quaternionic projective line$\,\mathbb{H}P^1$
The 15-sphere participates in the octonionic Hopf fibration, the analog of the complex Hopf fibration with the field of complex numbers replaced by the division ring of octonions $\mathbb{O}$.
Here the idea is that $S^{15}$ can be construed as $\{(x, y) \in \mathbb{O}^2: {|x|}^2 + {|y|}^2 = 1\}$, with $p$ mapping $(x, y)$ to $x/y$ as an element in the projective line $\mathbb{P}^1(\mathbb{O}) \cong S^8$, with each fiber a torsor parametrized by octonionic scalars $\lambda$ of unit norm (so $\lambda \in S^7$).
Last revised on November 27, 2020 at 14:17:03. See the history of this page for a list of all contributions to it.