The geometry of sedenion zero divisors

Abstract

The sedenion algebra 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 S can be viewed as the submanifold Z(S) subset S x S of normalized pairs whose product equals zero, or as the submanifold ZD(S) subset S$ of normalized elements with non-trivial annihilators. We prove that Z(S) is isometric to the excepcional Lie group G_2, equipped with a naturally reductive left-invariant metric. Moreover, Z(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 ZD(S) is isometric to the Stiefel manifold V_2(R^7), the space of orthonormal 2-frames in 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(R^7) with non-negative sectional curvature.