(\Bbb O)$, see Section 3.4, "$\Bbb O \Bbb P^2$ and the Exceptional Jordan Algebra," of the classic Baez article, "The Octonions":

John C. Baez, "The Octonions," Bull. Amer. Math. Soc. 39 (2002), 145-205. Erratum: Bull. Amer. Math. Soc. 42 (2005), 213-213. doi:10.1090/S0273-0979-01-00934-X. arXiv:math/0105155

For an algebraic topology description of $\Bbb O \Bbb P^2$ as the result of attaching a cell $e^{16}$ to $S^8$ via the Hopf map $S^{15} \to S^8$ (hence in analogy to the more familiar construction of $\Bbb C \Bbb P^2$ via the classical Hopf map $S^3 \to S^2$), see Example 4.47 of:

Hatcher, Allen. Algebraic Topology. Cambridge: Cambridge University, 2002.

Travis Willse
  • 108,056