On the unit sphere $\Bbb S$ in a real Hilbert space $\bold H$, we derive a binary operation $\odot $ such that $(\Bbb S,\odot )$ is a power-associative Kikkawa left loop with two-sided identity $\bold e_{0}$, i.e., it has the left inverse, automorphic inverse, and $A_l$ properties. The operation $\odot $ is compatible with the symmetric space structure of $\Bbb S$. $(\Bbb S,\odot )$ is not a loop, and the right translations which fail to be injective are easily characterized. $(\Bbb S,\odot )$ 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 $-\bold e_{0}$ where they have a nonremovable discontinuity. The orthogonal group $O(\bold H)$ is a semidirect product of $(\Bbb S,\odot )$ with its automorphism group. The left loop structure of $(\Bbb S,\odot )$ gives some insight into spherical geometry.
@article{119167,
author = {Michael K. Kinyon},
title = {Global left loop structures on spheres},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {41},
year = {2000},
pages = {325-346},
zbl = {1041.20044},
mrnumber = {1780875},
language = {en},
url = {http://dml.mathdoc.fr/item/119167}
}
Kinyon, Michael K. Global left loop structures on spheres. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 325-346. http://gdmltest.u-ga.fr/item/119167/
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