Octonionic projective line

Definition

For unit vectors $\xi,\eta\in S^{15}\subset\mathbb O^2$ , define

\xi\sim\eta\quad\Longleftrightarrow\quad \xi\xi^*=\eta\eta^*.

The octonionic projective line is the smooth quotient

\mathbb OP^1=S^{15}/{\sim}.

It is diffeomorphic to $S^8$ , equivalently to the one-point compactification $\mathbb O\cup\{\infty\}$ .

Octonionic Hopf fibration

The quotient map

S^{15}\longrightarrow\mathbb OP^1\cong S^8

has fibers diffeomorphic to $S^7$ and is the octonionic Hopf fibration. It is not a principal $S^7$ -bundle: the unit octonions form a Moufang loop, not an associative Lie group.

Lines in the octonionic plane

Each point of $\mathbb OP^1$ determines an eight-dimensional real linear subspace of $\mathbb O^2$ , called an octonionic line. Their translates are the affine octonionic lines used to test octonionic plurisubharmonicity.

Spin symmetry

The spin representation of $\operatorname{Spin}(9)$ on $\mathbb O^2$ makes the Hopf fibration equivariant and acts transitively on $\mathbb OP^1$ .

References

Semyon Alesker, “Plurisubharmonic functions on the octonionic plane and $\operatorname{Spin}(9)$ -invariant valuations on convex sets,” Journal of Geometric Analysis 18 (2008), 651–686. arXiv record. Relevant: §1.3.
John C. Baez, “The Octonions,” Bulletin of the American Mathematical Society 39 (2002), 145–205. DOI record. Relevant: §4.2.