The geometry of sedenion zero divisors

TL;DR

This paper explores the geometric structure of zero divisors in the sedenion algebra, revealing their connections to exceptional Lie groups and symmetric spaces, and constructing new Einstein and homogeneous metrics.

Contribution

It provides a detailed geometric analysis of sedenion zero divisors, linking them to $G_2$ and Stiefel manifolds, and introduces new Einstein and homogeneous metrics.

Findings

$ ext{Z}( ext{S})$ is isometric to the exceptional Lie group $G_2$.

$ ext{ZD}( ext{S})$ is isometric to the Stiefel manifold $V_2( extbf{R}^7)$.

Constructed new Einstein and homogeneous metrics with non-negative sectional curvature.

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.