Affine octonionic line

Definition

An affine octonionic line in $\mathbb O^2$ is a translate of one of the eight-dimensional real linear subspaces parametrized by $\mathbb OP^1$ . The space of all such affine lines is denoted $\mathcal A\mathbb OP^1$ .

Symmetry

The semidirect product

\mathbb O^2\rtimes\operatorname{Spin}(9)

acts transitively on $\mathcal A\mathbb OP^1$ : translations move the affine base point, while the spin action rotates the octonionic direction.

Analytic role

Affine octonionic lines are the test subspaces in the definition of an octonionic plurisubharmonic function. They also form the integration family for the octonionic Radon transform.

Nonassociativity warning

An octonionic line should not be treated naively as a free rank-one submodule of $\mathbb O^2$ : scalar multiplication over $\mathbb O$ is not associative. The projective-line construction supplies the well-defined eight-dimensional real subspaces needed here.

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 and 2.