Global Left Loop Structures on Spheres

Details Global Left Loop Structures on Spheres Global Left Loop Structures on Spheres

1999-10-21

arxiv.org/abs/math/9910111v2

Comment. Math. Univ. Carolinae 41 (2000), no. 2, 325-346

Mathematics

Group Theory

18 pages, no figures, 10pt, LaTeX2e, uses amsart.cls & tcilatex.tex. To appear in Comment. Math. Univ. Carolin. (special issue

Scientific paper

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.

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