Global Left Loop Structures on Spheres

Abstract

On the unit sphere $\mathbb{S}$ in a real Hilbert space $\mathbf{H}$, we derive a binary operation $\odot$ such that $(\mathbb{S},\odot)$ is a power-associative Kikkawa left loop with two-sided identity $\mathbf{e}_0$, i.e., it has the left inverse, automorphic inverse, and $A_l$ properties. The operation $\odot$ is compatible with the symmetric space structure of $\mathbb{S}$. $(\mathbb{S},\odot)$ is not a loop, and the right translations which fail to be injective are easily characterized. $(\mathbb{S},\odot)$ satisfies the left power alternative and left Bol identities ``almost everywhere'' but not everywhere. Left translations are everywhere analytic; right translations are analytic except at $-\mathbf{e}_0$ where they have a nonremovable discontinuity. The orthogonal group $O(\mathbf{H})$ is a semidirect product of $(\mathbb{S},\odot)$ with its automorphism group (cf. http://www.arxiv.org/abs/math.GR/9907085). The left loop structure of $(\mathbb{S},\odot)$ gives some insight into spherical geometry.