The geometry of sedenion zero divisors

Abstract

The sedenion algebra $\mathbb S$ is a non-commutative, non-associative, $16$-dimensional real algebra with zero divisors. It is obtained from the octonions through the Cayley-Dickson construction. The zero divisors of $\mathbb S$ can be viewed as the submanifold $\mathcal Z(\mathbb S) \subset \mathbb S \times \mathbb S$ of normalized pairs whose product equals zero, or as the submanifold $\operatorname{\mathit {ZD}}(\mathbb S) \subset \mathbb S$ of normalized elements with non-trivial annihilators. We prove that $\mathcal Z(\mathbb S)$ is isometric to the excepcional Lie group $G_2$, equipped with a naturally reductive left-invariant metric. Moreover, $\mathcal Z(\mathbb S)$ is the total space of a Riemannian submersion over the excepcional symmetric space of quaternion subalgebras of the octonion algebra, with fibers that are locally isometric to a product of two round $3$-spheres with different radii. Additionally, we prove that $\operatorname{\mathit {ZD}}(\mathbb S)$ is isometric to the Stiefel manifold $V_2(\mathbb R^7)$, the space of orthonormal $2$-frames in $\mathbb R^7$, endowed with a specific $G_2$-invariant metric. By shrinking this metric along a circle fibration, we construct new examples of an Einstein metric and a family of homogenous metrics on $V_2(\mathbb R^7)$ with non-negative sectional curvature.